**Previous months:**

2009 - 0909(1) - 0910(1) - 0912(1)

2010 - 1002(1) - 1003(8) - 1004(10) - 1005(2) - 1008(1) - 1010(2)

2011 - 1107(1) - 1108(2) - 1110(1)

2012 - 1202(1) - 1204(4) - 1206(1) - 1207(3) - 1209(2) - 1212(2)

2013 - 1302(2) - 1303(1) - 1307(1) - 1309(2) - 1310(4)

2014 - 1401(1) - 1404(1) - 1405(4) - 1406(2) - 1407(1) - 1408(2) - 1409(4) - 1410(2) - 1411(4) - 1412(8)

2015 - 1501(1) - 1502(1) - 1503(2) - 1504(1) - 1505(3) - 1506(2) - 1508(5) - 1510(2) - 1511(1) - 1512(1)

2016 - 1601(2) - 1602(2) - 1603(1) - 1604(3)

Any replacements are listed further down

[107] **viXra:1604.0340 [pdf]**
*submitted on 2016-04-25 07:04:43*

**Authors:** Vasile Pătraşcu

**Comments:** 10 Pages. Neutrosophic Sets and Systems, Volume 11, pp. 57-66, 2016

In this article, starting from primary representation of neutrosophic information, namely the triplet (μ, ω, ν) made up of the degree of truth μ, degree of indeterminacy ω and degree of falsity ν, we define a refined representation in a penta-valued fuzzy space, described by the index of truth t, index of falsity f, index of ignorance u, index of contradiction c and index of hesitation h. In the proposed penta-valued refined representation the indeterminacy was split into three sub-indeterminacies such as ignorance, contradiction and hesitancy. The set of the proposed five indexes represent the similarities of the neutrosophic information (μ, ω, ν) with these particular values: T=(1,0,0), F=(0,0,1), U=(0,0,0), C=(1,0,1) and H=(0.5,1,0.5). This representation can be useful when the neutrosophic information is obtained from bipolar information which is defined by the degree of truth and the degree of falsity to which is added the third parameter, its cumulative degree of imprecision.

**Category:** Set Theory and Logic

[106] **viXra:1604.0118 [pdf]**
*submitted on 2016-04-06 06:46:52*

**Authors:** Yakov A. Iosilevskii

**Comments:** 1134 Pages.

In contrast to Church, who proved in 1936, based on papers by Gödel, that a dual decision problem for the conventional axiomatic first-order predicate calculus is unsolvable, I have solved a trial decision problem algebraically (and hence analytically, not tabularily) for a properly designed axiomatic first-order algebraico-predicate calculus, called briefly the trial logic (TL), and have successfully applied the pertinent algebraic decision procedures to all conceivable logical relations of academic or practical interest, including the 19 categorical syllogisms. The structure of the TL is a synthesis of the structure of a conventional axiomatic first-order predicate calculus (briefly CAPC) and of the structure of an abstract integral domain. Accordingly, the TL contains as its autonomous parts the so-called Predicate-Free Relational Trial Logic (PFRTL), which is parallel to a conventional axiomatic sentential calculus (CASC), and the so-called Binder-Free Predicate Trial Logic (BFPTL), which is parallel to the predicate-free part of a pure CAPC. This treatise, presenting some of my findings, is alternatively called “the Theory of Trial Logic” (“the TTL”) or “the Trial Logic Theory” (“the TLT”). The treatise reopens the entire topic of symbolic logic that is called “decision problem” and that Church actually closed by the fact of synecdochically calling the specific dual decision problem, the insolvability of which he had proved, by the generic name “decision problem”, without the qualifier “dual”. Any additional axiom that is incompatible with the algebraic decision method of the trial logic and that is therefore detrimental for that method is regarded as one belonging to either to another logistic system or to mathematics.

**Category:** Set Theory and Logic

[105] **viXra:1603.0226 [pdf]**
*submitted on 2016-03-16 03:11:25*

**Authors:** Vasile Pătraşcu

**Comments:** 12 Pages.

Starting from the primary representation of neutrosophic information, namely the degree of truth, degree of indeterminacy and degree of falsity, we define a nuanced representation in a penta valued fuzzy space, described by the index of truth, index of falsity, index of ignorance, index of contradiction and index of hesitance. Also, it was constructed an associated penta valued logic and then using this logic, it was defined for the proposed penta valued structure the following operators: union, intersection, negation, complement and dual. Then, the penta valued representation is extended to a hexa valued one, adding the sixth component, namely the index of ambiguity.

**Category:** Set Theory and Logic

[104] **viXra:1602.0198 [pdf]**
*submitted on 2016-02-16 21:38:55*

**Authors:** Florentin Smarandache

**Comments:** 107 Pages.

Welcome into my scientific lab!
My lab[oratory] is a virtual facility with non-controlled conditions in which I mostly perform scientific chats.
I called the jottings herein scilogs (truncations of the words scientific, and gr. Λόγος – appealing rather to its original meanings "ground", "opinion", "expectation"), combining the welly of both science and informal (via internet) talks.
In this book, one may find new and old questions and ideas, some of them already put at work, others dead or waiting, referring to various fields of research (e.g. from neutrosophic algebraic structures to Zhang's degree of intersection, or from Heisenberg uncertainty principle to neutrosophic statistics) – email messages to research colleagues, or replies, notes about authors, articles or books, so on.
Feel free to budge in the lab or use the scilogs as open source for your own ideas.

**Category:** Set Theory and Logic

[103] **viXra:1602.0006 [pdf]**
*submitted on 2016-02-01 08:18:19*

**Authors:** Andrew Banks

**Comments:** 24 Pages.

Although for over two thousand years humans have been programmed to believe space and time are infinitely divisible, whereas matter is not, this article will produce a computing device capable of executing instructions in any given time that demonstrates space and time are finite just as matter is. Moreover, using this same device, it will also be proven Zermelo–Fraenkel set theory (ZF) is inconsistent.

**Category:** Set Theory and Logic

[102] **viXra:1601.0193 [pdf]**
*submitted on 2016-01-17 18:49:05*

**Authors:** Nikolaj Roerich

**Comments:** 3 Pages.

We show that Riemann Hypothesis is actually an Axiom.
Prooving it would mean knowing how to build the Universe.
That is the future that people evolution will lead to.
To build the Universee one needs to know the details about the corresponding Hilbert Space since Universe = Hilbert Space,
and Riemann Hypothesis solution is equivalent to the knowing the linear operator in that Hilbert Space that is called "L" which has the eigenvalues equal to the zeros of Riemann Function.

**Category:** Set Theory and Logic

[101] **viXra:1601.0023 [pdf]**
*submitted on 2016-01-04 06:29:47*

**Authors:** Janis Belov

**Comments:** 2 Pages.

We solve P vs NP Millenium problem.

**Category:** Set Theory and Logic

[100] **viXra:1512.0357 [pdf]**
*submitted on 2015-12-17 20:25:15*

**Authors:** Bolonkin A.A.

**Comments:** 8 Pages.

Предлагается принципиально новый метод оптимизации. В отличие от классической постановки задачи:
а) Дан функционал – найти его минималь.
Рассматриваются также задачи:
б) найти более «узкое» подмножество, содержащее абсолютную минималь;
в) найти подмножество решений лучших, чем данное;
г) найти оценки снизу данного функционала.
В настоящее время большинство исследователей, работающих в области оптимизации заняты решение задачи в классической постановке – отысканием точной минимали. Инженера же, как правило, в реальных задачах интересует подмножество квази-оптимальных решений, выбирая из которого, он заранее уверен, что получит значение функционала не хуже заданной величины (задача в) и оценка снизу, показывающая насколько он далек от точного оптимального решения (задача г). Кроме того у него есть много дополнительных соображений, которые нельзя учесть в математической модели или которые бы ее сильно усложнили. Постановка задачи в форме «в» дает ему определенную свободу выбора.
This method, called the “Method of Deformation of Functional (Extreme)”, solves for a total minimum and finds a solution set near the optimum. Solutions found by this method can be exact or approximate. Most other methods solve only for a unique local minimum. The ability to create a set of solutions rather than a unique solution has important practical ramifications in many designs, economic and scientific problems because a unique solution usually is difficult to realize in practice.
This method has the additional virtue of a simple proof, one that is useful for studying other methods of optimization, since most other methods can be delivered from the Method of Deformation.

**Category:** Set Theory and Logic

[99] **viXra:1511.0160 [pdf]**
*submitted on 2015-11-18 12:28:07*

**Authors:** Alex Patterson

**Comments:** 7 Pages.

Will be look at (data) type inference for the four major arithmetic types to search for symmetry-checks and factorization in the Lie algebra, using the multiplicative decomposition by such searches in the Lie Algebra to Poincare Group, Poincare Group important only for the theory check.

**Category:** Set Theory and Logic

[98] **viXra:1510.0133 [pdf]**
*submitted on 2015-10-16 03:30:37*

**Authors:** Takis Tsoukalas, Panagiotatos Mitropolitis Thessalonikis Anthimos Roussas

**Comments:** 8 Pages.

Assume Z ≤ D. Recent developments in applied singular oper- ator theory [11] have raised the question of whether t < π. We show that Borel’s criterion applies. In [11, 11, 28], the main result was the derivation of multiplicative, linearly tangential paths. In [11], the authors extended lines.

**Category:** Set Theory and Logic

[97] **viXra:1510.0041 [pdf]**
*submitted on 2015-10-05 06:44:34*

**Authors:** editors Florentin Smarandache, Mumtaz Ali

**Comments:** Pages. 98

This volume is a collection of fourteen papers, written by different authors and co-authors (listed in the order of the papers): F. Yuhua, K. Mandal, K. Basu, S. Pramanik, K. Mondal, S. Alkhazaleh, J. Nescolarde-Selva, J. L. Usó-Doménech, A. Betancourt-Vázquez, K. Pérez-Teruel, M. Leyva-Vázquez, A. Aydoğdu, I. Arockiarani, C. A. C. Sweety, F. Smarandache, L. Zhengda, S. Kar, S. Mukherjee, P. Das, and T. K. Kumar.

**Category:** Set Theory and Logic

[96] **viXra:1508.0309 [pdf]**
*submitted on 2015-08-30 19:26:33*

**Authors:** Minseong Kim

**Comments:** 2 Pages.

In computer science, a character set $\Sigma$ is often defined. Then, Kleene plus and Kleene star for formal language are defined. Then, $\Sigma^{+} = \Sigma^{*}\Sigma$ is proved, which means every string (set) in $\Sigma^{+}$ can be represented as a concatenation of a set in $\Sigma^{*}$ and a set in $\Sigma$. However, if one forms a set that cannot be defined by a formula but what people would believe as existing, then while the proof itself does not break down, it may be possible that state of matter is inconsistent. This paper explores this possibility.

**Category:** Set Theory and Logic

[95] **viXra:1508.0299 [pdf]**
*submitted on 2015-08-29 08:36:40*

**Authors:** Samuel Amok

**Comments:** 1 Page.

In this paper, I answer to a question that has been raised in http://www.les-mathematiques.net/phorum/read.php?16,1137927,1137947#msg-1137947

**Category:** Set Theory and Logic

[94] **viXra:1508.0284 [pdf]**
*submitted on 2015-08-27 01:08:39*

**Authors:** Shawkat Alkhazaleh, Emad Marei

**Comments:** 112 Pages.

In 1995 Smarandache introduced the concept of
neutrosophic set which is a mathematical tool for handling problems involving imprecise, indeterminacy and inconsistent data. In 2013 Maji introduced the concept of neutrosophic soft
set theory as a general mathematical tool for dealing with uncertainty.

**Category:** Set Theory and Logic

[93] **viXra:1508.0161 [pdf]**
*submitted on 2015-08-20 09:19:58*

**Authors:** Alex Patterson

**Comments:** 12 Pages. Special thanks to Michael J. Burns

This paper uses its own peculiar lettering system for each paragraph.
This paper proposes an overall solution to Godel’s incompleteness theorem and the Gödel sentence. Both are handled as one, by using Gödel numbers as the exemplary objects of incompleteness.
New terms and tools are introduced for quantification that creates a more synthetic (logical, reasonable, coherent) intervention and inter-weaving into these now classical problems of the assumptions in the Gödel material and literature.
Asymptotes are used within vertical and horizontal graphs to justify a future that need not be seen as a future in the sense of grammatical future-tense, but as a potential part such systems themselves that we deal with respect to incompleteness.
The thesis is that we can approach incompleteness by using theoretical reasoning and available tools that are allowed in theoretical reasoning to critique the very theory of incompleteness itself. That is the essential Abstract Thesis. It will be seen that a real attempt is attempted.

**Category:** Set Theory and Logic

[92] **viXra:1508.0089 [pdf]**
*submitted on 2015-08-11 16:25:07*

**Authors:** Peiman Ghasemi

**Comments:** 6 Pages.

Until the current moment, mankind is not realized that there is a diverse population of intelligent civilizations living in our universe. In the current article we will deduce the occurrence/existence of extraterrestrial life by mathematical proof. I would show you that even inside our galaxy, the Milky Way, a sufficient number of alien creatures are living. It's a mathematical proof for the extraterrestrial life debate, for the first time in mankind's history.

**Category:** Set Theory and Logic

[91] **viXra:1506.0165 [pdf]**
*submitted on 2015-06-22 20:30:53*

**Authors:** Takahiro Kato

**Comments:** 14 Pages.

This article provides yet another characterization of Boolean algebras and, using this characterization, establishes a more direct connection between propositional logic and Boolean algebras.

**Category:** Set Theory and Logic

[90] **viXra:1506.0147 [pdf]**
*submitted on 2015-06-19 08:23:22*

**Authors:** Thomas Colignatus

**Comments:** 12 Pages.

The Zermelo-Fraenkel-Axiom-of-Choice (ZFC) axioms for set theory appear to be inconsistent. A step in developing this proof is the observation that ZFC would be deductively incomplete if it were consistent. Both points are proven by means of the singleton. The axioms are still too lax on the notion of a 'well-defined set'.

**Category:** Set Theory and Logic

[89] **viXra:1505.0173 [pdf]**
*submitted on 2015-05-24 14:58:20*

**Authors:** A.D Albert

**Comments:** Pages.

Traditionally Fuzzy Logic is defined in the unit interval of [0,1]. These notes explore what will happen if this interval was larger than the unit interval (where normalization of truth values will violate the definition of a particular fuzzy set). These notes also derive suitable logical operators applicable to this extended interval utilizing first principles.

**Category:** Set Theory and Logic

[88] **viXra:1505.0122 [pdf]**
*submitted on 2015-05-16 12:35:47*

**Authors:** Misha Mikhaylov

**Comments:** 24 Pages.

It seems that statements determining features of some algebraic structures behavior are based on just intuitive assumptions or empiric observations and for sake of convenience (simplest example is the phrase: “let’s consider 0! =1”… perhaps, just because sir I. Newton entrusted, so, why not 2, 5, 7.65 – choose any). So, without logical explanation these are looking a little mysterious or sometimes even magic. This article is a humble attempt to get it straight rather formally. Some troubles may appear on the way – e.g. as it was shown earlier (in the ref. [2], for example), there are at least two binary relations hav-ing properties of idempotent equivalences – algebra’s elements that may aspire to be an identity. Apparently, probable obtaining of some well-known results in the text is not an attempt of their re-discovering, but it is rather “check-points” that confirm theory validity, more by token that it was made by using of the only exceptionally formal way, while usually they are obtained rather intuitively. Usually the notion of tensor product is determined for each kind of algebraic structure – especially for modulus (in group theory it is often called direct product – but this is a matter of semantics, so, it’s rather negligible). Here it is shown that tensor product may be introduced without defining of concrete algebraic structure. Without such introduction defining of algebraic operation is strongly complicated.

