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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ba63-1-1,
author = {Eleftherios Tachtsis},
title = {On the Existence of Free Ultrafilters on $\omega$ and on Russell-sets in ZF},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
volume = {63},
year = {2015},
pages = {1-10},
zbl = {1325.03060},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba63-1-1}
}
Eleftherios Tachtsis. On the Existence of Free Ultrafilters on ω and on Russell-sets in ZF. Bulletin of the Polish Academy of Sciences. Mathematics, Tome 63 (2015) pp. 1-10. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba63-1-1/