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## Bulletin of the Polish Academy of Sciences. Mathematics

2015 | Vol. 63, no. 1 | 1--10
Tytuł artykułu

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

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
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EN
Wydawca

Rocznik
Strony
1--10
Opis fizyczny
Bibliogr. 17 poz.
Twórcy
autor
• Department of Mathematics, University of the Aegean, Karlovassi 83200, Samos, Greece, ltah@aegean.gr
Bibliografia
• [1] A. Blass, A model without ultrafilters, Bull. Acad. Polon. Sci. Sér. Math. Astronom. Phys. 25 (1977), 329-331.
• [2] S. Feferman, Some applications of the notions of forcing and generic sets, Fund. Math. 55 (1965), 325-345.
• [3] E. J. Hall, K. Keremedis and E. Tachtsis, The existence of free ultrafilters on ω does not imply the extension of filters on ω to ultrafilters, Math. Logic Quart. 59 (2013), 258-267.
• [4] H. Herrlich, Axiom of Choice, Lecture Notes in Math. 1876, Springer, Berlin, 2006.
• [5] H. Herrlich, Binary partitions in the absence of choice or rearranging Russell's socks, Quaest. Math. 30 (2007), 465-470.
• [6] H. Herrlich, P. Howard and E. Tachtsis, The cardinal inequality α2 < 2α, Quaest. Math. 34 (2011), 35-66.
• [7] H. Herrlich, P. Howard and E. Tachtsis, On special partitions of Dedekind- and Russell-sets, Comment. Math. Univ. Carolin. 53 (2012), 105-122.
• [8] H. Herrlich, K. Keremedis and E. Tachtsis, On Russell and anti Russell-cardinals, Quaest. Math. 33 (2010), 1-9.
• [9] H. Herrlich and E. Tachtsis, On the number of Russell's socks or 2+2+2+…= ?, Comment. Math. Univ. Carolin. 47 (2006), 707-717.
• [10] H. Herrlich and E. Tachtsis, Odd-sized partitions of Russell-sets, Math. Logic Quart. 56 (2010), 185-190.
• [11] P. Howard and J. E. Rubin, Consequences of the Axiom of Choice, Math. Surveys Monogr. 59, Amer. Math. Soc., Providence, RI, 1998.
• [12] T. Jech, The Axiom of Choice, Stud. Logic Found. Math. 75, North-Holland, Amsterdam, 1973.
• [13] T. Jech, Set Theory, 3rd ed., Springer, Berlin, 2002.
• [14] K. Keremedis, Tychonoff products of two-element sets and some weakenings of the Boolean prime ideal theorem, Bull. Polish Acad. Sci. Math. 53 (2005), 349-359.
• [15] K. Keremedis, E. Felouzis and E. Tachtsis, On the compactness and countable compactness of 2R in ZF, Bull. Polish Acad. Sci. Math. 55 (2007), 293-302.
• [16] K. Kunen, Set Theory. An Introduction to Independence Proofs, Stud. Logic Found. Math. 102, North-Holland, Amsterdam, 1980.
• [17] E. Tachtsis, On the set-theoretic strength of countable compactness of the Tychonoff product 2R, Bull. Polish Acad. Sci. Math. 58 (2010), 91-107.
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