Ultraproducts and the axiom of choice. (English) Zbl 0642.03029

Brunner, Norbert

Arch. Math., Brno 22, 175-180 (1986).

The author’s aim is to present the following theorem. There is a model of ZF in which \({\mathcal P}(\omega)\) can be wellordered (whence there are nonprincipal ultrafilters on \(\omega)\) and there is a family \((F_ n)_{n\in \omega}\) of finite sets, such that their product \({\mathcal X}=\prod_{n\in \omega}F_ n\) contains no infinite, Dedekind-finite subset, but for every nonprincipal ultrafilter F on \(\omega\), the ultraproduct \({\mathcal X}/F\) is amorphous (an infinite set, every infinite subset of which is cofinite). A permutation model M of \(ZF^ 0+PW\) \((ZF^ 0\) \(=\) Zermelo-Fraenkel set theory without the axioms of foundation F or choice, PW is Rubin’s axiom that the power set of an ordinal can be wellordered). Based on a countable set of urelements, where the theorem holds, is constructed.
The theorem is applied for strengthening some results about the topology of \({\mathbb{R}}\) and \(T_ 2\)-spaces in ZF-models without full AC.

Reviewer: K.Wolfsdorf

MSC:

03E25	Axiom of choice and related propositions
03C20	Ultraproducts and related constructions
54A35	Consistency and independence results in general topology
54A25	Cardinality properties (cardinal functions and inequalities, discrete subsets)

Keywords:

Dedekind-finite; axiom of choice; amorphous set; topology of the reals; ultraproduct; permutation model; Rubin’s axiom; \(T_ 2\)-spaces; ZF- models without full AC