Q-points, selective ultrafilters, and idempotents, with an application to choiceless set theory
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| Format: | Preprint |
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2024
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| author | Fernández-Bretón, David Navarro-Castillo, Jareb Soria-Rojas, Jesús A. |
| author_facet | Fernández-Bretón, David Navarro-Castillo, Jareb Soria-Rojas, Jesús A. |
| contents | We study ultrafilters from the perspective of the algebra in the Čech-Stone compactification of the natural numbers, and idempotent elements therein. The first two results that we prove establish that, if $p$ is a Q-point (resp. a selective ultrafilter) and $\mathscr F^p$ (resp. $\mathscr G^p$) is the smallest family containing $p$ and closed under iterated sums (resp. closed under Blass--Frol\'ık sums and Rudin--Keisler images), then $\mathscr F^p$ (resp. $\mathscr G^p$) contains no idempotent elements. The second of these results about a selective ultrafilter has the following interesting consequence: assuming a conjecture of Blass, in models of the form $\mathbf{L}(\mathbb R)[p]$ where $\mathbf{L}(\mathbb R)$ is a Solovay model (of $\mathsf{ZF}$ without choice) and $p$ is a selective ultrafilter, there are no idempotent elements. In particular, the theory $\mathsf{ZF}$ plus the existence of a nonprincipal ultrafilter on $ω$ does not imply the existence of idempotent ultrafilters, which answers a question of DiNasso and Tachtsis (Proc. Amer. Math. Soc. 146, 397-411). Following the line of obtaining independence results in $\mathsf{ZF}$, we finish the paper by proving that $\mathsf{ZF}$ plus "every additive filter can be extended to an idempotent ultrafilter" does not imply the Ultrafilter Theorem over $\mathbb R$, answering another question of DiNasso and Tachtsis from the same paper. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_13499 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Q-points, selective ultrafilters, and idempotents, with an application to choiceless set theory Fernández-Bretón, David Navarro-Castillo, Jareb Soria-Rojas, Jesús A. Logic General Topology Primary 03E25, 54D35, Secondary 03E30, 22A15, 20M10 We study ultrafilters from the perspective of the algebra in the Čech-Stone compactification of the natural numbers, and idempotent elements therein. The first two results that we prove establish that, if $p$ is a Q-point (resp. a selective ultrafilter) and $\mathscr F^p$ (resp. $\mathscr G^p$) is the smallest family containing $p$ and closed under iterated sums (resp. closed under Blass--Frol\'ık sums and Rudin--Keisler images), then $\mathscr F^p$ (resp. $\mathscr G^p$) contains no idempotent elements. The second of these results about a selective ultrafilter has the following interesting consequence: assuming a conjecture of Blass, in models of the form $\mathbf{L}(\mathbb R)[p]$ where $\mathbf{L}(\mathbb R)$ is a Solovay model (of $\mathsf{ZF}$ without choice) and $p$ is a selective ultrafilter, there are no idempotent elements. In particular, the theory $\mathsf{ZF}$ plus the existence of a nonprincipal ultrafilter on $ω$ does not imply the existence of idempotent ultrafilters, which answers a question of DiNasso and Tachtsis (Proc. Amer. Math. Soc. 146, 397-411). Following the line of obtaining independence results in $\mathsf{ZF}$, we finish the paper by proving that $\mathsf{ZF}$ plus "every additive filter can be extended to an idempotent ultrafilter" does not imply the Ultrafilter Theorem over $\mathbb R$, answering another question of DiNasso and Tachtsis from the same paper. |
| title | Q-points, selective ultrafilters, and idempotents, with an application to choiceless set theory |
| topic | Logic General Topology Primary 03E25, 54D35, Secondary 03E30, 22A15, 20M10 |
| url | https://arxiv.org/abs/2412.13499 |