Set Theory and Logic

2107 Submissions

[4] viXra:2107.0143 [pdf] replaced on 2022-10-25 04:09:50

Invalidating Cantor’s Continuum Hypothesis and Solving Hilbert’s #1 Problem II

Authors: Stephane H. Maes
Comments: 5 Pages.

In this short paper, we provide a mathematical proof that in set theory, developed in a mathematical universe following the ZFC axioms, Cantor’s continuum hypothesis does not hold: the cardinality of the continuous set of all reals is ��, and not א1, i.e., there are infinity א1 (and maybe more than one) between ��, the cardinality of the continuum, and the cardinality of the infinite set of naturals, א0.The proof is derived from combinatorics, relying on ZFC solely for the model of Cantor and Gödel defining א0. It provides input to the still unresolved first of Hilbert famous 23 math problems of interest.This paper, resolves the first of the 23 Hilbert problems with invalidation of the continuum hypothesis.
Category: Set Theory and Logic

[3] viXra:2107.0062 [pdf] submitted on 2021-07-12 03:03:37

Modification of Cantor Sets With Potential Infinities

Authors: Anders Lindman
Comments: 3 Pages.

In continuous Euclidean space all lines have an infinite number of points, e.g. a line A = 10 cm has the same number of points as a line B = 5 cm. In this paper a new set theory (MST for modified set theory) is defined so that lines of different lengths always contain different numbers of points. Instead of allowing several actual infinities only one actual infinity is defined. All other sets are either finite or have potential infinite cardinality. This makes the logic of sets more straightforward than with Georg Cantor’s transfinite sets.
Category: Set Theory and Logic

[2] viXra:2107.0046 [pdf] replaced on 2022-08-09 04:45:35

One Fundamental Theorem Concerning Infinite Countable Sets

For A an infinite countable set containing infinitely many distinct natural integersand B an infinite countable set containing infinitely many distinct natural integerssuch that ∀n ∈ A, n ∈ B and ∀m ∈ B, m ∈ A, we demonstrate that it is possiblethat A≠B by exposing infinitely many counter-examples in which, for each counter-example, A and B are respectively two sample spaces of two probability spaces havingdifferent probabilities for similar events. We thus prove that the axiom of extensionality is false for infinite countable sets.
Category: Set Theory and Logic

[1] viXra:2107.0015 [pdf] submitted on 2021-07-03 21:19:46

Set Theoretic Proof ~~X ≠ X

Authors: Salvatore Gerard Micheal
Comments: 2 Pages. [Corrections are made by viXra Admin to comply with the rules of viXra.org]

We prove that ~~X ≠ X where ~ = "not" in a logical/set-theoretic context (ALL mathematics and logic), X represents ANY logical statement equivalent to a set of associated facts (which many times is countably infinite or more), ~X, read "not X", represents the logical / set-theoretic complement of X, which is comprised of the complementary set of associated facts with respect to X. We give a proof by contradiction and the solitary exception to the rule regarding phi, the null/empty set.
Category: Set Theory and Logic