**Category:** Set Theory and Logic

[87] **viXra:1505.0079 [pdf]**
*submitted on 2015-05-10 13:10:05*

**Authors:** Alexander S. Nudelman

**Comments:** 5 Pages.

In this paper we define an arithmetic theory PAM, which is an extension of Peano arithmetic PA, and prove that theory PAM has only one (up to isomorphism) model, which is the standard PA–model.

**Category:** Set Theory and Logic

[86] **viXra:1504.0086 [pdf]**
*submitted on 2015-04-11 03:11:45*

**Authors:** Vadim V Nazarenko

**Comments:** 1 Page.

As much as 0 is the opposite of प, every number of the Poorna series has it's opposite number from the Shoonya series.

**Category:** Set Theory and Logic

[85] **viXra:1503.0115 [pdf]**
*submitted on 2015-03-14 14:58:16*

**Authors:** Florentin Smarandache

**Comments:** 11 Pages.

In this paper we introduce for the first time a new type of structures, called (T, I, F)-Neutrosophic Structures, presented from a neutrosophic logic perspective, and we show particular cases of such structures in geometry and in algebra.
In any field of knowledge, each structure is composed from two parts: a space, and a set of axioms (or laws) acting (governing) on it. If the space, or at least one of its axioms (laws), has some indeterminacy, that structure is a (T, I, F)-Neutrosophic Structure.
The (T, I, F)-Neutrosophic Structures [based on the components T=truth, I=indeterminacy, F=falsehood] are different from the Neutrosophic Algebraic Structures [based on neutrosophic numbers of the form a+bI, where I=indeterminacy and I^n = I], that we rename as Neutrosophic I-Algebraic Structures (meaning algebraic structures based on indeterminacy “I” only). But we can combine both and obtain the (T, I, F)-Neutrosophic I-Algebraic Structures, i.e. algebraic structures based on neutrosophic numbers of the form a+bI, but also having indeterminacy related to the structure space (elements which only partially belong to the space, or elements we know nothing if they belong to the space or not) or indeterminacy related to at least one axiom (or law) acting on the structure space. Then we extend them to Refined (T, I, F)-Neutrosophic Refined I-Algebraic Structures.

**Category:** Set Theory and Logic

[84] **viXra:1503.0085 [pdf]**
*submitted on 2015-03-12 05:25:13*

**Authors:** Takahiro Kato

**Comments:** 323 Pages

Modules (also known as profunctors or distributors) and morphisms among them subsume categories and functors and provide more general and abstract framework to explore the theory of structures. In this book we generalize and redevelop the basic notions and results of category theory using this framework of modules.

**Category:** Set Theory and Logic

[83] **viXra:1502.0045 [pdf]**
*submitted on 2015-02-06 03:21:31*

**Authors:** Vasile Patrascu

**Comments:** 7 Pages. Neutrosophic Sets and Systems, pp. 40-46, Vol. 7, 2015.

This paper presents two variants of penta-valued representation for neutrosophic entropy. The first is an extension of Kaufmann's formula and the second is an extension of Kosko's formula.
Based on the primary three-valued information represented by the degree of truth, degree of falsity and degree of neutrality there are built some penta-valued representations that better highlights some specific features of neutrosophic entropy. Thus, we highlight five features of neutrosophic uncertainty such as ambiguity, ignorance, contradiction, neutrality and saturation. These five features are supplemented until a seven partition of unity by adding two features of neutrosophic certainty such as truth and falsity.
The paper also presents the particular forms of neutrosophic entropy obtained in the case of bifuzzy representations, intuitionistic fuzzy representations, paraconsistent fuzzy representations and finally the case of fuzzy representations.

**Category:** Set Theory and Logic

[82] **viXra:1501.0107 [pdf]**
*submitted on 2015-01-09 03:49:37*

**Authors:** Shrikrishna Jayraj Kalgaonkar

**Comments:** 33 Pages.

Almost everywhere “brotherhood relation” as non-reflexive, non-symmetric but transitive. I couldn’t agree with this. The reasons for this disagreement are explained in the article.
Brotherhood concept is discussed as a binary relation, types of binary relations, equivalence relations and its effect.
Finally suggesting that the brotherhood relation should be considered as a equivalence relation.

**Category:** Set Theory and Logic

[81] **viXra:1412.0269 [pdf]**
*submitted on 2014-12-29 20:01:52*

**Authors:** Jaykov Foukzon

**Comments:** 29 Pages.

In this paper paraconsistent first-order logic
LP^# with infinite hierarchy levels of contradiction is proposed. Corresponding paraconsistent set theory KSth^# is
proposed. Axiomatical system HST^#,as inconsistent generalization of Hrbacek set
theory HST is considered.

**Category:** Set Theory and Logic

[80] **viXra:1412.0235 [pdf]**
*submitted on 2014-12-25 05:46:46*

**Authors:** Thomas Colignatus

**Comments:** Short version of the argument (6 pages)

Paul of Venice (1369-1429) provides a consistency enhancer that resolves Russell's Paradox in naive set theory without using a theory of types. It allows a set of all sets. It also blocks the (diagonal) general proof of Cantor's Theorem on the power set. It is not unlikely that the Zermelo-Fraenkel (ZFC) axioms for set theory are still too lax on the notion of a "well-defined set". The transfinites of ZFC may be a mirage of still imperfect axiomatics w.r.t. the proper foundations for set theory.

**Category:** Set Theory and Logic

[79] **viXra:1412.0234 [pdf]**
*submitted on 2014-12-25 05:54:03*

**Authors:** Thomas Colignatus

**Comments:** 36 Pages. March 2012, written in Mathematica

> Context \[Bullet] In the philosophy of mathematics there is the distinction between platonism (realism), formalism, and constructivism. There seems to be no distinguishing experiment however to determine which approach is best according to which criteria. Philosophy finds a sounding board in the didactics of mathematics, as the relevant empirical context, instead of mathematics itself. Mathematicians are trained for abstract thought but in class they meet with real world students. Traditional mathematicians resolve their cognitive dissonance by relying on tradition. That tradition however is not targetted at didactic clarity and empirical relevance with respect to psychology. The mathematical curriculum is a mess. Philosophers can go astray when they don't realize the distinction between mathematics and the didactics of mathematics. > Problem \[Bullet] Aristotle distinguished between potential and actual infinite, Cantor proposed the transfinites, and Occam would want to reject those transfinites if they aren't really necessary. My book "A Logic of Exceptions" already refuted 'the' general proof of Cantor's Theorem on the power set, so that the latter holds only for finite sets but not for 'any' set. There still remains Cantor's diagonal argument on the real numbers. > Results \[Bullet] There is a 'bijection by abstraction' between \[DoubleStruckCapitalN] and \[DoubleStruckCapitalR]. Potential and actual infinity are two faces of the same coin. Potential infinity associates with counting, actual infinity with the continuum, but they would be 'equally large'. The notion of a limit in \[DoubleStruckCapitalR] cannot be defined independently from the construction of \[DoubleStruckCapitalR] itself. Occam's razor eliminates Cantor's transfinites. > Constructivist content \[Bullet] Constructive steps Subscript[S, 1], ..., Subscript[S, 5] are identified, where Subscript[S, 3] gives potential infinity and Subscript[S, 4] actual infinity. The latter is taken as "proper constructivism" and it contains abstraction. The confusions about Subscript[S, 6], nonconstructivism and the transfinites, derive rather from logic than from infinity.

**Category:** Set Theory and Logic

[78] **viXra:1412.0233 [pdf]**
*submitted on 2014-12-25 05:58:23*

**Authors:** Thomas Colignatus

**Comments:** 10 Pages. Paper of 2007, written in Mathematica

Adding some reasonable properties to the Gödelian system of Peano Arithmetic creates a new system for which Gödel's completeness theorems collapse and the Gödeliar becomes the Liar paradox again. Rejection of those properties is difficult since they are reasonable. Three-valued logic is a better option to deal with the Liar and its variants.

**Category:** Set Theory and Logic

[77] **viXra:1412.0201 [pdf]**
*submitted on 2014-12-19 02:03:14*

**Authors:** Karan Doshi

**Comments:** 14 Pages.

In this paper the author submits a proof using the Power Set relation for the existence of a transfinite cardinal strictly smaller than Aleph Zero, the cardinality of the Naturals. Further, it can be established taking these arguments to their logical conclusion that even smaller transfinite cardinals exist. In addition, as a lemma using these new found and revolutionary concepts, the author conjectures that some outstanding unresolved problems in number theory can be brought to heel. Specifically, a proof of the twin prime conjecture is given.

**Category:** Set Theory and Logic

[76] **viXra:1412.0155 [pdf]**
*submitted on 2014-12-09 20:21:48*

**Authors:** Florentin Smarandache

**Comments:** 500 Pages.

Neutrosophic Theory means Neutrosophy applied in many fields in order to solve problems related to indeterminacy.
Neutrosophy considers every entity <A> together with its opposite or negation <antiA>, and with their spectrum of neutralities <neutA> in between them (i.e. entities supporting neither nor <antiA>). Where

**Category:**

[75] **viXra:1412.0130 [pdf]**
*submitted on 2014-12-06 15:09:42*

**Authors:** Jaykov Foukzon

**Comments:** 17 Pages.

In recent years there has been a revitalised interest in non-classical solutions to the
semantic paradoxes. In this article the non-classical resolution of Curry’s Paradox and Shaw-Kwei's paradox without rejection any contraction postulate is proposed.

**Category:** Set Theory and Logic

[74] **viXra:1412.0006 [pdf]**
*submitted on 2014-12-01 09:42:44*

**Authors:** Vasile Patrascu

**Comments:** 10 Pages.

The paper presents some steps for multi-valued representation of neutrosophic information. These steps are provided in the framework of multi-valued logics using the following logical value: true, false, neutral, unknown and saturated. Also, this approach provides some calculus formulae for the following neutrosophic features: truth, falsity, neutrality, ignorance, under-definedness, over-definedness, saturation and entropy. In addition, it was defined net truth, definedness and neutrosophic score.

**Category:** Set Theory and Logic

[73] **viXra:1411.0529 [pdf]**
*submitted on 2014-11-21 07:00:54*

**Authors:** W. B. Vasantha Kandasamy, Florentin Smarandache, Ilanthenral K.

**Comments:** 242 Pages.

In this book the authors for the first time have ventured to study, analyse and investigate fuzzy and neutrosophic models and the experts opinion. To make such a study, innovative techniques and defined and developed. Several important conclusions about these models are derived using these new techniques. Open problems are suggested in this book.

**Category:** Set Theory and Logic

[72] **viXra:1411.0528 [pdf]**
*submitted on 2014-11-21 07:02:15*

**Authors:** Vasantha Kandasamy, Florentin Smarandache, Ilanthenral K.

**Comments:** 275 Pages.

In this book the authors for the first time have merged vertices and edges of lattices to get a new structure which may or may not be a lattice but is always a graph. This merging is done for graph too which will be used in the merging of fuzzy models. Further merging of graphs leads to the merging of matrices; both these concepts play a vital role in merging the fuzzy and neutrosophic models.
Several open conjectures are suggested.

**Category:** Set Theory and Logic

[71] **viXra:1411.0051 [pdf]**
*submitted on 2014-11-07 05:57:21*

**Authors:** Ricardo Alvira

**Comments:** 5 Pages.

It reviewes the difference between cocnepts involving Certainty/Uncertainty.

**Category:** Set Theory and Logic

[70] **viXra:1411.0009 [pdf]**
*submitted on 2014-11-01 23:51:59*

**Authors:** Karan Doshi

**Comments:** 8 Pages.

Well-ordering of the Reals@@ presents a major challenge in Set theory. Under the standard Zermelo Fraenkel Set theory (ZF) with the Axiom of Choice (ZFC), a well-ordering of the Reals is indeed possible. However the Axiom of Choice (AC) had to be introduced to the original ZF theory which is then shown equivalent to the well-ordering theorem. Despite the result however, no way has still been found of actually constructing a well-ordered Set of Reals. In this paper the author attempts to generate a well ordered Set of Reals without using the AC i.e. under ZF theory itself using the Axiom of the Power Set as the guiding principle.

**Category:** Set Theory and Logic

[69] **viXra:1410.0131 [pdf]**
*submitted on 2014-10-22 17:37:47*

**Authors:** A. A. Salama, Said Broumi

**Comments:** 5 Pages.

Since the world is full of indeterminacy, the neutrosophics found their place into
contemporary research. In this paper we define rough neutrosophic sets and study their
properties. Some propositions in this notion are proved. Possible application to computer
sciences is touched upon.

**Category:** Set Theory and Logic

[68] **viXra:1410.0022 [pdf]**
*submitted on 2014-10-04 20:57:05*

**Authors:** Yilun Shang

**Comments:** 6 Pages.

Let $U^{(n)}$ denote the maximal length arithmetic
progression in a non-uniform random subset of $\{0,1\}^n$, where $1$
appears with probability $p_n$. By using dependency graph and
Stein-Chen method, we show that $U^{(n)}-c_n\ln n$ converges in law
to an extreme type distribution with $\ln p_n=-2/c_n$. Similar
result holds for $W^{(n)}$, the maximal length aperiodic arithmetic
progression (mod $n$).

**Category:** Set Theory and Logic

[67] **viXra:1409.0174 [pdf]**
*submitted on 2014-09-25 14:54:50*

**Authors:** Misha Mikhaylov

**Comments:** 7 Pages.

Thoughts expressed in previous paper [3] were developed. There was shown formally that collection of sets’ properties may not appear chaotically and independently on each other. Presence or absence of one leads to rise or drop of another.

**Category:** Set Theory and Logic

[66] **viXra:1409.0056 [pdf]**
*submitted on 2014-09-08 16:49:27*

**Authors:** Misha Mikhaylov

**Comments:** 11 Pages.

There was an attempt to formalize an appearance of main relation properties in contrast to the usual one. There were paid an attention only for natural relations. Surely, such relations as “better than” are not observed here since the notion of “good” itself is still not defined.

**Category:** Set Theory and Logic

[65] **viXra:1409.0041 [pdf]**
*submitted on 2014-09-06 09:58:42*

**Authors:** Ihsan Raja Muda Nasution

**Comments:** 1 Page.

In this paper, we redefine the infinitesimals by using axiomatic method.

**Category:** Set Theory and Logic

[64] **viXra:1409.0006 [pdf]**
*submitted on 2014-09-02 01:36:05*

**Authors:** Felix M. Lev

**Comments:** 5 Pages.

Standard mathematics involves such notions as infinitely small/large, continuity and standard division. However, some of these notions are treated differently in traditional and constructive versions. This mathematics is usually treated as fundamental while finite mathematics is treated as inferior. Standard mathematics has foundational problems (as follows, for example, from G\"{o}del's incompleteness theorems) but people usually believe that this is less important than the fact that it describes many experimental data with high accuracy. We argue that the situation is the opposite: standard mathematics is only a special case of finite one in the
formal limit when the characteristic of the ring or field used in finite mathematics goes to infinity. Therefore foundational problems
in standard mathematics are not fundamental.

**Category:** Set Theory and Logic

[63] **viXra:1408.0211 [pdf]**
*submitted on 2014-08-29 10:11:03*

**Authors:** Ihsan Raja Muda Nasution

**Comments:** 1 Page.

Our result is the explicit form of the infinitesimals.

