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| rxiVerse > Set Theory and Logic |
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Previous months:
2026 - 2602(1) - 2607(1) - 2608(1)
Any replacements are listed farther down
[3] rxiVerse:2608.0029 [pdf] submitted on 2026-08-12 01:20:09
Authors: Juan Carlos Caso Alonso
Comments: 138 Pages. English. Contact: recursos dot clja at gmail dot com
This is not a paper, this is a letter, to convince others to create a formal academic document, exactly with the literary details, and style, international math community likes. This is all about how to build a system of relations. And I can provide all mathematical details, but not in academic style. Once that system is created we will define two different cardinal games, technics, based in the system, to compare infinite quantities of elements inside two concrete sets: P(N) and N. Both technics, existing and interacting, have an extraordinary property (8 years of experience, many different experts): experts tends to guarantee, very sure about themself, while they try to deny the weak point of one, that the other one is a perfect practical equivalence of an injectivity r: P(N) -> N. THEY, not me, guaranteed it. But they never have got patience to see the consequences of their elections, seeing the other technic. I give all the details about the system of relations and the two technics. In reallity we are creating a proof with two paths, where both paths drives to the same conclussion: Cantor's Theorem is false. This is just one concrete case, applied with all details, of a more general algorithm, for other sets or alephs. I am not going to explain it inside this document. This letter is too, to convince some people to help me to create proper documents, about the algorithm, and check the rest of my discoveries. For example, until WHICH aleph, until which set, we are able to prove it has a quantity of elements NOT BIGGER than N has, using concrete mathematical relations.
Category: Set Theory and Logic
[2] rxiVerse:2607.0052 [pdf] submitted on 2026-07-26 11:23:38
Authors: Mauro Avon
Comments: 172 Pages.
This paper introduces an alternative framework for formal logic, termed "Absolute Logic", designed to overcome the intrinsic limitations and structural relativity of standard First-Order Logic (FOL). While FOL relies on external Tarskian structures to assign meaning and restricts quantification to a single, predetermined domain, the proposed system establishes an invariant, self-contained semantics where every symbol possesses an intrinsic meaning. By unifying the traditional syntactic dichotomy between terms and formulas into a single concept of ``expression" and evaluating them relative to formalized variable-expression contexts, the system closely mirrors the natural, cumulative nature of human mathematical deduction. Furthermore, we address the critical balance between expressive power and constructive utility by integrating recursion-theoretic constraints, demonstrating how computability theory acts as a necessary bound to preserve foundational validity. The consistency and structural properties of the resulting language are formally established, providing a novel perspective on non-hierarchical, absolute logical systems.
Category: Set Theory and Logic
[1] rxiVerse:2602.0021 [pdf] submitted on 2026-02-06 07:18:50
Authors: Haoyong Long
Comments: 35 Pages.
This paper proposes a new axiomatic system for natural numbers — the Starting Pair System. The system begins with two mutually distinct basic objects (theEmpty Element 0 and the Existence Element 1) and establishes the foundation ofmathematical existence through the philosophical principle that "mutuality bringsexistence." The core feature of the system is the allocation mechanism of "operational existence": only the Existence Element 1 possesses the ability to activelygenerate subsequent numbers, while the Empty Element 0 serves as a static starting point. A key innovation is the definition of "1+0" as the meta-operation ofexistence-conferral; the generation of all natural numbers is the repetition of thismeta-operation. During the generation of numbers, operational laws are simultaneously verified through a "reification" mechanism, making each natural number anembodiment of specific operational relationships. This paper provides a completeaxiomatic formulation of the system, proves its equivalence to Peano’s axioms, anddemonstrates how it naturally gives rise to extended structures such as the fieldof fractions, positional numeral systems, and the imaginary number domain. Thesystem is mathematically compatible with classical arithmetic while offering a philosophical perspective of dynamic generation and relational ontology.
Category: Set Theory and Logic
[1] rxiVerse:2602.0021 [pdf] replaced on 2026-02-06 20:41:00
Authors: Yonghao Long
Comments: 35 Pages.
This paper proposes a new axiomatic system for natural numbers — the Start ing Pair System. The system begins with two mutually distinct basic objects (the Empty Element 0 and the Existence Element 1) and establishes the foundation ofmathematical existence through the philosophical principle that "mutuality brings existence." The core feature of the system is the allocation mechanism of "operational existence": only the Existence Element 1 possesses the ability to actively generate subsequent numbers, while the Empty Element 0 serves as a static starting point. A key innovation is the definition of "1+0" as the meta-operation of existence-conferral; the generation of all natural numbers is the repetition of this meta-operation. During the generation of numbers, operational laws are simultaneously verified through a "reification" mechanism, making each natural number anembodiment of specific operational relationships. This paper provides a completeaxiomatic formulation of the system, proves its equivalence to Peano’s axioms, anddemonstrates how it naturally gives rise to extended structures such as the fieldof fractions, positional numeral systems, and the imaginary number domain. The system is mathematically compatible with classical arithmetic while offering a philosophical perspective of dynamic generation and relational ontology.
Category: Set Theory and Logic