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Sandwiching the Consistency Strength of Two Global Choiceless Cardinal Patterns

Arthur W. Apter (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

We provide upper and lower bounds in consistency strength for the theories “ZF + ¬ A C ω + All successor cardinals except successors of uncountable limit cardinals are regular + Every uncountable limit cardinal is singular + The successor of every uncountable limit cardinal is singular of cofinality ω” and “ZF + ¬ A C ω + All successor cardinals except successors of uncountable limit cardinals are regular + Every uncountable limit cardinal is singular + The successor of every uncountable limit cardinal is singular...

Semiproper ideals

Hiroshi Sakai (2005)

Fundamenta Mathematicae

We say that an ideal I on κ λ is semiproper if the corresponding poset ℙ I is semiproper. In this paper we investigate properties of semiproper ideals on κ λ .

Singular cardinals and strong extenders

Arthur Apter, James Cummings, Joel Hamkins (2013)

Open Mathematics

We investigate the circumstances under which there exist a singular cardinal µ and a short (κ,µ)-extender E witnessing “κ is µ-strong”, such that µ is singular in Ult(V, E).

Singular Failures of GCH and Level by Level Equivalence

Arthur W. Apter (2014)

Bulletin of the Polish Academy of Sciences. Mathematics

We construct a model for the level by level equivalence between strong compactness and supercompactness in which below the least supercompact cardinal κ, there is an unbounded set of singular cardinals which witness the only failures of GCH in the universe. In this model, the structure of the class of supercompact cardinals can be arbitrary.

Some applications of Sargsyan's equiconsistency method

Arthur W. Apter (2012)

Fundamenta Mathematicae

We apply techniques due to Sargsyan to reduce the consistency strength of the assumptions used to establish an indestructibility theorem for supercompactness. We then show how these and additional techniques due to Sargsyan may be employed to establish an equiconsistency for a related indestructibility theorem for strongness.

Some Remarks on Tall Cardinals and Failures of GCH

Arthur W. Apter (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

We investigate two global GCH patterns which are consistent with the existence of a tall cardinal, and also present some related open questions.

Some variations on the partition property for normal ultrafilters on Pkl

Julius Barbanel (1993)

Fundamenta Mathematicae

Suppose κ is a supercompact cardinal and λ≥κ. In [3], we studied the relationship between the weak partition property and the partition property for normal ultrafilters on P κ λ . In this paper we study a hierarchy of properties intermediate between the weak partition property and the partition property. Given appropriate large cardinal assumptions, we show that these properties are not all equivalent.

Spectra of uniformity

Yair Hayut, Asaf Karagila (2019)

Commentationes Mathematicae Universitatis Carolinae

We study some limitations and possible occurrences of uniform ultrafilters on ordinals without the axiom of choice. We prove an Easton-like theorem about the possible spectrum of successors of regular cardinals which carry uniform ultrafilters; we also show that this spectrum is not necessarily closed.

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