[4] **viXra:1604.0392 [pdf]**
*submitted on 2016-04-30 17:18:06*

**Authors:** István Aggott Hönsch

**Comments:** 19 Pages.

A terminological framework is proposed for the mathematical examination and analysis of the Mandelbrot set's correlative ectocopial set. The Apeiropolis and anthropobrot multisets are defined and explained to be the mathematical entities underlying the well-known Buddhabrot visualization.
The definitions are presented as tools conducive to finding novel approaches and generating discoveries that might otherwise be missed via a primarily programmatic approach.
The anthropobrot multisets are introduced as a new, infinite repository of unique pareidolic figures as richly diverse as the Julia sets.

**Category:** Set Theory and Logic

[3] **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

[2] **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

[1] **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