Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2110.01489

A foundation for deductive mathematics

Frank Quinn
View original source ↗Read source document ↗Export metadata ↓

Abstract

Set theory is widely believed to provide a secure foundation for deductive mathematics, but current set theories do not quite do this. The mainstream essentially uses na\"\i ve set theory. After Russell's paradox showed this to be inconsistent, the patch ``don't say `set of all sets' '' was added. The resulting methodology has been extremely successful, but still lacks a consistent foundation. The set theory community extracted properties of na\"\i ve set theory to use as axioms, culminating in the Zermillo-Fraenkel-Choice (ZFC) axioms. Unfortunately they missed an axiom, and ZFC as it stands is not consistent with standard methodology. This paper addresses these issues. The first dozen pages (Sections 1--5) gives primitives, defines sets in this context, and verifies that these have the properties used in standard practice. Sections 6--7 relates this to traditional axiomatic set theory. We show the sets here correspond to the sets in a maximal model for the ZFC axioms. Section 8 gives the ``coherent limit axiom'', considered obviously true in mainstream practice, and shows it holds in the maximal model and fails in all others. There are several qualitative conclusions. First, standard mainstream practice implicitly takes place in the set theory described here. This also shows there are no ``hidden axioms'': we already have the full toolkit. Second, most of the axiomatic set theory of the last hundred years is irrelevant to standard mathematical practice. The ZFC models produced by forcing, for example, are essentially never maximal, and therefore do not constrain or inform standard practice.

Explore related subjects

math.LOmath.CTmath.HO

Keep this discovery

BibTeXRIS

Frank Quinn. 2021-10-04. A foundation for deductive mathematics. https://arxiv.org/abs/2110.01489

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

There is no maximal $K$-degree

The Kolmogorov complexity of a string characterize how complex it is to describe the string. If every prefix of a real $x$ is more complex to describe than every prefix (of the same length) of real $y$, then it is seen as $x$ is more complex to describe than $y$. It is wondered if there is a real $x$ so that no other reals are strictly more complex (to describe) than $x$. The behavior of Kolmogorov complexity functions generated by reals (namely $n\mapsto$ the minimal description length of the real) is quite chaos. Therefore, it is widely believed that there are many reals that are maximally complex to describe. For instance, it is conjectured that all random enough reals have maximal $K$-degree. In this paper, it is shown that there is no real with maximal $K$-degree. Actually, for almost all real $x$, we can uniformly computably find another real whose $K$-degree is strictly above $x$.

math.LO↗
arXiv2026

Quadruples and cubes

We prove, in $\mathsf{ZFC}$, that the $\lambda$-terraced cube relation fails whenever $\lambda$ is an uncountable cardinal. The corresponding terraced relation for quadruples fails for every $\lambda$. If $\lambda$ is $\aleph_0$ then the pretinent terraced relation has consistency strength of at least one Woodin cardinal. We prove positive polarized relations at a successor and a double successor from wondrous ideals. We show, however, that there are no such ideals over two consecutive cardinals simultaneously.

math.LO↗
arXiv2026

Possibilistic Logic over a Logic of Formal Inconsistency

In this article, we have introduced a new possibilistic logic on a logic of formal inconsistency with the aim of developing a possibility theoretic framework to deal with uncertainty and inconsistency meaningfully without leading to a system collapse. We have discussed the syntax and semantics for this logic and have proved the soundness and completeness theorems. A set of new measures of consistency, contradictoriness, and triviality of a set of formulas have been defined. These have then been put to use in an example to show that this framework can provide better means of machine reasoning.

math.LO↗