Some theorems on vector spaces and the axiom of choice
Spectra of uniformity
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.
Starkes und überstarkes Auswahlaxiom.
Stranger things about forcing without AC
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.
Subalgebra lattices of unary algebras and an axiom of choice
The axiom of choice and linearly ordered sets
The axiom of choice for linearly ordered families
The Axiom of Choice, the Löwenheim-Skolem Theorem and Borel models
The Banach-Tarski Paradox
The boolean prime ideal theorem holds iff maximal open filters exist
The cardinal equation 2m=m
The dependence of some logical axioms on disjoint transversals and linked systems
The distributivity numbers of finite products of P(ω)/fin
Generalizing [ShSp], for every n < ω we construct a ZFC-model where ℌ(n), the distributivity number of r.o., is greater than ℌ(n+1). This answers an old problem of Balcar, Pelant and Simon (see [BaPeSi]). We also show that both Laver and Miller forcings collapse the continuum to ℌ(n) for every n < ω, hence by the first result, consistently they collapse it below ℌ(n).
The even-odd hat problem
We answer a question of C. Hardin and A. Taylor concerning a hat-guessing game.
The Hahn-Banach theorem implies the Banach-Tarski paradox
The Hahn-Banach theorem implies the existence of a non-Lebesgue measurable set
The Hahn-Banach Theorem surveyed [Book]
The regularity properties on the real line
The relationship between two weak forms of the axiom of choice
The Tychonoff product theorem for compact Hausdorff spaces does not imply the axiom of choice: A new Proof. equivalent propositions.