Higher Cardinals are only a Convention

TL;DR

The paper argues that the concept of 'definite' in Zermelo's Axiom of Separation is a convention, and its interpretation affects the status of Cantor's theorem, suggesting it should be explicitly axiomatized.

Contribution

It clarifies the meaning of 'definite' in the axiom and highlights its role as a convention that impacts foundational theorems like Cantor's.

Findings

'definite' is a convention, not a trivial property

The interpretation of 'definite' affects the status of Cantor's theorem

Explicit axiomatization of 'definite' is necessary for clarity

Abstract

Zermelo's Axiom of Separation is: Exist x: Forall y: (y in x <==> y in a & E(y)) with definite(E) and parameter a. Thoralf Skolem suggested to characterize the terminus "definite" by "the property E should be representable by a FOL formula". But that is trivial. "definite" must mean more. The author claims that "definite" means "in accordance with the theory of definitions of logic". In this case the theorem of Cantor is no longer a theorem, but a undecidable sentence, and has to be established explicitly as axiom. This is not done by the community, but it is made a silent assumption that we can drop the appendix "definite(E)" from the axiom of separation at all. But this is a convention (even when it is silent) and it is nothing else than an axiom.