**Category:** Set Theory and Logic

[62] **viXra:1408.0039 [pdf]**
*submitted on 2014-08-07 22:42:05*

**Authors:** Oh Jung Uk

**Comments:** 8 Pages.

If B(P) is the truth value of proposition P then connectives could be translated to arithmetic operation in the congruent expression of 2 as described below.
B(~p)≡1+B(p)( mod 2 ),B(p∧q)≡B(p)B(q)( mod 2 )
B(p∨q)≡B(p)+B(q)+B(p)B(q)( mod 2 )
B(p⟶q)≡1+B(p)+B(p)B(q)( mod 2 )
B(p⟷q)≡1+B(p)+B(q)(mod 2)
By using this, logical laws could be proved, compound proposition could be simplified, and logical equation could be solved.

**Category:** Set Theory and Logic

[61] **viXra:1407.0033 [pdf]**
*submitted on 2014-07-04 06:53:06*

**Authors:** Subhajit Ganguly

**Comments:** 19 Pages.

Our aim is to help build a machine that can reduce the possibility of mishaps in navigation to zero. For that devise a new system of numbers, in which the real numbers are represented on the y-axis and complex numbers on the x-axis. Inside such a system, we incorporate the equivalent Ideal Fuzzy Logic that can be used by the machine to predict and avoid mishaps.

**Category:** Set Theory and Logic

[60] **viXra:1406.0115 [pdf]**
*submitted on 2014-06-18 12:02:18*

**Authors:** Mustafa A. Khan

**Comments:** 9 Pages.

This paper introduces a new type of set called the Sohraab-Hyder set or the SH set which is quite different from the well known and studied set of the Set Theory. This new set theory is logically consistent and leads to some fascinating results such as, (1) the necessary existence of an ultimate SH set that created itself in pre-eternity and will continue to exist to post-eternity, (2) the ultimate SH set is self sufficient, (2) the ultimate SH set must give rise to infinite number of other SH sets which represent logically consistent algebras and geometries, (3) the ultimate SH set must give rise to the existence of infinite universes, including our own, (4) the ultimate SH set must give rise to mathematical cells, organs and organisms which undergo must undergo evolution, (5) some of these mathematical cells, organs and organisms are same as the organic cells, organs and organisms which leads to the conclusion that the evolution of organisms is a mathematical fact and finally, (6) it is shown that every living and non-living entity in the universe, with the universe included of course, is a matrix of 0's and 1's.

**Category:** Set Theory and Logic

[59] **viXra:1406.0100 [pdf]**
*submitted on 2014-06-16 05:54:18*

**Authors:** Robert A. Herrmann

**Comments:** 4 Pages.

In 1963, nonstandard analysis was applied to the semantics of formal languages. In 1978, nonstandard analysis was first applied to the syntax of formal and informal languages via the notion of the consequence operator (operation) and later the logic-system. In order to incorporate both of these approaches into the Theory of Ultralogics, a foundational change is necessary.

**Category:** Set Theory and Logic

[58] **viXra:1405.0360 [pdf]**
*submitted on 2014-05-30 12:59:54*

**Authors:** Ricardo Alvira

**Comments:** 42 Pages.

Presentation based on book expected to be published in 1-2 months: "A Unified Theory of Complexity". ISBN: 978-1499335859. 250 pages aprox.

**Category:** Set Theory and Logic

[57] **viXra:1405.0227 [pdf]**
*submitted on 2014-05-14 03:52:18*

**Authors:** Leszek W. Guła

**Comments:** 2 Pages. no

**Category:** Set Theory and Logic

[56] **viXra:1405.0225 [pdf]**
*submitted on 2014-05-13 07:06:04*

**Authors:** Jailton C. Ferreira

**Comments:** 3 Pages.

A counterexample to Cantor's Diagonal Argument is revisited.

**Category:** Set Theory and Logic

[55] **viXra:1405.0202 [pdf]**
*submitted on 2014-05-10 01:05:38*

**Authors:** Bertrand Wong

**Comments:** 3 Pages.

This article raises some important points about logic, e.g., mathematical logic.

**Category:** Set Theory and Logic

[54] **viXra:1404.0454 [pdf]**
*submitted on 2014-04-24 08:43:58*

**Authors:** Ricardo Alvira

**Comments:** 14 Pages. To appear as “Un acercamiento lógico al diseño de indicadores de sostenibilidad urbana” en Urbanismo y Sostenibilidad en La Ciudad. Edited by BREEAM ES and ITG [currently in press]

The present text proposes a methodology for designing urban sustainability indicators based on set theory / fuzzy logic. The aim is to provide a sound basis that increases ‘understandability’ and ‘shareability’ of sustainability indicators. While the analysis focuses on urban sustainability indicators, its conclusions can be applied to many different type of indicators, not only referred to sustainability but to many different issues.

**Category:** Set Theory and Logic

[53] **viXra:1401.0149 [pdf]**
*submitted on 2014-01-22 05:43:50*

**Authors:** Antonio Leon

**Comments:** 14 Pages.

It is proved in this paper the undecidable formula involved in Gödel's first incompleteness theorem would be inconsistent if the formal system where it is defined were complete. So, before proving the formula is undecidable it is necessary to assume the system is not complete in order to ensure the formula is not inconsistent. Consequently, Gödel proof does not prove the formal system is incomplete but that, once assumed it is incomplete, it is possible to define an undecidable formula within the system. This conclusion makes Gödel's incompleteness theorems devoid of substance.

**Category:** Set Theory and Logic

[52] **viXra:1310.0242 [pdf]**
*submitted on 2013-10-28 02:44:24*

**Authors:** Antonio Leon

**Comments:** 190 Pages.

Selected set theory and supertask arguments on the formal consistency of the actual infinity hypothesis subsumed by the Axiom of Infinity.

**Category:** Set Theory and Logic

[51] **viXra:1310.0221 [pdf]**
*submitted on 2013-10-24 16:37:16*

**Authors:** A. A. Salama, Mohamed eisa, Florentin Smarandache

**Comments:** 1 Page.

The purpose of this paper is to introduce and study the characteristic
function of a neutrosophic set. After given the fundamental definitions of
neutrosophic set operations generated by, we obtain several properties, and discussed
the relationship between neutrosophic sets generated by Ng and others. Finally, we
introduce the neutrosophic topological spaces generated by Ng . Possible application
to GIS topology rules are touched upon.

**Category:** Set Theory and Logic

[50] **viXra:1309.0028 [pdf]**
*submitted on 2013-09-05 20:36:23*

**Authors:** Florentin Smarandache

**Comments:** 9 Pages.

In this paper we present a short history of logics: from particular cases of 2-symbol or numerical
valued logic to the general case of n-symbol or numerical valued logic. We show generalizations
of 2-valued Boolean logic to fuzzy logic, also from the Kleene’s and Lukasiewicz’ 3-symbol
valued logics or Belnap’s 4-symbol valued logic to the most general n-symbol or numerical
valued refined neutrosophic logic. Two classes of neutrosophic norm (n-norm) and neutrosophic
conorm (n-conorm) are defined. Examples of applications of neutrosophic logic to physics are
listed in the last section.
Similar generalizations can be done for n-Valued Refined Neutrosophic Set, and respectively n-
Valued Refined Neutrosopjhic Probability.

**Category:** Set Theory and Logic

[49] **viXra:1309.0013 [pdf]**
*submitted on 2013-09-04 07:32:54*

**Authors:** Robert A. Herrmann

**Comments:** 19 Pages.

Relative to universal logic, it is demonstrated how useful it is to utilize general logic-systems to investigate finite consequence operators (operations). Among many other examples relative to the lattice of finite consequence operators, a general characterization for the lattice-theoretic supremum for a nonempty collection of finite consequence operators is given. Further, it is shown that for any denumerable language there is rather simple collection of finite consequence operators and for a propositional language, three simple modifications to the finitary rules of inference that demonstrate that the lattice of finite consequence operators is not meet-complete.

**Category:** Set Theory and Logic

[48] **viXra:1307.0124 [pdf]**
*submitted on 2013-07-23 18:54:31*

**Authors:** editor Linfan Mao

**Comments:** 135 Pages.

The First International Conference on Smarandache Multispace and Multistructure was organized by Prof. Linfan Mao, and it was held in the Beijing University of Civil Engineering and Architecture of P. R. China on June 28-30, 2013. There were 46 researchers from China, India, Iran, Nigeria, and USA that have taken part in this conference with 14 papers on Smarandache multispace and geometry, birings, neutrosophy,neutrosophic groups, regular maps and topological graphs with applications to non-solvable equation systems.
Definition.
In any domain of knowledge, a Smarandache multispace (or S-multispace) with its multistructure is a finite or infinite (countable or uncountable) union of many spaces that have various
structures. The spaces may overlap. The notions of multispace (also spelt multi-space) and multi-
structure (also spelt multi-structure) were introduced by Smarandache in 1969 under his idea of hybrid science: combining different fields into a unifying field, which is closer to our real life world since we live in a heterogeneous space. Today, this idea is widely accepted by the world of sciences. S-multispace is a qualitative notion, since it is too large and includes both metric and non-metric spaces. It is believed that the smarandache multispace with its multistructure is the best candidate for 21st century Theory of Everything in any domain. It unifies many knowledge fields.
Applications.
A such multispace can be used for example in physics for the Unified Field Theory that tries to unite the gravitational, electromagnetic, weak and strong interactions. Or in the parallel quantum computing and in the mu-bit theory, in multi-entangled states or particles and up to multi-entangles objects. We also mention: the algebraic multispaces (multi-groups, multi-rings, multi-vector spaces, multi-operation systems and multi-manifolds, also multi-voltage graphs, multi-embedding of a graph in an n-manifold, etc.), geometric multispaces (combinations of Euclidean and Non-Euclidean geometries into one space as in Smarandache geometries), theoretical physics, including the relativity theory, the M-theory and the cosmology, then multi-space models for p-branes and cosmology, etc.
- The multispace and multistructure were first used in the Smarandache geometries (1969), which are combinations of different geometric spaces such that at least one geometric axiom behaves differently in each such space.
- In paradoxism (1980), which is a vanguard in literature, arts, and science, based on finding
common things to opposite ideas [i.e. combination of contradictory fields].
- In neutrosophy (1995), which is a generalization of dialectics in philosophy, and takes into con-
sideration not only an entity < A > and its opposite < antiA > as dialectics does, but also the
neutralities < neutA > in between. Neutrosophy combines all these three < A >,< antiA > and
< neutA > together. Neutrosophy is a metaphilosophy.
- Then in neutrosophic logic (1995), neutrosophic set (1995), and neutrosophic probability (1995), which have, behind the classical values of truth and falsehood, a third component called indeterminacy (or neutrality, which is neither true nor false, or is both true and false simultaneously - again a combination of opposites: true and false in indeterminacy).
- Also used in Smarandache algebraic structures (1998), where some algebraic structures are in-
cluded in other algebraic structures.

**Category:** Set Theory and Logic

[47] **viXra:1306.0209 [pdf]**
*submitted on 2013-06-25 11:06:36*

**Authors:** Andrew Nassif

**Comments:** 15 Pages. May need formatting, but not sure because they are notes.

A 15 page collection of my notes on mathematical articles I have created.

**Category:** Set Theory and Logic

[46] **viXra:1303.0084 [pdf]**
*submitted on 2013-03-11 13:38:33*

**Authors:** Mosayyeb Fakhreslam

**Comments:** 2 Pages.

In this paper we offer a solution to the unexpected examination paradox by introducing the unexpected examinationless day paradox.

**Category:** Set Theory and Logic

[45] **viXra:1302.0137 [pdf]**
*submitted on 2013-02-20 13:01:57*

**Authors:** Joachim Derichs

**Comments:** 44 Pages.

The outline of a programme for restructuring mathematical logic. We explain what we mean by ‘restructuring’ and carry out exemplary parts of the programme.

**Category:** Set Theory and Logic

[44] **viXra:1302.0048 [pdf]**
*submitted on 2013-02-08 12:33:56*

**Authors:** Jaykov Foukzon

**Comments:** 13 Pages.

In this article we derived an importent example of the inconsistent
countable set. Main result is: ~con(ZFC+E(\omega-model of ZFC)).

**Category:** Set Theory and Logic

[43] **viXra:1212.0127 [pdf]**
*submitted on 2012-12-20 14:29:35*

**Authors:** Colin Naturman, Henry Rose

**Comments:** 10 Pages.

The concept of ultra-universal algebras in varieties is generalized to models of first order theories. Characterizations of theories which have ulta-universal models are found and general examples of ultra-universal models are investigated. In particular we show that a theory has an ultra-universal model iff it is consistent and its class of models satisfies the joint embedding property.

**Category:** Set Theory and Logic

[42] **viXra:1212.0088 [pdf]**
*submitted on 2012-12-13 06:50:57*

**Authors:** Qiu Kui Zhang

**Comments:** 8 Pages.

In this article some difficulties are deduced from the set of natural numbers. The demonstrated difficulties suggest that if the set of natural numbers exists it would conflict with the axiom of regularity. As a result, we have the conclusion that the class of natural numbers is not a set but a proper class.

**Category:** Set Theory and Logic

[41] **viXra:1209.0070 [pdf]**
*submitted on 2012-09-20 17:26:27*

**Authors:** Nader Vakil

**Comments:** 4 Pages.

In this paper we show the consistency of the essential part of Sergeyev's numerical methodology (\cite{Yarov 1}, \cite{Yarov 2}) by constructing a model of it within the framework of an ultrapower of the ordinary real number system.

**Category:** Set Theory and Logic

[40] **viXra:1207.0064 [pdf]**
*submitted on 2012-07-17 02:27:13*

**Authors:** Pierre-Yves Gaillard

**Comments:** 1 Page.

We give a short proof of Zorn's Lemma.

**Category:** Set Theory and Logic

[39] **viXra:1207.0039 [pdf]**
*submitted on 2012-07-11 06:01:20*

**Authors:** Pierre-Yves Gaillard

**Comments:** 2 Pages.

The book "Categories and Sheaves" by Kashiwara and Schapira starts with a few statements which are not proved, a reference being given instead. We spell out the proofs in a short and self-contained way.

**Category:** Set Theory and Logic

[38] **viXra:1207.0009 [pdf]**
*submitted on 2012-07-03 22:07:09*

**Authors:** Pierre-Yves Gaillard

**Comments:** 2 Pages.

We give definitions in the spirit of Bourbaki's Set Theory for the basic notions of category theory. The goal is to avoid using either Grothendieck's universes axiom, or ``classes'' (or ``collections'') of sets which are not sets.

**Category:** Set Theory and Logic

[37] **viXra:1206.0106 [pdf]**
*submitted on 2012-07-01 00:39:37*

**Authors:** Pierre-Yves Gaillard

**Comments:** 6 Pages.

This is the beginning of an attempt at rewriting the book "Categories and Sheaves" by Kashiwara and Schapira without using Grothendieck's universes axiom.

**Category:** Set Theory and Logic

[36] **viXra:1204.0030 [pdf]**
*submitted on 2012-04-08 14:42:29*

**Authors:** Wilber Valgusbitkevyt

**Comments:** 2 Pages.

Reverse Modus Ponens followed by set theory using lines followed by considering the maximum number of colours that can be used using graph homomorphism.

