Set Theory and Logic

2210 Submissions

[2] viXra:2210.0144 [pdf] submitted on 2022-10-29 00:36:29

Rebuilding Set Theory

Authors: Hongyi Li
Comments: 12 Pages.

This paper gives a new definition of set and introduces the concept of elastic set with variable extension. Without any unprovable or unreliable proposition, hypothesis or conjecture, it eliminates Russell's paradox, Cantor's paradox, Galileo's paradox and Infinite Hotel's paradox. Reveals the essence of infinityand gives the expression of the number of elements of infinite set, Corrects the error that an infinite set can correspond to its proper subset one by one. At the same time, the hypothesis that there is an unique natural numbers set that has included all natural numbers has been refuted. It is pointed that the diagonal Argument and Cantor theorem have not proved that real numbers can not be one-to-one corresponded to natural numbers. Since the set is variable, we can also use the variable things such as variables or functions as the elements of the set.There are so many mistakes in the most strict mathematics, let alone the humanities? It shows that the wisdom of modern Westerners, who lack the gene of cultural diversity, is far less than that of the ancient Greek era, or even degenerated to the barbaric era. No wonder a good world would fight for some differences that may not be meaningful, and even foolishly walked to the edge of nuclear war!
Category: Set Theory and Logic

[1] viXra:2210.0095 [pdf] submitted on 2022-10-21 01:42:43

One First Crucial Theorem for the Re-foundation of Elementary Set Theory and the Teaching of That Discipline to Future Generations

Authors: Nhat-Anh Phan
Comments: 14 Pages. Copyright. All rights reserved.

For a given infinite countable set A = U_{i∈N∗}{a_i }, ∀i ∈ N∗ , a_i≠∅, we demonstrate that it is legitimate to partition A in a finite or infinite number of infinite countable sub-sets when the initial order of indexation of the elements of A maintains a strictly increasing order in each sub-set. At this occasion we introduce two new formalism allowing to signify the fact that it is A that has been partitioned and the fact that A and the latter partition of A belong to the same class comprising infinitely many infinite countable sets constituted by the same infinite countable sub-sets that constitute the partition of A.
Category: Set Theory and Logic