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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

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2015

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Vol. 63, no. 1

1--10

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.

2

On a Certain Notion of Finite and a Finiteness Class in Set Theory without Choice

Herrlich H., Howard P., Tachtsis E.

Bulletin of the Polish Academy of Sciences. Mathematics

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2015

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Vol. 63, no. 2

89--112

EN

We study the deductive strength of properties under basic set-theoretical operations of the subclass E-Fin of the Dedekind finite sets in set theory without the Axiom of Choice (AC), which consists of all E-finite sets, where a set X is called E-finite if for no proper subset Y of X is there a surjection f:Y→X.

3

Remarks on the Stone Spaces of the Integers and the Reals without AC

Herrlich H., Keremedis K., Tachtsis E.

Bulletin of the Polish Academy of Sciences. Mathematics

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2011

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Vol. 59, no 2

101--114

EN

In ZF, i.e., the Zermelo–Fraenkel set theory minus the Axiom of Choice AC, we investigate the relationship between the Tychonoff product 2P(X), where 2 is 2 = f0; 1g with the discrete topology, and the Stone space S(X) of the Boolean algebra of all subsets of X, where X =ω,R. We also study the possible placement of well-known topological statements which concern the cited spaces in the hierarchy of weak choice principles.

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On the set-theoretic strength of countable compactness of the Tychonoff product 2R

Tachtsis E.

Bulletin of the Polish Academy of Sciences. Mathematics

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2010

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Vol. 58, no 2

91-107

EN

We work in ZF set theory (i.e., Zermelo-Fraenkel set theory minus the Axiom of Choice AC) and show the following: 1. The Axiom of Choice for well-ordered families of non-empty sets (ACWO) does not imply "the Tychonoff product 2R, where 2 is the discrete space {0,1}, is countably compact" in ZF. This answers in the negative the following question from Keremedis, Felouzis, and Tachtsis [Bull. Polish Acad. Sci. Math. 55 (2007)]: Does the Countable Axiom of Choice for families of non-empty sets of reals imply 2R is countably compact in ZF? 2. Assuming the Countable Axiom of Multiple Choice (CMC), the statements "every infinite subset of 2R has an accumulation point", "every countably infinite subset of 2R has an accumulation point", "2R is countably compact", and UF(ω) = "there is a free ultrafilter on ω" are pairwise equivalent. 3. The statements "for every infinite set X, every countably infinite subset of 2X has an accumulation point", "every countably infinite subset of 2R has an accumulation point", and UF(ω) are, in ZF, pairwise equivalent. Hence, in ZF, the statement "2R is countably compact "implies UF(ω). 4. The statement "every infinite subset of 2R has an accumulation point" implies "every countable family of 2-element subsets of the powerset Ρ(R) of R has a choice function". 5. The Countable Axiom of Choice restricted to non-empty finite sets, (CACfin), is, in ZF, strictly weaker than the statement "for every infinite set X, 2X is countably compact".

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On the compactness and countable compactness of 2R in ZF

Keremedis K., Felouzis E., Tachtsis E.

Bulletin of the Polish Academy of Sciences. Mathematics

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2007

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Vol. 55, no 4

293-302

EN

In the framework of ZF (Zermelo-Fraenkel set theory without the Axiom of Choice) we provide topological and Boolean-algebraic characterizations of the statements "2R is countably compact" and "2R is compact".

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Countable compact scattered T2 spaces and weak forms of AC

Keremedis K., Felouzis E., Tachtsis E.

Bulletin of the Polish Academy of Sciences. Mathematics

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2006

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Vol. 54, no 1

75-84

EN

We show that: (1) It is provable in ZF (i.e., Zermelo-Fraenkel set theory minus the Axiom of Choice AC) that every compact scattered T2 topological space is zero-dimensional. (2) If every countable union of countable sets of reals is countable, then a countable compact T2 space is scattered iff it is metrizable. (3) If the real line R can be expressed as a well-ordered union of well-orderable sets, then every countable compact zero-dimensional T2 space is scattered. (4) It is not provable in ZF+¬AC that there exists a countable compact T2 space which is dense-in-itself.