Sacks forcing collapses to
We shall prove that Sacks algebra is nowhere -distributive, which implies that Sacks forcing collapses to .
Sandwiching the Consistency Strength of Two Global Choiceless Cardinal Patterns
We provide upper and lower bounds in consistency strength for the theories “ZF + + 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 + + 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...
Set theories incorporating Hilbert's ε-symbol [Book]
Set-theoretic constructions of two-point sets
A two-point set is a subset of the plane which meets every line in exactly two points. By working in models of set theory other than ZFC, we demonstrate two new constructions of two-point sets. Our first construction shows that in ZFC + CH there exist two-point sets which are contained within the union of a countable collection of concentric circles. Our second construction shows that in certain models of ZF, we can show the existence of two-point sets without explicitly invoking the Axiom of Choice....
Sobre Una Definicion De Cardinales Finitos En Un Topos Arbitrario.
Some propositions equivalent to the Sikorski Extension Theorem for Boolean algebras
Some remarks on selectors
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