Displaying similar documents to “Inaccessible cardinals without the axiom of choice”

The Wholeness Axioms and the Class of Supercompact Cardinals

Arthur W. Apter (2012)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

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

Stranger things about forcing without AC

Martin Goldstern, Lukas D. Klausner (2020)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

Typically, set theorists reason about forcing constructions in the context of Zermelo--Fraenkel set theory (ZFC). We show that without the axiom of choice (AC), several simple properties of forcing posets fail to hold, one of which answers Miller's question from the work: Arnold W. Miller, {Long Borel hierarchies}, MLQ Math. Log. Q. {54} (2008), no. 3, 307--322.

α-Properness and Axiom A

Tetsuya Ishiu (2005)

Fundamenta Mathematicae

Similarity:

We show that under ZFC, for every indecomposable ordinal α < ω₁, there exists a poset which is β-proper for every β < α but not α-proper. It is also shown that a poset is forcing equivalent to a poset satisfying Axiom A if and only if it is α-proper for every α < ω₁.

On partial orderings having precalibre-ℵ₁ and fragments of Martin's axiom

Joan Bagaria, Saharon Shelah (2016)

Fundamenta Mathematicae

Similarity:

We define a countable antichain condition (ccc) property for partial orderings, weaker than precalibre-ℵ₁, and show that Martin's axiom restricted to the class of partial orderings that have the property does not imply Martin's axiom for σ-linked partial orderings. This yields a new solution to an old question of the first author about the relative strength of Martin's axiom for σ-centered partial orderings together with the assertion that every Aronszajn tree is special. We also answer...

On the Leibniz-Mycielski axiom in set theory

Ali Enayat (2004)

Fundamenta Mathematicae

Similarity:

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)...