A Finitely Axiomatized Non-Classical First-Order Theory Incorporating Category Theory and Axiomatic Set Theory

Marcoen J T F Cabbolet

doi:10.3390/axioms10020119

Abstract

It is well known that Zermelo-Fraenkel Set Theory (ZF), 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 Löwenheim–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 non-classical first-order language with countably 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 schema, 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.

Highlights

Zermelo–Fraenkel set theory, which emerged from the axiomatic set theory proposed by Zermelo in [1] by implementing improvements suggested independently by Skolem and Fraenkel, is without a doubt the most widely accepted foundational theory for mathematics
The purpose of this paper is to present the axioms one has to accept such that the axioms of Zermelo-Fraenkel Set Theory (ZF) can be derived from the new axioms—we omit a discussion of axiom of choice (AC)—but such
Proceeding, we prove that the infinite axiom schema REP of ZF is a theorem of our theory T: Theorem 4

Summary

Introduction

Let us, in accordance with common convention, use ZFC to denote the full theory, ZF for the full theory minus the axiom of choice (AC), and let us use ZF(C) in statements that are to hold for both ZF and ZFC; in a sentence, the two pathological features are the following: with regard to jurisdictional claims in (i) and Alexander Šostak

Objectives

Discussion

Conclusion