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| viXra.org > Set Theory and Logic > 2303 |
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[5] viXra:2303.0168 [pdf] submitted on 2023-03-30 02:54:04
Authors: Jincheng Zhang
Comments: 9 Pages.
Nature of a proposition constructed by diagonal method of proof is a paradox, so it is an unclosed term and an extra-field proposition. There are two kinds of infinities, standard infinity and non-standard infinity, and we will explore the diagonal problem in each of the two kinds of infinities below.We conclude that: (1) In standard infinity, Cantor's diagonal number can metamorphose into real number and the contradiction vanishes. (2) In nonstandard infinity, Cantor's diagonal numbers become hyperreal number . Essentially both are unclosed terms of the calculation.Therefore, Cantor's diagonal method proves that "the real numbers are not countable" is wrong.
Category: Set Theory and Logic
[4] viXra:2303.0160 [pdf] submitted on 2023-03-29 02:14:49
Authors: Albert Henrik Preiser
Comments: 12 Pages.
A natural set never contains itself. The existence or non-existence of a natural set is decided using a modified "Axiom of Comprehension". This modified axiom keeps everything that contains itself out of set formation. This also includes the property of not containing itself, although every natural set has this property. If it were possible to form a set with this property, then the antinomy "contains itself" and "does not contain itself", named after Bertrand Russell, would apply to this set. While this is paradoxical, it is still a corollary when trying to form a set with the property "does not contain itself". The modified "Axiom of Comprehension" takes this fact into account and decides on the non-existence of a set to be formed with the property "does not contain itself" because it would have the property "contains itself". There can therefore be no antinomy like that of Bertrand Russell in the case of natural sets. The modified "Axiom of Comprehension" states that a natural set does not contain itself. This means that the set to be formed cannot have the property or condition required for the set to be formed. This, together with the elimination of Russell's antinomy, provides an existence criterion for natural sets. With this, essential statements about the natural sets can be proved.
Category: Set Theory and Logic
[3] viXra:2303.0105 [pdf] replaced on 2023-03-25 16:53:00
Authors: Jim Rock
Comments: 1 Page. Contains a reply to a common objection made about 2303.0105 v1.
Two contradictory arguments are developed from a hierarchy of sets in [0, 1]. One argument is a proof by contradiction and its conclusion is true. The other argument is an existence argument and while its conclusion is not true, it follows logically from the a valid assumption followed by three true statements that precede the conclusion.
Category: Set Theory and Logic
[2] viXra:2303.0051 [pdf] submitted on 2023-03-08 02:04:23
Authors: Theophilus Agama
Comments: 6 Pages.
In this paper, we study the topology of problems and their solution spaces developed introduced in our first paper. We introduce and study the notion of separability and quotient problem and solution spaces. This notions will form a basic underpinning for further studies on this topic.
Category: Set Theory and Logic
[1] viXra:2303.0008 [pdf] submitted on 2023-03-01 12:42:05
Authors: Jincheng Zhang
Comments: 15 Pages.
What is infinity? What principles should be adhered to when researching infinity? For a predicate on a natural number system, is it true that finite holds, and infinite holds? This paper reexamines the nature of infinity and proposes two opposite infinite axioms (δ+1=δ or δ+1≠δ). Based on these two infinite axioms, the "infinite induction" of the identity formula is proved; it is found that the infinite axioms in the ZF system do not satisfy the equality axiom, and there are many contradictions in the reasoning of the Cantor ordinal number. The ordinal theory of set theory ZF system is not strict. It is hoped that the mathematics community will pay attention to these questions and give a convincing answer.
Category: Set Theory and Logic
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