Set Theory and Logic

Presentation of the Unlimited Transference of Pairs Method: an Alternative to Bijections

Authors: Juan Carlos Caso Alonso, Francisco Mario Cruz Almeida
Comments: 61 Pages. In Spanish - I can be contacted at recursos.clja@gmail.com

First of all, each point of this project, has been checked as OK, unofficially, for at least, two different persons that said they were mathematicians. The problem is that they are not the same group of persons, and each one believes the mistake is in a different point, while others consider that concrete point, as totally correct. Each comment, each guessing you can made, probably have been taken into account. The unique problem of this project is its crazy goal and the incapacity to put in the same room all the people I have talked sometime.We are going to present an alternative technic to compare infinite cardinalities. Different from bijections or injections, and not totally equivalent in potential, able to work when the previous ones can't. Applied to an obvious case we are going to create. After that, show the equivalents points between the obvious example and the real example we really want to study: P(N) vs N. To show how P(N) has not a cardinality bigger than N.I would like to remember now the first paragraph.We can apply the technic to different sets, with different natures and transfinite cardinalities. Always compared with N.The injectivity concept says to us that is impossible to create a relation r: A -> B, where |B| < |A|, and r being injective. In every possible relation, we must have a minimum quantity of pairs of elements from AXA, with the same image. At least one pair. The core of the idea of the TPI technic (Unlimited Transference of Pairs) is to create a mathematically correct process that shows that minimum is not bigger than 0. Considering the case of 0 pairs with same image, the case of a perfect injection. Remember: It will not be the same as an injective relation.The technic must be considered as valid. And after that, other documents will be recovered, each one per each equivalence, between the technic I will show in this document, and the real case we want to study really. Each equivalence must be considered as valid too. Finally, we will apply the same conclussion we obtained in the obvious example, to the real case.
Category: Set Theory and Logic

A Formal Grammar of Propositional Calculus

Authors: Elmar Guseinov
Comments: 7 Pages.

Практическое значение синтаксического определения теоремы как элемента множества, порождаемого дедуктивной системой, заключается в возможности доказательства любой теоремы компьютером с неограниченным временем работы. Одним из способов непосредственной реализации данной идеи является построение формальной грамматики дедуктивной системы. Так, при автоматическом доказательстве теорем методами машинного обучения в качестве обучающей выборки мы могли бы указать множество пар вида (X,Y), где X — теорема, а Y — порождающая X последовательность правил вывода формальной грамматики. В данной статье мы ограничимся построением грамматики исчисления высказываний. Полученная грамматика типа 0 иерархии Хомского порождает язык, словами которого являются все тавтологии.

Defined syntactically, a theorem is a word generated by a deductive system. In practice, this means that each theorem can be proven by a sufficiently long working computer. A possible implementation of this idea is based on the construction of a formal grammar of the deductive system. In automated theorem proving based on the machine learning, we then can use pairs (X,Y) as the training data, where X stands for a theorem and Y is a sequence of inference rules of the formal grammar corresponding to X. This article provides the formal grammar of propositional calculus, namely, type 0 grammar producing all tautologies.
Category: Set Theory and Logic

A Formal Proof of De Morgan's Laws for Boolean Algebras

Authors: Elmar Guseinov
Comments: 5 Pages.

В статье приведено формальное доказательство законов де Моргана для булевых алгебр.

The article provides a formal proof of de Morgan’s laws for Boolean algebras.
Category: Set Theory and Logic

Correct Setting of Mapping Function in One-to-One Correspondence

Authors: Hongyi Li
Comments: 3 Pages.

In order to strictly discuss the one-to-one correspondence between the elements of sets, the number of elements that can participate in the correspondence was first discussed, and then the necessary conditions for the formation of injection and bijection were discussed. According to these conditions, it wss found that some mapping functions do not satisfy these conditions. For example, it is impossible to obtain a mapping function satisfying these conditions between any infinite set and its any proper subset.
Category: Set Theory and Logic

Correct Definition of the Set of Natural Numbers

Authors: Hongyi Li
Comments: 2 Pages.

It was proved that a set that already contains all natural numbers does not exist in real mathematical world because the concept of all natural numbers does not exist in the world. The correct definition of the set of natural numbers is a set that contains infinite natural numbers 1,2,3.... but can never contain all natural numbers. There are many sets of infinite natural numbers with different sizes. Thus, it is true that the subsets of N cannot correspond one-to-one with N, but it is false that the subsets cannot correspond one-to-one with any set of natural numbers.
Category: Set Theory and Logic

A Method to Check the Reliability of One-to-One Correspondence

Authors: Hongyi Li
Comments: 2 Pages.

The method of checking whether the established one-to-one correspondence is correct was given. It is proved by the verification that any infinite set cannot be in one-to-one correspondence with any of its proper subsets.Key words: mathematic fundation; infinite set; proper subset; one-to-one correspondence; check method
Category: Set Theory and Logic