On countability and representations

Sam Sanders

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arXiv · 2602.06728

On countability and representations

Sam Sanders

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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.

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Sam Sanders. 2026-02-06. On countability and representations. https://arxiv.org/abs/2602.06728

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