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On the Existence of Free Ultrafilters on ω and on Russell-sets in ZF

Tachtsis E.

Bulletin of the Polish Academy of Sciences. Mathematics

2015

Vol. 63, no. 1

1--10

In ZF (i.e. Zermelo–Fraenkel set theory without the Axiom of Choice AC), we investigate the relationship between UF(ω) (there exists a free ultrafilter on ω) and the statements ‶there exists a free ultrafilter on every Russell-set″ and ‶there exists a Russell-set A and a free ultrafilter F on A″. We establish the following results: 1. UF(ω) implies that there exists a free ultrafilter on every Russell-set. The implication is not reversible in ZF. 2. The statement there exists a free ultrafilter on every Russell-set″ is not provable in ZF. 3. If there exists a Russell-set A and a free ultrafilter on A, then UF(ω) holds. The implication is not reversible in ZF. 4. If there exists a Russell-set A and a free ultrafilter on A, then there exists a free ultrafilter on every Russell-set. We also observe the following: (a) The statements BPI(ω) (every proper filter on ω can be extended to an ultrafilter on ω) and ‶there exists a Russell-set A and a free ultrafilter F on A″ are independent of each other in ZF. (b) The statement ‶there exists a Russell-set and there exists a free ultrafilter on every Russell-set″ is, in ZF, equivalent to ‶there exists a Russell-set A and a free ultrafilter F on A″. Thus, ‶there exists a Russell-set and there exists a free ultrafilter on every Russell-set″ is also relatively consistent with ZF.