Variants of Baumgartner's Axiom for Lipschitz Functions on Baire and Cantor Space
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916998379732992 |
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| author | Switzer, Corey Bacal |
| author_facet | Switzer, Corey Bacal |
| contents | We consider several variants of Baumgartner's axiom for $\aleph_1$-dense sets defined on the Baire and Cantor spaces in terms of Lipschitz functions with respect to the usual metric. A variation of Baumgartner's original argument shows that these variants are consistent. However, unlike in the case of the classical $\mathsf{BA}$, we are able to give many applications for which the corresponding fact for linear orders is open. In particular we show that there are provable implications from the $ω^ω$ variants to the $2^ω$ variants and that some of these principles imply all the cardinals in the Cichoń's diagram are large. We also show, similar to (but not the same as) $\mathsf{BA}$, that none of the Lipschitz variants follow from a large fragment of $\mathsf{MA}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_07917 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Variants of Baumgartner's Axiom for Lipschitz Functions on Baire and Cantor Space Switzer, Corey Bacal Logic 03E17, 03E35, 03E50 We consider several variants of Baumgartner's axiom for $\aleph_1$-dense sets defined on the Baire and Cantor spaces in terms of Lipschitz functions with respect to the usual metric. A variation of Baumgartner's original argument shows that these variants are consistent. However, unlike in the case of the classical $\mathsf{BA}$, we are able to give many applications for which the corresponding fact for linear orders is open. In particular we show that there are provable implications from the $ω^ω$ variants to the $2^ω$ variants and that some of these principles imply all the cardinals in the Cichoń's diagram are large. We also show, similar to (but not the same as) $\mathsf{BA}$, that none of the Lipschitz variants follow from a large fragment of $\mathsf{MA}$. |
| title | Variants of Baumgartner's Axiom for Lipschitz Functions on Baire and Cantor Space |
| topic | Logic 03E17, 03E35, 03E50 |
| url | https://arxiv.org/abs/2510.07917 |