Variants of Baumgartner's Axiom for Lipschitz Functions on Baire and Cantor Space

2025/10/09 by Switzer, Corey Bacal
#03E17 #03E35 #03E50 #FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.2510.07917

Abstract

We consider several variants of Baumgartner's axiom for ℵ₁-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 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) BA, that none of the Lipschitz variants follow from a large fragment of MA.

Citations

• Weak Baumgartner axioms and universal spaces
• The Special Tree Number
• Continuous images of sets of reals

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• Weak Baumgartner axioms and universal spaces
• Baumgartner's Axiom and Small Posets
• Some stability properties for the Bishop--Phelps--Bollobás property for Lipschitz maps
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