On consistency and existence in mathematics

TL;DR

This paper explores the relationship between consistency and existence in mathematics, analyzing historical debates and using logical tools to clarify their connection and the difficulty of proving each.

Contribution

It provides a nuanced analysis of the consistency-existence relationship, linking historical perspectives with formal logical results like G"odel's Completeness Theorem.

Findings

Demonstrating consistency is as difficult as possible.

Proving existence from consistency is as straightforward as possible.

Clarifies the historical debate between Frege and Hilbert.

Abstract

This paper engages the question "Does the consistency of a set of axioms entail the existence of a model in which they are satisfied?" within the frame of the Frege-Hilbert controversy. The question is related historically to the formulation, proof, and reception of G\"odel's Completeness Theorem. Tools from mathematical logic are then used to argue that there are precise senses in which Frege was correct to maintain that demonstrating consistency is as difficult as it can be but also in which Hilbert was correct to maintain that demonstrating existence given consistency is as easy as it can be.