On countability and representations

TL;DR

This paper explores the foundational aspects of countability and representations, revealing how the definition depends on equivalence relations and connecting classical principles to higher-order Reverse Mathematics.

Contribution

It establishes the dependence of countability on equivalence relations and links classical principles to strong axioms in higher-order Reverse Mathematics.

Findings

Countability definition hinges on the associated equivalence relation.

Classical principles are equivalent to strong axioms like countable choice.

Results connect basic set principles to advanced axioms in higher-order Reverse Mathematics.

Abstract

The topic of this paper is the subtle interplay between countability and representations. In particular, we establish that the definition of countability of a certain set $X$ crucially hinges on the associated equivalence relation $=_{X}$. Armed with this knowledge, we study well-known and basic principles about countable sets, going back to Cantor, Sierpi\'nski, and K\"{o}nig, working in Kohlenbach's higher-order Reverse Mathematics. While these principles are relatively weak in second-order Reverse Mathematics, we obtain equivalences involving countable choice and Feferman's projection principle. The latter are essentially the strongest axioms studied in higher-order Reverse Mathematics and usually only come to the fore when dealing with the uncountable.