**Category:** Set Theory and Logic

[35] **viXra:1204.0012 [pdf]**
*submitted on 2012-04-03 20:00:22*

**Authors:** Wilber Valgusbitkevyt

**Comments:** 3 Pages. I may upload another draft for more descriptive and elaborative explanations.

If there is something I am not explaining very elaborately or descriptively, let me know. I have had a math professor who showed me her inconsistent equations saying "these equations have no solution" although my equations are consistent and homogeneous which always have a solution. Also, I have had another math professor who told me it is wrong to assign specific values to variables although I was assigning factor variables to composite variables not to mention how there is an underlying condition how these numbers are positive integers excluding zero.
In this paper, I am creating a new theorem called Victoria Hayanisel Theorem dedicated to Princess Eugenie of York to describe the state of numbers, circles, and lines.
Followed by the theorem, I am using the set theory and Fermat's Infinite Descent Method (if my method is different, I will name it) to show how the conjecture is true.

**Category:** Set Theory and Logic

[34] **viXra:1204.0011 [pdf]**
*submitted on 2012-04-03 20:03:25*

**Authors:** Wilber Valgusbitkevyt

**Comments:** 2 Pages.

There is a pattern for the arbitary sequence which can be divided into four groups to be formalized as a recursive formula which shows the conjecture is true.

**Category:** Set Theory and Logic

[33] **viXra:1204.0009 [pdf]**
*submitted on 2012-04-03 20:08:46*

**Authors:** Wilber Valgusbitkevyt

**Comments:** 2 Pages.

Translate the graph into sets by using the shared vertexes as indexes. Then consider several different cases to see how the conjecture is true in each case.

**Category:** Set Theory and Logic

[32] **viXra:1202.0070 [pdf]**
*submitted on 2012-02-21 04:37:25*

**Authors:** Thierry Delort

**Comments:** 120 Pages.

This document, written in French, contains 2 parts:
In the 1st part, Théorie mathématique Platoniste (Platonic mathematical theory) we expose a complete Platonic theory, covering all the fiels of logic and foundation of mathematics, including a complete set theory.
In the 2nd part, Théorie aléatoire des nombres (random theory of number)we expose a theory of random in mathematics, that can be considered as as a branch of logic as well as a branch of number theory. In particular we show that it gives a theoretical justification of the Goldbach conjecture (weak and strong) as well of the twin prime Conjecture.

**Category:** Set Theory and Logic

[31] **viXra:1110.0055 [pdf]**
*submitted on 18 Oct 2011*

**Authors:** Thomas Evans

**Comments:** 19 pages

It is the underlying purpose of the author throughout this and subsequent related
papers to consider the examination of conjectures such as the Birch-Swinnerton-Dyer
conjecture, the Riemann Hypotheses, as well as a number of other misunderstood or
unacknowledged phenomena. It is the author's hope that through such considerations,
both autonomous and presented herein, that it may become evident that the introduction
of fundamental, new practices is a necessity to any advancement in the directions of the
aforementioned. This represents the first in a series of eight (8) papers regarding these
materials. Throughout the remaining 7 the author presents, to a much greater degree of
rigor, the basic theory of analytic gauge functions, associated phenomenology, and there
from a solution to the (two) above conjectures. This paper facilitates an introduction to
the theory of analytic gauges. In the first section the author presents a re-examination of
the concepts of geometries of connections. Very briefly introduced are the basic concepts
of analytic numbers, analytic fields, analytic gauge functions, etc.

**Category:** Set Theory and Logic

[30] **viXra:1108.0025 [pdf]**
*submitted on 19 Aug 2011*

**Authors:** Thomas Evans

**Comments:** 11 pages

I present extensions to logic theory whose utilitarian application contains itself
in the form of a developmental, logical framework determinant of all being, and then
derive several applications thereof to areas of general quantum theory and pure
mathematics, providing solutions to 2 longstanding relevant problems: P vs NP and the
Riemann Hypothesis.

**Category:** Set Theory and Logic

[29] **viXra:1108.0011 [pdf]**
*submitted on 4 Aug 2011*

**Authors:** Andrew Schumann

**Comments:** 23 pages

We present a general way that allows to construct systematically analytic
calculi for a large family of non-Archimedean many-valued logics:
hyperrational-valued, hyperreal-valued, and p-adic valued logics characterized
by a special format of semantics with an appropriate rejection
of Archimedes' axiom. These logics are built as different extensions of
standard many-valued logics (namely, Lukasiewicz's, Gödel's, Product,
and Post's logics). The informal sense of Archimedes' axiom is that anything
can be measured by a ruler. Also logical multiple-validity without
Archimedes' axiom consists in that the set of truth values is infinite and
it is not well-founded and well-ordered. We consider two cases of
non-Archimedean multi-valued logics: the first with many-validity in the interval
[0; 1] of hypernumbers and the second with many-validity in the
ring Zp of p-adic integers. On the base of non-Archimedean valued logics,
we construct non-Archimedean valued interval neutrosophic logics by
which we can describe neutrality phenomena.

**Category:** Set Theory and Logic

[28] **viXra:1107.0045 [pdf]**
*submitted on 23 Jul 2011*

**Authors:** Mauro Avon

**Comments:** 158 pages

This paper outlines an approach to mathematical logic which is different from the standard one. We
list the most relevant features of the system. In first-order logic there exist two different concepts of
term and formula, in place of these two concepts in our approach we have just one notion of
expression. The set-builder notation is enclosed as an expression-building pattern. In our system we
can easily express second-order and all-order conditions (the set to which a quantifier refers is
explicitly written in the expression). The meaning of a sentence will depend solely on the meaning
of the symbols it contains, it will not depend on external 'structures'. Our deductive system is based
on a very simple definition of proof and provides a good model of human mathematical deductive
process. The soundness and consistency of the system are proved, as well as the fact that our system
is not affected by the most known types of paradox. The paper provides both the theoretical
material and two fully documented examples of deduction. The author has built the whole system
with the idea to provide a faithful model of human mathematical deductive process. He believes this
objective has been achieved but obviously the reader is free to examine the system and get his own
opinion about it.

**Category:** Set Theory and Logic

[27] **viXra:1010.0052 [pdf]**
*submitted on 20 Mar 2010*

**Authors:** Florentin Smarandache

**Comments:** 3 pages

A paradox is called a statement <P> which is true and false in the same time.
Therefore, if we suppose that statement <P> is true, it results that <P> is false; and reciprocally,
if we suppose that <P> is false, it results that <P> is true.

**Category:** Set Theory and Logic

[26] **viXra:1010.0002 [pdf]**
*submitted on 1 Oct 2010*

**Authors:** Dm. Vatolin

**Comments:** 11 pages, Russian.

In this paper the question is examined that a incompleteness
of enough advanced theories of arithmetics does not follow from the Gëdel statements.

**Category:** Set Theory and Logic

[25] **viXra:1008.0091 [pdf]**
*submitted on 31 Aug 2010*

**Authors:** Florentin Smarandache

**Comments:** 157 pages

It was a surprise for me when in 1995 I received a manuscript from the mathematician,
experimental writer and innovative painter Florentin Smarandache, especially because the
treated subject was of philosophy - revealing paradoxes - and logics.
He had generalized the fuzzy logic, and introduced two new concepts:
a) "neutrosophy" - study of neutralities as an extension of dialectics;
b) and its derivative "neutrosophic", such as "neutrosophic logic", "neutrosophic set",
"neutrosophic probability", and "neutrosophic statistics" and thus opening new ways
of research in four fields: philosophy, logics, set theory, and probability/statistics.

**Category:** Set Theory and Logic

[24] **viXra:1005.0059 [pdf]**
*submitted on 14 May 2010*

**Authors:** Dm. Vatolin

**Comments:** 15 pages, Russian.

This article formulates three geometrical axioms from which it follows
that the continuum power is greater then any well-ordered set power.

**Category:** Set Theory and Logic

[23] **viXra:1005.0006 [pdf]**
*submitted on 10 Mar 2010*

**Authors:** Andrew Schumann, Florentin Smarandache

**Comments:** 121 pages

This book written by A. Schumann & F. Smarandache is devoted to advances
of non-Archimedean multiple-validity idea and its applications to logical reasoning.
Leibnitz was the first who proposed Archimedes' axiom to be rejected.
He postulated infinitesimals (infinitely small numbers) of the unit interval [0, 1]
which are larger than zero, but smaller than each positive real number. Robinson
applied this idea into modern mathematics in [117] and developed so-called
non-standard analysis. In the framework of non-standard analysis there were
obtained many interesting results examined in [37], [38], [74], [117].

**Category:** Set Theory and Logic

[22] **viXra:1004.0112 [pdf]**
*submitted on 21 Apr 2010*

**Authors:** V. Veeramani, Roque Batulan

**Comments:** 7 pages

This paper contains the Basic Definitions of an Intuitionstic Fuzzy Set theory and
operations on it. Mainly we discussed the basic concepts of α - cut with examples
and Characterisations.

**Category:** Set Theory and Logic

[21] **viXra:1004.0096 [pdf]**
*submitted on 19 Apr 2010*

**Authors:** Cheng-Gui Huang

**Comments:** 1 pages.

I claim that Neutrosophy, by Professor Florentin Smarandache, is a deep thought in human culture.
That gives advantage to break the mechanical understanding of human culture. For example,
according to the mechanical theory: existence and non-existence could not be simultaneously.
Actually existence and non-existence are simultaneously. Everyone knows that human life is
like a way in the empty space of a bird flying. Everyone can not see himself a second ago,
everyone can not see himself for the time being and everyone can not see himself a second
future. Everyone could not know what is the existence of self. Everyone is also difficult to
say the non-existence of self. So the existence and non-existence of self are simultaneously.
And the existence and nonexistence of everything are simultaneously, where, the law of excluded
middle does not apply. These basic facts express the depth of Smarandache's Neutrosophy. He
has a lot of friends in ancient and in nowadays, in the West and in the East.

**Category:** Set Theory and Logic

[20] **viXra:1004.0092 [pdf]**
*submitted on 19 Apr 2010*

**Authors:** Feng Liu

**Comments:** 8 pages.

Logic should have been defined as the unity of contradiction between logic director and logic
implementation. Chinese Daoism asserts that everything is defined in the unity of opposites,
namely yin and yang, accordingly yang conducts change and yin brings it up (I-Ching, also known
as Book of Changes). In this way logic is redefined in an indeterminate style to facilitate
"both A and Anti-A" etc. in neutrosophics of logic. The unity of opposites is also described
as neutrality in neutrosophy. An intermediate multi-referential model of excitation and inhibition
is developed to derive a multiagent architecture of logic, based on Chinese yin-yang philosophy.
This methodology of excitation/inhibition suggests a rhymed way of logic, leading to a dynamic
methodology of weight strategy that links logic with neural network approach. It also confirms
the crucial role of indeterminacy in logic as a fatal criticism to classical mathematics and
current basis of science.

**Category:** Set Theory and Logic

[19] **viXra:1004.0091 [pdf]**
*submitted on 19 Apr 2010*

**Authors:** Feng Liu

**Comments:** 12 pages.

As a philosophical analysis of some fatal paradoxes, the paper distinguishes the conceptual difference
between representation of truth and source of truth, and leads to the conclusion that in order to acquire
the genuine source of truth, independently of specific representations possibly belonging to different
worlds, one is necessary to ignore all the ideas, logics, conceptions, philosophies and representable
knowledge even himself belonging to those misleading worlds, returning to his infant nature, as a
preliminary step for his cultivation of unconstrained wisdom. It also carries out some coordinative
crucial issues as natural-doctrine, minded-unwitting, logic-infancy, conception-deconception,
determinacy-indeterminacy. The paper tries to verify the role of neutrosophy and neutrosophic logic in
religious issues and open a gateway toward the oriental classics, excavating the lost treasure.

**Category:** Set Theory and Logic

[18] **viXra:1004.0065 [pdf]**
*submitted on 10 Apr 2010*

**Authors:** Florentin Smarandache

**Comments:** 8 pages

In this paper we introduce the operators of validation and invalidation of a proposition, and we
extend the operator of S-denying a proposition, or an axiomatic system, from the geometric space
to respectively any theory in any domain of knowledge, and show six examples in geometry, in
mathematical analysis, and in topology.

**Category:** Set Theory and Logic

[17] **viXra:1004.0051 [pdf]**
*submitted on 8 Mar 2010*

**Authors:** Haibin Wang, Florentin Smarandache, Yan-Qing Zhang, Rajshekhar Sunderraman

**Comments:** 4 pages

Neutrosophic set is a part of neutrosophy which
studies the origin, nature, and scope of neutralities, as
well as their interactions with different ideational
spectra. Neutrosophic set is a powerful general formal
framework that has been recently proposed. However,
neutrosophic set needs to be specified from a technical
point of view. To this effect, we define the settheoretic
operators on an instance of neutrosophic set,
we call it single valued neutrosophic set (SVNS). We
provide various properties of SVNS, which are
connected to the operations and relations over SVNS.

**Category:** Set Theory and Logic

[16] **viXra:1004.0026 [pdf]**
*submitted on 3 Apr 2010*

**Authors:** Florentin Smarandache

**Comments:** 14 pages

These paradoxes are called "neutrosophic" since they are based on indeterminacy (or neutrality,
i.e. neither true nor false), which is the third component in neutrosophic logic. We generalize the
Venn Diagram to a Neutrosophic Diagram, which deals with vague, inexact, ambiguous, illdefined
ideas, statements, notions, entities with unclear borders. We define the neutrosophic truth
table and introduce two neutrosophic operators (neuterization and antonymization operators)
give many classes of neutrosophic paradoxes.

**Category:** Set Theory and Logic

[15] **viXra:1004.0016 [pdf]**
*submitted on 8 Mar 2010*

**Authors:** Florentin Smarandache

**Comments:** 11 pages

In this paper we present the N-norms/N-conorms in neutrosophic logic and set as extensions of
T-norms/T-conorms in fuzzy logic and set.
Also, as an extension of the Intuitionistic Fuzzy Topology we present the Neutrosophic
Topologies.

**Category:** Set Theory and Logic

[14] **viXra:1004.0013 [pdf]**
*submitted on 8 Mar 2010*

**Authors:** Florentin Smarandache, V. Christianto

**Comments:** 15 pages

We extend Knuth's 16 Boolean binary logic operators to fuzzy logic and neutrosophic
logic binary operators. Then we generalize them to n-ary fuzzy logic and neutrosophic logic
operators using the smarandache codification of the Venn diagram and a defined vector
neutrosophic law. In such way, new operators in neutrosophic logic/set/probability are built.

**Category:** Set Theory and Logic

[13] **viXra:1004.0010 [pdf]**
*submitted on 8 Mar 2010*

**Authors:** Florentin Smarandache

**Comments:** 6 pages

The paper presents an initial explorations on T, I, F operations based on genetic concept
hierarchy and genetic referential hierarchy, as a novel proposal to the indeterminacy issue in
neutrosophic logic, in contrast to the T, I, F values inherited from conventional logics in which
those values would fail to demonstrate the genetic aspect of a concept and accordingly loose the
connection between generality and practicality. Based on the novel definition of logic and on the
relativity of T, F concept, it illustrates that T, F are hierarchical operations which inter-consist and
inter-complement each other, that "I" relates to a learning behavior profiled by an inspiration from
I-ching, and that the neutralization operation, as the means to solve contradictions, will eventually
come to the unification of opposites, leading to the fundamental issues in Buddhism and such alike.
It also implies that Buddhism and Daoism are not religions.

**Category:** Set Theory and Logic

[12] **viXra:1004.0006 [pdf]**
*submitted on 8 Mar 2010*

**Authors:** Florentin Smarandache

**Comments:** 10 pages

Based on the authors intensive investigation on the oriental dialectics, the paper presents a novel
theoretical frame of matter element in the world leading science, extenics dealing with inconsistency or
incompatibility, covering the widest range of application area from informatics, system engineering to
management and finance. The dialectic matter-element is defined as the integral of all existing and prospecting
ones based on all the infinite possible cognitive models. The novel model serves as the origin of constraint
matter elements, the unity of both state description and cognitive action (cognition force with respect to neural
science), a latent part of extenics, and possibly as essence of matter element. It explains, in a novel perspective,
the origin of a name, and uncovers the source of contradiction and even the impetus of cognition.

