Displaying similar documents to “Concerning a proof of ℵ α + 1 ≦ 2 ℵ α without axiom of choice: A correction”

Inaccessible cardinals without the axiom of choice

Andreas Blass, Ioanna M. Dimitriou, Benedikt Löwe (2007)

Fundamenta Mathematicae

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We consider four notions of strong inaccessibility that are equivalent in ZFC and show that they are not equivalent in ZF.

On the Leibniz-Mycielski axiom in set theory

Ali Enayat (2004)

Fundamenta Mathematicae

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Motivated by Leibniz’s thesis on the identity of indiscernibles, Mycielski introduced a set-theoretic axiom, here dubbed the Leibniz-Mycielski axiom LM, which asserts that for each pair of distinct sets x and y there exists an ordinal α exceeding the ranks of x and y, and a formula φ(v), such that ( V α , ∈ ) satisfies φ(x) ∧¬ φ(y). We examine the relationship between LM and some other axioms of set theory. Our principal results are as follows: 1. In the presence of ZF, the following are equivalent: (a)...

The Wholeness Axioms and the Class of Supercompact Cardinals

Arthur W. Apter (2012)

Bulletin of the Polish Academy of Sciences. Mathematics

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We show that certain relatively consistent structural properties of the class of supercompact cardinals are also relatively consistent with the Wholeness Axioms.

Sandwiching the Consistency Strength of Two Global Choiceless Cardinal Patterns

Arthur W. Apter (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

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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...

Strong compactness, measurability, and the class of supercompact cardinals

Arthur W. Apter (2001)

Fundamenta Mathematicae

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We prove two theorems concerning strong compactness, measurability, and the class of supercompact cardinals. We begin by showing, relative to the appropriate hypotheses, that it is consistent non-trivially for every supercompact cardinal to be the limit of (non-supercompact) strongly compact cardinals. We then show, relative to the existence of a non-trivial (proper or improper) class of supercompact cardinals, that it is possible to have a model with the same class of supercompact cardinals...