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Measurable cardinals and category bases

Andrzej Szymański (1991)

Commentationes Mathematicae Universitatis Carolinae

We show that the existence of a non-trivial category base on a set of regular cardinality with each subset being Baire is equiconsistent to the existence of a measurable cardinal.

Measurable cardinals and fundamental groups of compact spaces

Adam Przeździecki (2006)

Fundamenta Mathematicae

We prove that all groups can be realized as fundamental groups of compact spaces if and only if no measurable cardinals exist. If the cardinality of a group G is nonmeasurable then the compact space K such that G = π₁K may be chosen so that it is path connected.

Measurable cardinals and the cofinality of the symmetric group

Sy-David Friedman, Lyubomyr Zdomskyy (2010)

Fundamenta Mathematicae

Assuming the existence of a P₂κ-hypermeasurable cardinal, we construct a model of Set Theory with a measurable cardinal κ such that 2 κ = κ and the group Sym(κ) of all permutations of κ cannot be written as the union of a chain of proper subgroups of length < κ⁺⁺. The proof involves iteration of a suitably defined uncountable version of the Miller forcing poset as well as the “tuning fork” argument introduced by the first author and K. Thompson [J. Symbolic Logic 73 (2008)].

Mixed Levels of Indestructibility

Arthur W. Apter (2015)

Bulletin of the Polish Academy of Sciences. Mathematics

Starting from a supercompact cardinal κ, we force and construct a model in which κ is both the least strongly compact and least supercompact cardinal and κ exhibits mixed levels of indestructibility. Specifically, κ 's strong compactness, but not its supercompactness, is indestructible under any κ -directed closed forcing which also adds a Cohen subset of κ. On the other hand, in this model, κ 's supercompactness is indestructible under any κ -directed closed forcing which does not add a Cohen subset...

More Easton theorems for level by level equivalence

Arthur W. Apter (2012)

Colloquium Mathematicae

We establish two new Easton theorems for the least supercompact cardinal that are consistent with the level by level equivalence between strong compactness and supercompactness. These theorems generalize Theorem 1 in our earlier paper [Math. Logic Quart. 51 (2005)]. In both our ground model and the model witnessing the conclusions of our present theorems, there are no restrictions on the structure of the class of supercompact cardinals.

More set-theory around the weak Freese–Nation property

Sakaé Fuchino, Lajos Soukup (1997)

Fundamenta Mathematicae

We introduce a very weak version of the square principle which may hold even under failure of the generalized continuum hypothesis. Under this weak square principle, we give a new characterization (Theorem 10) of partial orderings with κ-Freese-Nation property (see below for the definition). The characterization is not a ZFC theorem: assuming Chang’s Conjecture for ω , we can find a counter-example to the characterization (Theorem 12). We then show that, in the model obtained by adding Cohen reals,...

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