**Category:** Set Theory and Logic

[11] **viXra:1003.0269 [pdf]**
*submitted on 8 Mar 2010*

**Authors:** C. Le

**Comments:** 4 pages, edited by C. Le, and translated into German by Bernd Hutschenreuther

The Smarandache's Class of Paradoxes are semantic paradoxes of the form "All is <A>, the
<nonA> too!", where <nonA> is what is not <A>. As a particular case, replacing <A> but an
attribute (or, in general, by an idea) it is well know the Smarandache semantic paradox:
"All is possible, the impossible too!" which is the motto of the Paradoxism movement in arts,
letters, and sciences.

**Category:** Set Theory and Logic

[10] **viXra:1003.0224 [pdf]**
*submitted on 7 Mar 2010*

**Authors:** Charles Ashbacher

**Comments:** 145 pages

As someone who works heavily in both math and computers, I can truly appreciate the
role that logic plays in our modern world. One cannot understand the foundations of
mathematics while lacking knowledge of the basics of logic and how proofs are
constructed. Two of the first classes I took as a graduate student in mathematics were in
the foundations of mathematics, and hardly a day goes by where I do not use some topic
from those courses.

**Category:** Set Theory and Logic

[9] **viXra:1003.0171 [pdf]**
*submitted on 6 Mar 2010*

**Authors:** Florentin Smarandache

**Comments:** 4 pages

In this paper we give two theorems from the Propositional Calculus of the
Boolean Logic with their consequences and applications and we prove them
axiomatically.

**Category:** Set Theory and Logic

[8] **viXra:1003.0167 [pdf]**
*submitted on 6 Mar 2010*

**Authors:** Florentin Smarandache

**Comments:** 7 pages

In this article one builds a class of recursive sets, one establishes
properties of these sets and one proposes applications. This article widens
some results of [1].

**Category:** Set Theory and Logic

[7] **viXra:1003.0119 [pdf]**
*submitted on 6 Mar 2010*

**Authors:** Florentin Smarandache

**Comments:** 6 pages

Thirty original and collected problems, puzzles, and paradoxes in mathematics and physics are
explained in this paper, taught by the author to the elementary and high school teachers at the
University of New Mexico - Gallup in 1997-8 and afterwards. They have a more educational
interest because make the students think different!

**Category:** Set Theory and Logic

[6] **viXra:1003.0117 [pdf]**
*submitted on 6 Mar 2010*

**Authors:** Florentin Smarandache

**Comments:** 12 pages

Classes of linguistic paradoxes are introduced with examples and
explanations. They are part of the author's work on the Paradoxist
Philosophy based on mathematical logic.
The general cases exposed below are modeled on the English
language structure in a rigid way. In order to find nice
particular examples of such paradoxes one grammatically adjusts the
sentences.

**Category:** Set Theory and Logic

[5] **viXra:1003.0065 [pdf]**
*submitted on 6 Mar 2010*

**Authors:** Feng Liu, Florentin Smarandache

**Comments:** 10 pages

The paper presents a fresh new comprehensive ideology on Neutrosophic Logic based
on contradiction study in a broad sense: general critics on conventional logic by examining the essence of
logic, fresh insights on logic definition based on Chinese philosophical survey, and a novel and genetic
logic model as the elementary cell against Von Neumann oriented ones based on this novel definition. As
for the logic definition, the paper illustrates that logic is rather a tradeoff between different factors than
truth and false abstraction. It is stressed that the kernel of any intelligent system is exactly a contradiction
model. The paper aims to solve the chaos of logic and exhibit the potential power of neutrosophy: a new
branch of scientific philosophy.

**Category:** Set Theory and Logic

[4] **viXra:1003.0062 [pdf]**
*submitted on 6 Mar 2010*

**Authors:** Feng Liu, Florentin Smarandache

**Comments:** 7 pages

The paper presents a fresh new start on the neutrality of neutrosophy in
that "both A and Non-A" as an alternative to describe Neuter-A in that we conceptualize
things in both intentional and unintentional background. This unity of opposites
constitutes both objective world and subjective world. The whole induction of such
argument is based on the intensive study on Buddhism and Daoism including I-ching. In
addition, a framework of contradiction oriented learning philosophy inspired from the
Later Trigrams of King Wen in I-ching is meanwhile presented. It is shown that although
A and Non-A are logically inconsistent, but they are philosophically consistent in the
sense that Non-A can be the unintentionally instead of negation that leads to confusion. It
is also shown that Buddhism and Daoism play an important role in neutrosophy, and
should be extended in the way of neutrosophy to all sciences according to the original
intention of neutrosophy.

**Category:** Set Theory and Logic

[3] **viXra:0912.0017 [pdf]**
*submitted on 8 Dec 2009*

**Authors:** Feng Xu

**Comments:** 6 pages, first published in 2006 in Hadronic Journal, volume 29, page 227

The set of all the subsets of a set is its power set, and the cardinality of the power set is always larger than the set and its subsets. Based on the definition and the inequality in cardinality, a set cannot include its power set as element, and a power set cannot include itself as element. "Russell's set" is a putative set of all the sets that don't include themselves as element. It can be shown, however, that "Russell's set" can never take in all such sets. This is because its own power set, which (like any power set) is a set that doesn't include itself (thus qualifies as an element for "Russell's set"), cannot (although should) be taken in due to the cardinality inequality. Thus "Russell's set" can never be formed. Without it, Russell's paradox, which forced the modification of Cantor's intuitive set theory into a more restricted axiomatic theory, can never be formulated. The reported approach to resolve Russell's paradox is fundamentally different from the conventional approaches. It may restore the self-consistency of Cantor's original set theory, make the Axiom of Regularity unnecessary, and expand the coverage of set to assemblies that include themselves as element.

**Category:** Set Theory and Logic

[2] **viXra:0910.0041 [pdf]**
*submitted on 21 Oct 2009*

**Authors:** Amrit S. Sorli

**Comments:** 2 pages

In 1949, Gödel postulated a theorem that stated: "In any universe described by the theory of relativity,
time cannot exist". Gödel idea was that forth coordinate of space-time is not time. Fourth coordinate is
spatial too. In this article will be shown that on the base of elementary perception and experimental data
Gödel theorem is right. With eyes one observes universe is in a continuous change. A change n gets
transformed into a change n+1, the change n+1 into a change n+2 and so on. Clocks measure a
frequency, velocity and numerical order of change. Experimental date confirms that changes and clocks
do not run time; they run in space only. Time is not a part of space. Fourth coordinate of space-time is
spatial too. Space itself is timeless. Physical time that is clocks run is man created physical reality.
Fundamental arena of the universe is timeless space. In the timeless space into which massive bodies
and elementary particles move there is no past and no future. Past and future belong to the inner
neuronal space-time that is a result of neuronal activity of the brain.

**Category:** Set Theory and Logic

[1] **viXra:0909.0039 [pdf]**
*submitted on 16 Sep 2009*

**Authors:** Victor Porton

**Comments:** 2 Pages.

In the framework of ZF formally considered generalizations, such as whole numbers generalizing
natural number, rational numbers generalizing whole numbers, real numbers generalizing
rational numbers, complex numbers generalizing real numbers, etc. The formal consideration
of this may be especially useful for computer proof assistants.

**Category:** Set Theory and Logic

[83] **viXra:1604.0104 [pdf]**
*replaced on 2016-04-10 18:42:42*

**Authors:** Allen D Allen

**Comments:** Abstract contains 200 words, ms runs 6 pages

By proving that his “last theorem” (FLT) is true for the integral exponent n = 3, Fermat took the first step in a standard method of proving there exists no greatest lower bound on n for which FLT is true, thus proving the theorem. Unfortunately, there are two reasons why the standard method of proof is not available for FLT. First, transitive inequality lies at the heart of that method. Secondly, FLT admits to a change from > to < rendering their transitive natures unavailable. A related, self evident symmetry illustrates another problem that would have plagued Fermat and centuries of successors. FLT asserts such a narrow proposition, it is difficult to find an antecedent while easy to find a non equivalent consequence. For example, if FLT asserted that the exponent n is even, then FLT would be equivalent to the proposition that Fermat’s equation has two solutions, one for positive bases and one for their negative counterparts. This could be addressed with conservative transformations. The example provided by FLT motivates the use of an early paper by the author to prove a theorem on theorems. The theorem on theorems demonstrates there are infinitely many theorems as difficult to prove as FLT.

**Category:** Set Theory and Logic

[82] **viXra:1603.0226 [pdf]**
*replaced on 2016-03-17 02:40:06*

**Authors:** Vasile Pătraşcu

**Comments:** 12 Pages.

Starting from the primary representation of neutrosophic information, namely the degree of truth, degree of indeterminacy and degree of falsity, we define a nuanced representation in a penta valued fuzzy space, described by the index of truth, index of falsity, index of ignorance, index of contradiction and index of hesitance. Also, it was constructed an associated penta valued logic and then using this logic, it was defined for the proposed penta valued structure the following operators: union, intersection, negation, complement and dual. Then, the penta valued representation is extended to a hexa valued one, adding the sixth component, namely the index of ambiguity.

**Category:** Set Theory and Logic

[81] **viXra:1603.0226 [pdf]**
*replaced on 2016-03-17 02:40:06*

**Authors:** Vasile Pătraşcu

**Comments:** 12 Pages.

Starting from the primary representation of neutrosophic information, namely the degree of truth, degree of indeterminacy and degree of falsity, we define a nuanced representation in a penta valued fuzzy space, described by the index of truth, index of falsity, index of ignorance, index of contradiction and index of hesitance. Also, it was constructed an associated penta valued logic and then using this logic, it was defined for the proposed penta valued structure the following operators: union, intersection, negation, complement and dual. Then, the penta valued representation is extended to a hexa valued one, adding the sixth component, namely the index of ambiguity.

**Category:** Set Theory and Logic

[80] **viXra:1602.0006 [pdf]**
*replaced on 2016-04-15 09:03:00*

**Authors:** Andrew Banks

**Comments:** 19 Pages.

Cantor’s infinity (CI) depends on the capability of completing an infinite collection of successive steps (ICSS). Otherwise, CI does not solve Zeno’s dichotomy paradox and all natural numbers cannot be upward constructed using the successor function. However, there is no evidence in this universe whatsoever that demonstrates the existence of ICSS nor is there any direct proof that supports its theoretical viability. Regardless of these facts, ICSS rests at the foundation of theoretical mathematics. As such, this article will produce a computing device that is capable of executing instructions in any given time. This device will demonstrate that ICSS contradicts the assumption that the natural numbers are unbounded (NNU). Therefore, adding ICSS to mathematics that includes NNU causes it to be inconsistent. Hence, CI is not consistent since it contains both. Furthermore, using this computing device and other machinery, it will be proven that space and time are only finitely divisible.

**Category:** Set Theory and Logic

[79] **viXra:1508.0089 [pdf]**
*replaced on 2016-02-13 08:42:50*

**Authors:** Peiman Ghasemi

**Comments:** 7 Pages.

Until the current moment, mankind is not realized that there is a diverse population of intelligent civilizations living in our universe. In the current article we will deduce the occurrence/existence of extraterrestrial life by mathematical proof. I would show you that even inside our galaxy, the Milky Way, a sufficient number of alien creatures are living. The first section includes an algebraic probabilistic proof when the event of life is not highly biased and the second section includes a proof by contradiction that describes the event fundamentally. It's a mathematical proof for the extraterrestrial life debate, for the first time in mankind's history.

**Category:** Set Theory and Logic

[78] **viXra:1508.0089 [pdf]**
*replaced on 2016-02-13 03:43:30*

**Authors:** Peiman Ghasemi

**Comments:** 7 Pages.

Until the current moment, mankind is not realized that there is a diverse population of intelligent civilizations living in our universe. In the current article we will deduce the occurrence/existence of extraterrestrial life by mathematical proof. I would show you that even inside our galaxy, the Milky Way, a sufficient number of alien creatures are living. The first section includes an algebraic probabilistic proof when the event of life is not highly biased and the second section includes a proof by contradiction that describes the event fundamentally. It's a mathematical proof for the extraterrestrial life debate, for the first time in mankind's history.

**Category:** Set Theory and Logic

[77] **viXra:1508.0089 [pdf]**
*replaced on 2015-10-13 11:16:10*

**Authors:** Peiman Ghasemi

**Comments:** 6 Pages.

Until the current moment, mankind is not realized that there is a diverse population of intelligent civilizations living in our universe. In the current article we will deduce the occurrence/existence of extraterrestrial life by mathematical proof. I would show you that even inside our galaxy, the Milky Way, a sufficient number of alien creatures are living. It's a mathematical proof for the extraterrestrial life debate, for the first time in mankind's history.

**Category:** Set Theory and Logic

[76] **viXra:1506.0165 [pdf]**
*replaced on 2015-09-10 05:47:49*

**Authors:** Takahiro Kato

**Comments:** 16 Pages.

This article provides yet another characterization of Boolean algebras and, using this characterization, establishes a more direct connection between propositional logic and Boolean algebras.

**Category:** Set Theory and Logic

[75] **viXra:1506.0147 [pdf]**
*replaced on 2015-07-26 23:12:31*

**Authors:** Thomas Colignatus

**Comments:** 13 Pages.

The Zermelo-Fraenkel-Axiom-of-Choice (ZFC) system of axioms for set theory appears to be inconsistent. A step in developing this proof is the observation that ZFC would be deductively incomplete if it were consistent. Both points are proven by means of the singleton. The axioms are still too lax on the notion of a well-defined set.

**Category:** Set Theory and Logic

[74] **viXra:1506.0147 [pdf]**
*replaced on 2015-06-27 03:05:22*

**Authors:** Thomas Colignatus

**Comments:** 11 Pages.

The Zermelo-Fraenkel-Axiom-of-Choice (ZFC) system of axioms for set theory
appears to be inconsistent. A step in developing this proof is the observation
that ZFC would be deductively incomplete if it were consistent. Both points are
proven by means of the singleton. The axioms are still too lax on the notion of
a 'well-defined set'.

**Category:** Set Theory and Logic

[73] **viXra:1506.0147 [pdf]**
*replaced on 2015-06-24 10:54:12*

**Authors:** Thomas Colignatus

**Comments:** 11 Pages.

The Zermelo-Fraenkel-Axiom-of-Choice (ZFC) system of axioms for set theory appears to be inconsistent. A step in developing this proof is the observation that ZFC would be deductively incomplete if it were consistent. Both points are proven by means of the singleton. The axioms are still too lax on the notion of a 'well-defined set'.

**Category:** Set Theory and Logic

[72] **viXra:1505.0122 [pdf]**
*replaced on 2016-03-02 13:16:32*

**Authors:** Misha Mikhaylov

**Comments:** 24 Pages.

It seems that statements determining features of some algebraic structures behavior are based on just intuitive assumptions or empiric observations and for sake of convenience (simplest example is the phrase: “let’s consider 0! =1”… perhaps, just because Sir Isaac Newton entrusted, so, why not choose any: e.g. 2, 5, or 7.65). So, without logical explanation these are looking a little mysterious or sometimes even magic. This article is a humble attempt to get it straight rather formally. Some troubles may appear on the way – e.g. as it was shown earlier (in the ref. [2], for example), there are at least two binary relations having properties of idempotent equivalences – algebra’s elements that may aspire to be an identity. Apparently, probable obtaining of some well-known results in the text is not an attempt of their re-discovering, but it is rather “check-points” that confirm theory validity, more by token that it was made by using of the only exceptionally formal way, while usually they are obtained rather intuitively. Usually the notion of tensor product is determined for each kind of algebraic structure – especially for modulus (in group theory it is often called direct product – but this is a matter of semantics, so, it’s rather negligible). Here it is shown that tensor product may be introduced without defining of concrete algebraic structure. Without such introduction defining of algebraic operation is strongly complicated.

