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A new large cardinal and Laver sequences for extendibles

Paul Corazza (1997)

Fundamenta Mathematicae

We define a new large cardinal axiom that fits between A 3 and A 4 in the hierarchy of axioms described in [SRK]. We use this new axiom to obtain a Laver sequence for extendible cardinals, improving the known large cardinal upper bound for the existence of such sequences.

A note on a question of Abe

Douglas Burke (2000)

Fundamenta Mathematicae

Assuming large cardinals, we show that every κ-complete filter can be generically extended to a V-ultrafilter with well-founded ultrapower. We then apply this to answer a question of Abe.

A Note on Indestructibility and Strong Compactness

Arthur W. Apter (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

If κ < λ are such that κ is both supercompact and indestructible under κ-directed closed forcing which is also (κ⁺,∞)-distributive and λ is 2 λ supercompact, then by a result of Apter and Hamkins [J. Symbolic Logic 67 (2002)], δ < κ | δ is δ⁺ strongly compact yet δ is not δ⁺ supercompact must be unbounded in κ. We show that the large cardinal hypothesis on λ is necessary by constructing a model containing a supercompact cardinal κ in which no cardinal δ > κ is 2 δ = δ supercompact, κ’s supercompactness...

A note on strong compactness and resurrectibility

Arthur Apter (2000)

Fundamenta Mathematicae

We construct a model containing a proper class of strongly compact cardinals in which no strongly compact cardinal ĸ is ĸ + supercompact and in which every strongly compact cardinal has its strong compactness resurrectible.

Adding a lot of Cohen reals by adding a few. II

Moti Gitik, Mohammad Golshani (2015)

Fundamenta Mathematicae

We study pairs (V, V₁), V ⊆ V₁, of models of ZFC such that adding κ-many Cohen reals over V₁ adds λ-many Cohen reals over V for some λ > κ.

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