Filter spaces: towards a unified theory of large cardinal and embedding axioms

Abstract

AbstractWe construct a parametrized framework, at the center of which is a space D and the notion of a fine, normal ultrafilter on that space. The assertion that such a filter exists is a statement of varying power. By adjusting the parameters, this statement can be made equivalent to a wide range of large cardinal axioms. Examples include all standard forms: inaccessibility, measurability, weak, strong and super-compactness, hugeness, etc., but also axioms formed from embedding properties such as ‘almost λ-supercompactness’, μ-measurability, and more ad hoc axioms such as appear in relative consistency results, e.g. “κ is λ-supercompact and λ is measurable”, or “κ is measurable with at least two normal measures”