**Category:** Set Theory and Logic

[71] **viXra:1504.0086 [pdf]**
*replaced on 2015-04-12 00:44:59*

**Authors:** Vadim V Nazarenko

**Comments:** 1 Page.

As much as 0 is the opposite of प, every number of the Poorna series has it's opposite number from the Shoonya series.

**Category:** Set Theory and Logic

[70] **viXra:1503.0085 [pdf]**
*replaced on 2016-02-02 04:09:25*

**Authors:** Takahiro Kato

**Comments:** 365 Pages.

Modules (also known as profunctors or distributors) and morphisms among them subsume categories and functors and provide more general and abstract framework to explore the theory of structures. In this book we generalize and redevelop the basic notions and results of category theory using this framework of modules.

**Category:** Set Theory and Logic

[69] **viXra:1412.0235 [pdf]**
*replaced on 2015-07-28 04:51:49*

**Authors:** Thomas Colignatus

**Comments:** 2 Pages. The paper refers to the book FMNAI that supersedes the paper

Paul of Venice (1369-1429) provides a consistency condition that resolves Russell's Paradox in naive set theory without using a theory of types. It allows a set of all sets. It also blocks the (diagonal) general proof of Cantor's Theorem (in Russell's form, for the power set). The Zermelo-Fraenkel-Axiom-of-Choice (ZFC) axioms for set theory appear to be inconsistent. They are still too lax on the notion of a well-defined set. The transfinites of ZFC may be a mirage, and a consequence of still imperfect axiomatics in ZFC for the foundations of set theory. For amendment of ZFC two alternatives are mentioned: ZFC-PV (amendment of de Axiom of Separation) or BST (Basic Set Theory).

**Category:** Set Theory and Logic

[68] **viXra:1412.0235 [pdf]**
*replaced on 2015-06-27 03:07:27*

**Authors:** Thomas Colignatus

**Comments:** 30 Pages.

Paul of Venice (1369-1429) provides a consistency condition that resolves Russell's Paradox in naive set theory without using a theory of types. It allows a set of all sets. It also blocks the (diagonal) general proof of Cantor's Theorem (in Russell's form, for the power set). The Zermelo-Fraenkel-Axiom-of-Choice (ZFC) axioms for set theory appear to be inconsistent. They are still too lax on the notion of a 'well-defined set'. The transfinites of ZFC may be a mirage, and a consequence of still imperfect axiomatics in ZFC for the foundations of set theory. For amendment of ZFC two alternatives are mentioned: ZFC-PV (amendment of de Axiom of Separation) or BST (Basic Set Theory). Theorems 2.5 & 2.7 show for the singleton that ZFC is inconsistent. Lemma 3.2 shows that a Cantorian reading of ZFC implies the possibility of the weaker Pauline reading. Theorem 3.3 gives a constructive proof of the existence of a Pauline set. Appendix D deproves Cantor's Theorem.

**Category:** Set Theory and Logic

[67] **viXra:1412.0235 [pdf]**
*replaced on 2015-06-17 11:59:17*

**Authors:** Thomas Colignatus

**Comments:** 30 Pages.

Paul of Venice (1369-1429) provides a consistency condition that resolves Russell's Paradox in naive set theory without using a theory of types. It allows a set of all sets. It also blocks the (diagonal) general proof of Cantor's Theorem (in Russell's form, for the power set). The Zermelo-Fraenkel-Axiom-of-Choice (ZFC) axioms for set theory appear to be inconsistent. They are still too lax on the notion of a 'well-defined set'. The transfinites of ZFC may be a mirage, and a consequence of still imperfect axiomatics in ZFC for the foundations of set theory.
For amendment of ZFC two alternatives are mentioned: ZFC-PV (amendment of de Axiom of Separation) or BST (Basic Set Theory).
Theorems 2.5 & 2.7 show for the singleton that ZFC is inconsistent. Lemma 3.2 shows that a Cantorian reading of ZFC implies the possibility of the weaker Pauline reading. Theorem 3.3 gives a constructive proof of the existence of a Pauline set. Appendix D deproves Cantor's Theorem.

**Category:** Set Theory and Logic

[66] **viXra:1412.0235 [pdf]**
*replaced on 2015-06-12 16:10:45*

**Authors:** Thomas Colignatus

**Comments:** 40 Pages.

Paul of Venice (1369-1429) provides a consistency condition that resolves Russell's Paradox in naive set theory without using a theory of types. It allows a set of all sets. It also blocks the (diagonal) general proof of Cantor's Theorem (in Russell's form, for the power set). It is not unlikely that the Zermelo-Fraenkel (ZFC) axioms for set theory are still too lax on the notion of a 'well-defined set'. The transfinites of ZFC may be a mirage, and a consequence of still imperfect axiomatics in ZFC for the foundations for set theory.
For amendment of ZFC two alternatives are mentioned: ZFC-PV (amendment of de Axiom of Separation) or BST (Basic Set Theory).
Lemma 3.2 shows that a Cantorian reading of ZFC implies the possibility of the weaker Pauline reading. Theorem 3.3 gives the existence of a Pauline set. It has a fundamental constructive proof and a compact non-constructive proof. Theorem 1.1.6 shows that ZFC has an inconsistency.

**Category:** Set Theory and Logic

[65] **viXra:1412.0235 [pdf]**
*replaced on 2015-06-05 02:16:20*

**Authors:** Thomas Colignatus

**Comments:** 38 Pages.

Paul of Venice (1369-1429) provides a consistency condition that resolves Russell's Paradox in naive set theory without using a theory of types. It allows a set of all sets. It also blocks the (diagonal) general proof of Cantor's Theorem (in Russell's form, for the power set). It is not unlikely that the Zermelo-Fraenkel (ZFC) axioms for set theory are still too lax on the notion of a 'well-defined set'. The transfinites of ZFC may be a mirage, and a consequence of still imperfect axiomatics in ZFC for the foundations for set theory.
For amendment of ZFC two alternatives are mentioned: ZFC-PV (amendment of de Axiom of Separation) or BST (Basic Set Theory).
Lemma 3.2 shows that a Cantorian reading of ZFC implies the possibility of the weaker Pauline reading. Theorem 3.3 gives the existence of a Pauline set. It has a fundamental constructive proof and a compact non-constructive proof. Theorem 1.1.6 shows that ZFC has an anomaly. Corollary 3.3 turns that anomaly into an inconsistency.

**Category:** Set Theory and Logic

[64] **viXra:1412.0235 [pdf]**
*replaced on 2015-05-20 11:16:57*

**Authors:** Thomas Colignatus

**Comments:** 18 Pages.

Paul of Venice (1369-1429) provides a consistency condition that resolves Russell's Paradox in naive set theory without using a theory of types. It allows a set of all sets. It also blocks the (diagonal) general proof of Cantor's Theorem (in Russell's form, for the power set). It is not unlikely that the Zermelo-Fraenkel (ZFC) axioms for set theory are still too lax on the notion of a 'well-defined set'. The transfinites of ZFC may be a mirage, and a consequence of still imperfect axiomatics in ZFC w.r.t. the proper foundations for set theory. For amendment of ZFC two alternatives are mentioned: ZFC-PV (amendment of de Axiom of Separation) or BST (Basic Set Theory).

**Category:** Set Theory and Logic

[63] **viXra:1412.0235 [pdf]**
*replaced on 2015-05-01 04:33:25*

**Authors:** Thomas Colignatus

**Comments:** 13 Pages. Corrects a wrong bracket in Notation on p2 of the April 30 version

Paul of Venice (1369-1429) provides a consistency condition that resolves Russell's Paradox in naive set theory without using a theory of types. It allows a set of all sets. It also blocks the (diagonal) general proof of Cantor's Theorem (in Russell's form, for the power set). It is not unlikely that the Zermelo-Fraenkel (ZFC) axioms for set theory are still too lax on the notion of a 'well-defined set'. The transfinites of ZFC may be a mirage, and a consequence of still imperfect axiomatics in ZFC w.r.t. the proper foundations for set theory. For amendment of ZFC two alternatives are mentioned: ZFC-PV (amendment of de Axiom of Separation) or BST (Basic Set Theory).

**Category:** Set Theory and Logic

[62] **viXra:1412.0235 [pdf]**
*replaced on 2015-04-30 10:33:05*

**Authors:** Thomas Colignatus

**Comments:** 13 Pages.

Paul of Venice (1369-1429) provides a consistency condition that resolves Russell's Paradox in naive set theory without using a theory of types. It allows a set of all sets. It also blocks the (diagonal) general proof of Cantor's Theorem (in Russell's form, for the power set). It is not unlikely that the Zermelo-Fraenkel (ZFC) axioms for set theory are still too lax on the notion of a 'well-defined set'. The transfinites of ZFC may be a mirage, and a consequence of still imperfect axiomatics in ZFC w.r.t. the proper foundations for set theory. For amendment of ZFC two alternatives are mentioned: ZFC-PV (amendment of de Axiom of Separation) or BST (Basic Set Theory).

**Category:** Set Theory and Logic

[61] **viXra:1412.0235 [pdf]**
*replaced on 2014-12-31 02:45:55*

**Authors:** Thomas Colignatus

**Comments:** 8 Pages.

Paul of Venice (1369-1429) provides a consistency enhancer that resolves Russell's Paradox in naive set theory without using a theory of types. It allows a set of all sets. It also blocks the (diagonal) general proof of Cantor's Theorem on the power set. It is not unlikely that the Zermelo-Fraenkel (ZFC) axioms for set theory are still too lax on the notion of a 'well-defined set'. The transfinites of ZFC may be a mirage, and a consequence of still imperfect axiomatics in ZFC w.r.t. the proper foundations for set theory. For amendment of ZFC two alternatives are mentioned: ZFC-PV (amendment of de Axiom of Separation) or BST (Basic Set Theory).

**Category:** Set Theory and Logic

[60] **viXra:1412.0234 [pdf]**
*replaced on 2015-07-28 05:05:15*

**Authors:** Thomas Colignatus

**Comments:** 2 Pages. The paper refers to the book FMNAI that supersedes the paper

> Context • In the philosophy of mathematics there is the distinction between platonism (realism), formalism, and constructivism. There seems to be no distinguishing or decisive experiment to determine which approach is best according to non-trivial and self-evident criteria. As an alternative approach it is suggested here that philosophy finds a sounding board in the didactics of mathematics rather than mathematics itself. Philosophers can go astray when they don’t realise the distinction between mathematics (possibly pure modeling) and the didactics of mathematics (an empirical science). The approach also requires that the didactics of mathematics is cleansed of its current errors. Mathematicians are trained for abstract thought but in class they meet with real world students. Traditional mathematicians resolve their cognitive dissonance by relying on tradition. That tradition however is not targetted at didactic clarity and empirical relevance with respect to psychology. The mathematical curriculum is a mess. Mathematical education requires a (constructivist) re-engineering. Better mathematical concepts will also be crucial in other areas, such as e.g. brain research. > Problem • Aristotle distinguished between potential and actual infinite, Cantor proposed the transfinites, and Occam would want to reject those transfinites if they aren’t really necessary. My book “A Logic of Exceptions” already refuted ‘the’ general proof of Cantor's Conjecture on the power set, so that the latter holds only for finite sets but not for ‘any’ set. There still remains Cantor’s diagonal argument on the real numbers. > Results • There is a bijection by abstraction between N and R. Potential and actual infinity are two faces of the same coin. Potential infinity associates with counting, actual infinity with the continuum, but they would be ‘equally large’. The notion of a limit in R cannot be defined independently from the construction of R itself. Occam’s razor eliminates Cantor’s transfinites. > Constructivist content • Constructive steps S1, ..., S5 are identified while S6 gives non-constructivism (possibly the transfinites). Here S3 gives potential infinity and S4 actual infinity. The latter is taken as ‘proper constructivism with abstraction'. The confusions about S6 derive rather from logic than from infinity.

**Category:** Set Theory and Logic

[59] **viXra:1412.0201 [pdf]**
*replaced on 2015-01-27 21:14:44*

**Authors:** Karan Doshi

**Comments:** 11 Pages.

In this paper the author submits a proof using the Power Set relation for the existence of a transfinite cardinal strictly smaller than Aleph Zero, the cardinality of the Naturals. Further, it can be established taking these arguments to their logical conclusion that even smaller transfinite cardinals exist. In addition, as a lemma using these new found and revolutionary concepts, the author conjectures that some outstanding unresolved problems in number theory can be brought to heel. Specifically, a proof of the twin prime conjecture is given.

**Category:** Set Theory and Logic

[58] **viXra:1412.0155 [pdf]**
*replaced on 2015-01-30 13:43:13*

**Authors:** Florentin Smarandache

**Comments:** 480 Pages.

Neutrosophic Theory means Neutrosophy applied in many fields in order to solve problems related to indeterminacy.
Neutrosophy considers every entity <A> together with its opposite or negation <antiA>, and with their spectrum of neutralities <neutA> in between them (i.e. entities supporting neither nor <antiA>). Where

**Category:**

[57] **viXra:1412.0130 [pdf]**
*replaced on 2015-02-27 12:54:44*

**Authors:** Jaykov Foukzon

**Comments:** 8 Pages. DOI: 10.11648/j.pamj.s.2015040101.12

In 1942 Haskell B. Curry presented what is now called Curry's paradox which can be found in a logic independently of its stand on negation. In recent years there has been a revitalised interest in non-classical solutions to the semantic paradoxes. In this article the non-classical resolution of Curry’s Paradox and Shaw-Kwei's paradox without rejection any contraction postulate is proposed.

**Category:** Set Theory and Logic

[56] **viXra:1409.0174 [pdf]**
*replaced on 2016-02-27 04:59:59*

**Authors:** Misha Mikhaylov

**Comments:** 7 Pages.

It was shown here that just transitive relation may be considered as closure and its presence is necessary and sufficient to order a set linearly, and it is not possible to do this by using other relation’s property – neither reflexivity nor symmetry. By interaction, it occurs due to ambiguity of their definition – it was shown earlier (ref. [3]) they have various appearances. Among them just the only transitivity is determined uniquely. At the same time the last one doesn’t exist separately from any others. Circumstances of their joint existence are clarifying in this article.

**Category:** Set Theory and Logic

[55] **viXra:1409.0174 [pdf]**
*replaced on 2016-02-25 11:57:16*

**Authors:** Misha Mikhaylov

**Comments:** 7 Pages.

It was shown here that just transitive relation may be considered as closure and its presence is necessary and sufficient to order a set linearly. It is not impossible to do this by using other relation’s property – neither reflexivity nor symmetry. By interaction, it occurs due to ambiguity of their definition – it was shown earlier (ref. [3]) they have various appearances. Among them just the only transitivity is determined uniquely. At the same time the last one doesn’t exist separately from any others. Circumstances of their joint existence are clarifying in this article.

**Category:** Set Theory and Logic

[54] **viXra:1409.0174 [pdf]**
*replaced on 2014-09-28 13:34:49*

**Authors:** Misha Mikhaylov

**Comments:** 7 Pages.

Thoughts expressed in previous paper [3] were developed. There was shown formally that collection of sets’ properties may not appear chaotically and independently on each other. Presence or absence of one leads to rise or drop of another.

