A marriage of category theory and set theory: a finitely axiomatized nonstandard first-order theory implying ZF

TL;DR

This paper introduces a finitely axiomatized nonstandard first-order theory that implies ZF set theory, avoiding infinite axioms and countable models, and also supports category theory as an ontological basis.

Contribution

It formulates a finite set of axioms in a nonstandard language that derive ZF and eliminate countable models, providing a new foundational framework.

Findings

Finite axiomatization of ZF derived from nonstandard axioms

The theory does not have a countable model if it has any model at all

Category theory axioms hold within this universe

Abstract

It is well known that ZFC, despite its usefulness as a foundational theory for mathematics, has two unwanted features: it cannot be written down explicitly due to its infinitely many axioms, and it has a countable model due to the L\"{o}wenheim-Skolem theorem. This paper presents the axioms one has to accept to get rid of these two features. For that matter, some twenty axioms are formulated in a nonstandard first-order language with countable many constants: to this collection of axioms is associated a universe of discourse consisting of a class of objects, each of which is a set, and a class of arrows, each of which is a function. The axioms of ZF are derived from this finite axiom scheme, and it is shown that it does not have a countable model--if it has a model at all, that is. Furthermore, the axioms of category theory are proven to hold: the present universe may therefore serve as an ontological basis for category theory. However, it has not been investigated whether any of the soundness and completeness properties hold for the present theory: the inevitable conclusion is therefore that only further research can establish whether the present results indeed constitute an advancement in the foundations of mathematics.