On countability and representations
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866911428006379520 |
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| author | Sanders, Sam |
| author_facet | Sanders, Sam |
| contents | 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, Sierpiński, and König, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_06728 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On countability and representations Sanders, Sam Logic 03B30, 03F35 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, Sierpiński, and König, 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. |
| title | On countability and representations |
| topic | Logic 03B30, 03F35 |
| url | https://arxiv.org/abs/2602.06728 |