**Category:** Set Theory and Logic

[53] **viXra:1409.0056 [pdf]**
*replaced on 2016-02-23 12:30:09*

**Authors:** Misha Mikhaylov

**Comments:** 11 Pages.

Usually itemizing relations’ properties those of them are always pointed out – some appearance of reflexivity, symmetry and transitivity. Also it is not so clear whether they are introduced artificially – i.e. axiomatically, rather for the sake of convenience or it may be done due to inartificial reasons. At the same time an origin of them is not so clear – whether they appear chaotically and independently on each other or there should be rigorous association between them. It is shown here that request of relation’s reversibility leads to these properties’ presence or absence. Often symmetry appearances are defined by using of ambiguous way. In fact, anti-symmetry is not direct negation for symmetry – there is also something that may be called as asymmetry or it may be something else. To avoid it here there was found the unified method of their definition. The same thing may be told about reflexivity and it was shown that just the only intransitivity may be represented as direct negation of transitivity.

**Category:** Set Theory and Logic

[52] **viXra:1409.0056 [pdf]**
*replaced on 2014-09-14 04:03:45*

**Authors:** Misha Mikhaylov

**Comments:** 11 Pages.

Usually itemizing relations’ properties those of them are always pointed out – some appearance of reflexivity, symmetry and transitivity. Also it is not so clear whether they are introduced artificially – i.e. axiomatically, rather for the sake of convenience or it may be done due to inartificial reasons. At the same time an origin of them is not so clear – whether they appear chaotically and independently on each other or there should be rigorous association between them. It is shown here that request of relation’s reversibility leads to these properties’ presence or absence. Often symmetry appearances are defined by using of ambiguous way. In fact, anti-symmetry is not direct negation for symmetry – there is also something that may be called as asymmetry or it may be something else. To avoid it here there was found the unified method of their definition. The same thing may be told about reflexivity and it was shown that just the only intransitivity may be represented as direct negation of transitivity.

**Category:** Set Theory and Logic

[51] **viXra:1409.0041 [pdf]**
*replaced on 2015-06-01 08:54:57*

**Authors:** Ihsan Raja Muda Nasution

**Comments:** 1 Page.

In this paper, we redefine the infinitesimals by using axiomatic method.

**Category:** Set Theory and Logic

[50] **viXra:1409.0041 [pdf]**
*replaced on 2015-05-31 18:06:55*

**Authors:** Ihsan Raja Muda Nasution

**Comments:** 1 Page.

In this paper, we redefine the infinitesimals by using axiomatic method.

**Category:** Set Theory and Logic

[49] **viXra:1409.0041 [pdf]**
*replaced on 2014-11-06 09:08:11*

**Authors:** Ihsan Raja Muda Nasution

**Comments:** 1 Page.

**Category:** Set Theory and Logic

[48] **viXra:1409.0006 [pdf]**
*replaced on 2014-10-20 00:20:20*

**Authors:** Felix M. Lev

**Comments:** 7 Pages. A figure added

Standard mathematics involves such notions as infinitely small/large, continuity and standard division. This mathematics is usually treated as fundamental while finite mathematics is treated as inferior. Standard mathematics has foundational problems (as follows, for example, from G\"{o}del's incompleteness theorems) but it is usually believed that this is less important than the fact that it describes many experimental data with high accuracy. We argue that the situation is the opposite: standard mathematics is only a degenerate case of finite one in the formal limit when the characteristic of the ring or field used in finite mathematics goes to infinity. Therefore foundational problems in standard mathematics are not fundamental.

**Category:** Set Theory and Logic

[47] **viXra:1408.0211 [pdf]**
*replaced on 2014-08-29 22:58:49*

**Authors:** Ihsan Raja Muda Nasution

**Comments:** 2 Pages.

Our result is the explicit form of the infinitesimals.

**Category:** Set Theory and Logic

[46] **viXra:1405.0360 [pdf]**
*replaced on 2014-11-16 10:51:17*

**Authors:** Ricardo Alvira

**Comments:** 42 Pages.

Presentation based on Chapter 2.4 of the book 'A Unified Complexity Theory'. ISBN: 978-1499335859. 274 pages

**Category:** Set Theory and Logic

[45] **viXra:1405.0360 [pdf]**
*replaced on 2014-07-23 14:52:39*

**Authors:** Ricardo Alvira

**Comments:** 42 Pages.

Presentation based on Chapter 2.4 of the book 'A Unified Complexity Theory'. ISBN: 978-1499335859. 274 pages.

**Category:** Set Theory and Logic

[44] **viXra:1405.0360 [pdf]**
*replaced on 2014-06-30 05:41:35*

**Authors:** Ricardo Alvira

**Comments:** 42 Pages.

Presentation based on Chapter 2.4 of the book 'A Unified Theory of Complexity'. ISBN: 978-1499335859. 270 pages aprox. The book is currently available in spanish as 'Una Teoria Unificada de la Complejidad'

**Category:** Set Theory and Logic

[43] **viXra:1405.0360 [pdf]**
*replaced on 2014-06-23 06:15:21*

**Authors:** Ricardo Alvira

**Comments:** 42 Pages.

Presentation based on Chapter 2.4 of the book 'A Unified Theory of Complexity'. ISBN: 978-1499335859. 270 pages aprox. The book is currently available in spanish as 'Una Teoria Unificada de la Complejidad'

**Category:** Set Theory and Logic

[42] **viXra:1405.0360 [pdf]**
*replaced on 2014-05-30 17:14:51*

**Authors:** Ricardo Alvira

**Comments:** 42 Pages.

Presentation based on Chapter 2.4 of the book 'A Unified Theory of Complexity'. ISBN: 978-1499335859. 250 pages aprox. The book is expected to be published soon.

**Category:** Set Theory and Logic

[41] **viXra:1405.0202 [pdf]**
*replaced on 2014-05-13 09:26:34*

**Authors:** Bertrand Wong

**Comments:** 3 Pages.

This article raises some important points about logic, e.g., mathematical logic.

**Category:** Set Theory and Logic

[40] **viXra:1404.0454 [pdf]**
*replaced on 2014-06-16 09:48:58*

**Authors:** Ricardo Alvira

**Comments:** 14 Pages.

The present text proposes a methodology for designing urban sustainability indicators based on set theory / fuzzy logic. The aim is to provide a sound basis that increases ‘understandability’ and ‘shareability’ of sustainability indicators. While the analysis focuses on urban sustainability indicators, its conclusions can be applied to many different type of indicators, not only referred to sustainability but to many different issues.
The article is the english translation of an article to appear as “Un acercamiento lógico al diseño de indicadores de sostenibilidad urbana” en Urbanismo y Sostenibilidad en La Ciudad. Edited by BREEAM ES and ITG [currently in press]

**Category:** Set Theory and Logic

[39] **viXra:1310.0242 [pdf]**
*replaced on 2015-08-30 00:04:14*

**Authors:** Antonio Leon

**Comments:** 238 Pages.

Selected set theory and supertask arguments on the formal consistency of the actual infinity hypothesis subsumed by the Axiom of Infinity.

**Category:** Set Theory and Logic

[38] **viXra:1310.0075 [pdf]**
*replaced on 2014-04-18 11:11:16*

**Authors:** Daniel Cordero Grau

**Comments:** 3 Pages.

In this paper we prove the continuum hypothesis by categorical logic, proving that the theory of initial ordinals and the theory of cardinals are isomorphic. To prove that the theorems of the theory of cardinals are theorems of the theory of initial ordinals, and conversely, the theorems of the theory of initial ordinals are theorems of the theory of cardinals, and so, since isomorphic structures are isomorphic theories by the fundamental theorem of mathematical logic, cardinals and initial ordinals are isomorphic structures, we use the definition of a theory, the definition of an isomorphism of structures in its equivalent form, the definition of an isomorphism of categories, the definition of a structure, the definition of a formal language, the definition of a functor, the definition of a category, the axioms of mathematical logic and the axioms of the theory of categories, which include the Gödel-Bernays-von Neumann axioms, so as to apply both the theorem on the comparablity of ordinals to the theory of cardinals and the fundamental theorem of cardinal arithmetic to the theory of ordinals.

**Category:** Set Theory and Logic

[37] **viXra:1310.0075 [pdf]**
*replaced on 2014-03-18 20:34:36*

**Authors:** Daniel Cordero Grau

**Comments:** 3 Pages.

In this paper we prove the continuum hypothesis by categorical logic, proving that the theory of initial ordinals and the theory of cardinals are isomorphic. To prove that the theorems of the theory of cardinals are theorems of the theory of initial ordinals, and conversely, the theorems of the theory of initial ordinals are theorems of the theory of cardinals, and so, since isomorphic structures are isomorphic theories by the fundamental theorem of mathematical logic, cardinals and initial ordinals are isomorphic structures, we use the definition of a theory, the definition of an isomorphism of structures in its equivalent form, the definition of an isomorphism of categories, the definition of a structure, the definition of a formal language, the definition of a functor, the definition of a category, the axioms of mathematical logic and the axioms of the theory of categories, which include the Gödel-Bernays-von Neumann axioms, so as to apply both the theorem on the comparablity of ordinals to the theory of cardinals and the fundamental theorem of cardinal arithmetic to the theory of ordinals.

**Category:** Set Theory and Logic

[36] **viXra:1310.0075 [pdf]**
*replaced on 2014-03-18 13:30:15*

**Authors:** Daniel Cordero Grau

**Comments:** 3 Pages.

In this paper we prove the continuum hypothesis by categorical logic, proving that the theory of initial ordinals and the theory of cardinals are isomorphic. To prove that the theorems of the theory of cardinals are theorems of the theory of initial ordinals, and conversely, the theorems of the theory of initial ordinals are theorems of the theory of cardinals, and so, since isomorphic structures are isomorphic theories by the fundamental theorem of mathematical logic, cardinals and initial ordinals are isomorphic structures, we use the definition of a theory, the definition of an isomorphism of structures, in its equivalent form, the definition of an isomorphism of categories, the definition of a structure, the definition of a formal language, the definition of a functor, the definition of a category, the axioms of mathematical logic and the axioms of the theory of categories which include the Gödel-Bernays-von Neumann axioms, so as to apply both the theorem on the comparablity of ordinals to the theory of cardinals and the fundamental theorem of cardinal arithmetic to the theory of ordinals.

**Category:** Set Theory and Logic

[35] **viXra:1310.0075 [pdf]**
*replaced on 2013-12-19 21:43:23*

**Authors:** Daniel Cordero Grau

**Comments:** 3 Pages.

In this paper we prove the continuum hypothesis with categorical mathematical logic by proving that the theory of initial ordinals and the theory of cardinals are isomorphic. We use the fact that isomorphic structures are isomorphic theories known as the fundamental theorem of mathematical logic, the definition of an isomorphism of structures, in its equivalent form, the definition of an isomorphism of categories, the definition of a theory, the definition of a structure, the definition of a formal language, the definition of a functor, the definition of a category, the axioms of mathematical logic, and the axioms of the theory of categories, which include the Gödel-Bernays-von Neumann axioms for classes and sets. And thus, applying both the theorem of comparablity of ordinals to the theory of cardinals, and the fundamental theorem of cardinal arithmetic to the theory of ordinals, we prove the theorem.

**Category:** Set Theory and Logic

[34] **viXra:1310.0075 [pdf]**
*replaced on 2013-12-18 18:06:04*

**Authors:** Daniel Cordero Grau

**Comments:** 3 Pages.

In this paper we prove the continuum hypothesis with categorical mathematical logic, by proving that the theory of initial ordinals and the theory of cardinals are isomorphic. We use the fact that isomorphic structures are isomorphic theories known as the fundamental theorem of mathematical logic, the definition of an isomorphism of structures, in its equivalent form, the definition of an isomorphism of categories, the definition of a functor, the definition of a category, the axioms of mathematical logic, and the axioms of the theory of categories, which include the Gödel-Bernays-von Neumann axioms for classes and sets. And thus, applying both the theorem of comparablity of ordinals to the theory of cardinals, and the fundamental theorem of cardinal arithmetic to the theory of ordinals, we prove the theorem.

**Category:** Set Theory and Logic

[33] **viXra:1310.0075 [pdf]**
*replaced on 2013-12-12 18:31:32*

**Authors:** Daniel Cordero Grau

**Comments:** 3 Pages.

In this paper we prove the continuum hypothesis with categorical mathematical logic, by proving that the theory of initial ordinals and the theory of cardinals are isomorphic. We use the definition of an isomorphism of theories, in its equivalent form, the definition of an isomorphism of categories, the definition of a functor, the definition of a category, the axioms of mathematical logic, and the axioms of the theory of categories, which include the Gödel-Bernays-von Neumann axioms for classes and sets. And so, applying both the theorem of comparablity of ordinals to the theory of cardinals, and the fundamental theorem of cardinal arithmetic to the theory of ordinals, we prove the theorem.

**Category:** Set Theory and Logic

[32] **viXra:1310.0075 [pdf]**
*replaced on 2013-12-10 12:36:03*

**Authors:** Daniel Cordero Grau

**Comments:** 3 Pages.

In this paper we prove the continuum hypothesis with categorical mathematical logic, by proving that the theory of initial ordinals and the theory of cardinals are equivalent. We use the definition of an isomorphism of theories in mathematical logic, in its equivalent form, the definition of an isomorphism of categories from the theory of categories, and also, we use the definition of a functor, the definition of a category, the axioms of mathematical logic, and the axioms of the theory of categories, which include the Gödel-Bernays-von Neumann axioms for classes and sets, and so, applying both the theorem of comparablity of ordinals to the theory of cardinals, and the fundamental theorem of cardinal arithmetic to the theory of ordinals, we prove the theorem.

**Category:** Set Theory and Logic

[31] **viXra:1310.0075 [pdf]**
*replaced on 2013-12-09 19:21:45*

**Authors:** Daniel Cordero Grau

**Comments:** 3 Pages.

In this paper we prove the continuum hypothesis, by proving that the theory of initial ordinals and the theory of cardinals are equivalent. We use the definition of an isomorphism of theories in mathematical logic, in its equivalent form, the definition of an isomorphism of categories from the theory of categories, and also, we use the definition of a functor, the definition of a category, the axioms of mathematical logic, and the axioms of the theory of categories, which include the Gödel-Bernays-von Neumann axioms for classes and sets, and so, applying both the theorem of comparablity of ordinals to the theory of cardinals, and the fundamental theorem of cardinal arithmetic to the theory of ordinals, we prove the theorem.

**Category:** Set Theory and Logic

[30] **viXra:1310.0075 [pdf]**
*replaced on 2013-12-04 11:07:24*

**Authors:** Daniel Cordero Grau

**Comments:** 2 Pages.

In this paper we give a categorical proof of the continuum hypothesis.

**Category:** Set Theory and Logic

[29] **viXra:1310.0006 [pdf]**
*replaced on 2016-03-12 20:09:52*

**Authors:** Daniel Cordero Grau

**Comments:** 2 Pages.

In this paper we prove the Space Time Computational Intractability, the Fundamental Theorem of Computational Complexity, proving the Quantum Computation Intractability

**Category:** Set Theory and Logic

[28] **viXra:1309.0013 [pdf]**
*replaced on 2015-11-30 06:58:58*

**Authors:** Robert A. Herrmann

**Comments:** 19 Pages.

Relative to universal logic, it is demonstrated how useful it is to utilize general logic-systems to investigate finite consequences operators (operations). Among many other examples relative to the lattice of finite consequence operators, a general characterization for the lattice-theoretic supremum for a nonempty collection of finite consequence operators is given. Further, it is shown that for any denumerable language there is a rather simple collection of finite consequence operators and for a propositional language, three simple modifications to the finitary rule of inference that demonstrate that the lattice of finite consequence operators is not meet-complete.

**Category:** Set Theory and Logic

[27] **viXra:1302.0048 [pdf]**
*replaced on 2015-06-02 18:13:52*

**Authors:** Jaykov Foukzon

**Comments:** 15 Pages. British Journal of Mathematics & Computer Science DOI: 10.9734/BJMCS/2015/16849

In this article we derived an importent example of the inconsistent countable set in second order ZFC (ZFC_2)with the full second-order semantics. Main results is:(i) ~Con(ZFC_2),(ii) let k be an inaccessible cardinal and H_k is a set of all sets having hereditary size less then k, then ~Con(ZFC+V=H_k).

**Category:** Set Theory and Logic

[26] **viXra:1302.0048 [pdf]**
*replaced on 2015-04-15 12:21:21*

**Authors:** Jaykov Foukzon

**Comments:** 18 Pages.

In this article we derived an importent example of the inconsistent countable set in second order ZFC (ZFC_2) with the full second-order semantic.
Main results is:(i) ~Con(ZFC_2),(ii) let k be an inaccessible cardinal and H_k is a
set of all sets having hereditary size less then k, then ~Con(ZFC+V=H_k).

**Category:** Set Theory and Logic

[25] **viXra:1302.0048 [pdf]**
*replaced on 2015-04-14 15:45:13*

**Authors:** Jaykov Foukzon

**Comments:** 18 Pages.

In this article we derived an importent example of the inconsistent countable set in second order ZFC (ZFC_2) with the full second-order semantic.
Main results is:(i) ~Con(ZFC_2),(ii) let k be an inaccessible cardinal and H_k is a
set of all sets having hereditary size less then k, then ~Con(ZFC+V=H_k).

**Category:** Set Theory and Logic

[24] **viXra:1302.0048 [pdf]**
*replaced on 2015-02-20 23:04:38*

**Authors:** Jaykov Foukzon

**Comments:** 9 Pages.

In this article we derived an importent example of the inconsistent
countable set in second order ZFC (ZFC_2). Main result is: ~con(ZFC_2+∃(omega-model of ZFC_2).

**Category:** Set Theory and Logic

[23] **viXra:1302.0048 [pdf]**
*replaced on 2013-02-20 04:58:47*

**Authors:** Jaykov Foukzon

**Comments:** 15 Pages. Advances in Pure Mathematics.DOI: 10.4236/apm.2013.33053

In this article we derived an importent example of the inconsistent countable set. Main result is: ~con(ZFC+E(\omega-model of ZFC)).

**Category:** Set Theory and Logic

[22] **viXra:1302.0048 [pdf]**
*replaced on 2013-02-16 11:11:07*

**Authors:** Jaykov Foukzon

**Comments:** 15 Pages.

In this article we derived an importent example of the inconsistent countable set. Main result is: ~con(ZFC+E(\omega-model of ZFC)).

**Category:** Set Theory and Logic

[21] **viXra:1212.0088 [pdf]**
*replaced on 2013-06-04 05:22:15*

**Authors:** Qiu Kui Zhang

**Comments:** 10 Pages.

In this article some difficulties are deduced from the set of natural numbers. The demonstrated difficulties suggest that if the set of natural numbers exists it would conflict with the axiom of regularity. As a result, we have the conclusion that the class of natural numbers is not a set but a proper class.

**Category:** Set Theory and Logic

[20] **viXra:1212.0088 [pdf]**
*replaced on 2013-02-17 04:14:16*

**Authors:** Qiu Kui Zhang

**Comments:** 10 Pages.

In this article some difficulties are deduced from the set of natural numbers. The demonstrated difficulties suggest that if the set of natural numbers exists it would conflict with the axiom of regularity. As a result, we have the conclusion that the class of natural numbers is not a set but a proper class.

**Category:** Set Theory and Logic

[19] **viXra:1212.0088 [pdf]**
*replaced on 2012-12-28 03:13:00*

**Authors:** Qiu Kui Zhang

**Comments:** 9 Pages.

**Category:** Set Theory and Logic

[18] **viXra:1212.0088 [pdf]**
*replaced on 2012-12-22 07:01:34*

**Authors:** Qiu Kui Zhang

**Comments:** 9 Pages.

In this article some dificulties are deduced from the set of natural numbers. The demonstrated dificulties suggest that if the set of natural numbers exists it would confict with the axiom of regularity. As a result, we have the conclusion that the class of natural numbers is not a set but a proper class.

**Category:** Set Theory and Logic

[17] **viXra:1209.0103 [pdf]**
*replaced on 2014-06-07 09:17:38*

**Authors:** Fernando Sanchez-Escribano

**Comments:** Spanish original(11 pages) and English translation (11 pages)

Hereby it is presented a new set theory (generalized as the so-called system theory, fully respectful with the dictates of intuition (including the existence and numerability of the universal set, of which all beings are elements) and capable to overcome all the logical difficulties that forced logicians in last century to accept never desired axioms for avoiding contradictions.
The somewhat philosophical nature of the essay, imposed by the need to distinghish between concepts before not sufficiently clear, if not confused, and now denoted with distinct names (some neologisms), such as entema, concept, reflexo, set, system, aente, uente…, should not prevent the appreciation of its mathematical significance.

**Category:** Set Theory and Logic

[16] **viXra:1209.0103 [pdf]**
*replaced on 2014-06-07 09:05:43*

**Authors:** Fernando Sanchez-Escribano

**Comments:** Spanish original (11 pages) and English translation (11 pages)

Hereby it is presented a new set theory (generalized as the so-called system theory, fully respectful with the dictates of intuition (including the existence and numerability of the universal set, of which all beings are elements) and capable to overcome all the logical difficulties that forced logicians in last century to accept never desired axioms for avoiding contradictions.
The somewhat philosophical nature of the essay, imposed by the need to distinghish between concepts before not sufficiently clear, if not confused, and now denoted with distinct names (some neologisms), such as entema, concept, reflexo, set, system, aente, uente…, should not prevent the appreciation of its mathematical significance.

**Category:** Set Theory and Logic

[15] **viXra:1209.0103 [pdf]**
*replaced on 2013-05-29 12:49:35*

**Authors:** Fernando Sánchez-Escribano

**Comments:** Pages.

Hereby it is presented a new set theory (generalized as the so-called system theory, fully respectful with the dictates of intuition (including the existence and numerability of the universal set, of which all beings are elements) and capable to overcome all the logical difficulties that forced logicians in last century to accept never desired axioms for avoiding contradictions.
The somewhat philosophical nature of the essay, imposed by the need to distinghish between concepts before not sufficiently clear, if not confused, and now denoted with distinct names (some neologisms), such as entema, concept, reflexo, set, system, aente, uente…, should not prevent the appreciation of its mathematical significance.

**Category:** Set Theory and Logic

[14] **viXra:1209.0103 [pdf]**
*replaced on 2013-02-09 03:54:15*

**Authors:** Fernando Sánchez-Escribano

**Comments:** 11 pages for Spanish original; 11 pages for English translation.

A new set theory (generalized as the so-called system theory) is presented, that respects all the dictates of intuition (including the existence of a universal set, the one including all beings, and its numerability) and is able not only to overcome all the logical difficulties that forced logicians in last century to accept axioms never desired, in order to avoid contradictions, but also to make obvious the mistakes that caused these.

**Category:** Set Theory and Logic

[13] **viXra:1207.0009 [pdf]**
*replaced on 2012-07-09 00:35:18*

**Authors:** Pierre-Yves Gaillard

**Comments:** 3 Pages.

We unsuccessfully try to give definitions in the spirit of Bourbaki's set theory for the basic notions of category theory. The goal is to avoid using either Grothendieck's universes axiom, or "classes" (or "collections") of sets which are not sets. We explain why our attempt fails.

**Category:** Set Theory and Logic

[12] **viXra:1207.0009 [pdf]**
*replaced on 2012-07-07 01:06:47*

**Authors:** Pierre-Yves Gaillard

**Comments:** 2 Pages.

We give definitions in the spirit of Bourbaki's Set Theory for the basic notions of category theory. The goal is to avoid using either Grothendieck's universes axiom, or ``classes'' (or ``collections'') of sets which are not sets.

**Category:** Set Theory and Logic

[11] **viXra:1204.0012 [pdf]**
*replaced on 2012-04-04 14:58:56*

**Authors:** Wilber Valgusbitkevyt

**Comments:** 5 Pages. Further elaboration on Victoria Hayanisel Theorems were explained.

In this paper, I am creating three new theorems called Victoria Hayanisel Theorem dedicated to Princess Eugenie of York to describe the state of numbers, circles, and lines.
Followed by the theorem, I am using the set theory and Fermat's Infinite Descent Method (if my method is different, I will name it) to show how the conjecture is true.

**Category:** Set Theory and Logic

[10] **viXra:1107.0045 [pdf]**
*replaced on 2013-12-08 11:38:28*

**Authors:** Mauro Avon

**Comments:** 201 Pages.

The paper is about an approach to logic that differs from the standard first-order logic and other known approaches. It should be a new approach the author has created proposing to obtain a general and unifying approach to logic and a faithful model of human mathematical deductive process. We list the most relevant features of the system. In first-order logic there exist two different concepts of term and formula, in place of these two concepts in our approach we have just one notion of expression. The set-builder notation is enclosed as an expression-building pattern. In our system we
can easily express second-order and all-order conditions (the set to which a quantifier refers is explicitly written in the expression). The meaning of a sentence will depend solely on the meaning of the symbols it contains, it will not depend on external `structures'. Our deductive system is based on a very simple definition of proof and provides a good model of human mathematical deductive process. The soundness and consistency of the system are proved, as well as the fact that our system is not affected by the most known types of paradox. The paper provides both the theoretical material and two fully documented examples of deduction. The author believes his aims have been achieved, obviously the reader is free to examine the system and get his own opinion about it.

**Category:** Set Theory and Logic

[9] **viXra:1107.0045 [pdf]**
*replaced on 2011-12-11 13:41:47*

**Authors:** Mauro Avon

**Comments:** 159 Pages.

The paper is about an approach to logic that differs from the standard first-order logic and other known approaches. It should be a new approach the author has created proposing to obtain a general and unifying approach to logic and a faithful model of human mathematical deductive process. We list the most relevant features of the system. In first-order logic there exist two different concepts of term and formula, in place of these two concepts in our approach we have just one notion of expression. The set-builder notation is enclosed as an expression-building pattern. In our system we can easily express second-order and all-order conditions (the set to which a quantifier refers is explicitly written in the expression). The meaning of a sentence will depend solely on the meaning of the symbols it contains, it will not depend on external 'structures'. Our deductive system is based on a very simple definition of proof and provides a good model of human mathematical deductive process. The soundness and consistency of the system are proved, as well as the fact that our system is not affected by the most known types of paradox. The paper provides both the theoretical material and two fully documented examples of deduction. The author believes his aims have been achieved but obviously the reader is free to examine the system and get his own opinion about it.

**Category:** Set Theory and Logic

[8] **viXra:1107.0045 [pdf]**
*replaced on 8 Sep 2011*

**Authors:** Mauro Avon

**Comments:** 159 pages

The paper is about an approach to logic that differs from the standard first-order logic and other
known approaches. It should be a new approach the author has created proposing to obtain a general
and unifying approach to logic and a faithful model of human mathematical deductive process. We
list the most relevant features of the system. In first-order logic there exist two different concepts of
term and formula, in place of these two concepts in our approach we have just one notion of
expression. The set-builder notation is enclosed as an expression-building pattern. In our system we
can easily express second-order and all-order conditions (the set to which a quantifier refers is
explicitly written in the expression). The meaning of a sentence will depend solely on the meaning
of the symbols it contains, it will not depend on external 'structures'. Our deductive system is based
on a very simple definition of proof and provides a good model of human mathematical deductive
process. The soundness and consistency of the system are proved, as well as the fact that our system
is not affected by the most known types of paradox. The paper provides both the theoretical
material and two fully documented examples of deduction. The author believes his aims have been
achieved but obviously the reader is free to examine the system and get his own opinion about it.

**Category:** Set Theory and Logic

[7] **viXra:1107.0045 [pdf]**
*replaced on 12 Aug 2011*

**Authors:** Mauro Avon

**Comments:** 159 pages

This paper outlines an approach to mathematical logic which is different from the standard one. We
list the most relevant features of the system. In first-order logic there exist two different concepts of
term and formula, in place of these two concepts in our approach we have just one notion of
expression. The set-builder notation is enclosed as an expression-building pattern. In our system we
can easily express second-order and all-order conditions (the set to which a quantifier refers is
explicitly written in the expression). The meaning of a sentence will depend solely on the meaning
of the symbols it contains, it will not depend on external 'structures'. Our deductive system is based
on a very simple definition of proof and provides a good model of human mathematical deductive
process. The soundness and consistency of the system are proved, as well as the fact that our system
is not affected by the most known types of paradox. The paper provides both the theoretical
material and two fully documented examples of deduction. The author has built the whole system
with the idea to provide a faithful model of human mathematical deductive process. He believes this
objective has been achieved but obviously the reader is free to examine the system and get his own
opinion about it.

**Category:** Set Theory and Logic

[6] **viXra:1010.0002 [pdf]**
*replaced on 19 Apr 2011*

**Authors:** Dm. Vatolin

**Comments:** 9 pages, in Russian.

In this paper the question is examined that an incompleteness
of advanced enough theories of arithmetic does not
follow from the Gëdel statements.

**Category:** Set Theory and Logic

[5] **viXra:1010.0002 [pdf]**
*replaced on 6 Apr 2011*

**Authors:** Dm. Vatolin

**Comments:** 9 pages, Russian.

In this paper the question is examined that a incompleteness
of enough advanced theories of arithmetics does not follow from the Gëdel statements.

**Category:** Set Theory and Logic

[4] **viXra:1010.0002 [pdf]**
*replaced on 31 Oct 2010*

**Authors:** Dm. Vatolin

**Comments:** 11 pages, Russian.

In this paper the question is examined that a incompleteness
of enough advanced theories of arithmetics does not follow from the Gëdel statements.

**Category:** Set Theory and Logic

[3] **viXra:1005.0059 [pdf]**
*replaced on 22 Nov 2010*

**Authors:** Dm. Vatolin

**Comments:** 14 pages, Russian.

This article formulates three geometrical axioms from which it follows
that the continuum power is greater then any well-ordered set power.

**Category:** Set Theory and Logic

[2] **viXra:1004.0026 [pdf]**
*replaced on 22 Apr 2010*

**Authors:** Florentin Smarandache

**Comments:** 14 pages

These paradoxes are called "neutrosophic" since they are based on indeterminacy (or neutrality,
i.e. neither true nor false), which is the third component in neutrosophic logic. We generalize the
Venn Diagram to a Neutrosophic Diagram, which deals with vague, inexact, ambiguous, illdefined
ideas, statements, notions, entities with unclear borders. We define the neutrosophic truth
table and introduce two neutrosophic operators (neuterization and antonymization operators)
give many classes of neutrosophic paradoxes.

**Category:** Set Theory and Logic

[1] **viXra:1002.0003 [pdf]**
*replaced on 15 Mar 2010*

**Authors:** Willi Penker

**Comments:** 3 pages

To shift assignments between infinite sets is to create a
disturbance within the assignment itself that cannot be
removed. An assignment carrying such a disturbance cannot be
regarded as static.

**Category:** Set Theory and Logic