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1
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On the Existence of Free Ultrafilters on ω and on Russell-sets in ZF

100%
Tachtsis E.
Bulletin of the Polish Academy of Sciences. Mathematics
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2015
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tom 63
|
nr 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
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On the Set-Theoretic Strength of Countable Compactness of the Tychonoff Product $2^{ℝ}$

100%
Tachtsis E.
Bulletin of the Polish Academy of Sciences. Mathematics
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2010
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tom 58
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nr 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 ($AC^{WO}$) does not imply "the Tychonoff product $2^{ℝ}$, 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 $2^{ℝ}$ is countably compact in ZF? 2. Assuming the Countable Axiom of Multiple Choice (CMC), the statements "every infinite subset of $2^{ℝ}$ has an accumulation point", "every countably infinite subset of $2^{ℝ}$ has an accumulation point", "$2^{ℝ}$ 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 $2^{X}$ has an accumulation point", "every countably infinite subset of $2^{ℝ}$ has an accumulation point", and UF(ω) are, in ZF, pairwise equivalent. Hence, in ZF, the statement "$2^{ℝ}$ is countably compact" implies UF(ω). 4. The statement "every infinite subset of $2^{ℝ}$ has an accumulation point" implies "every countable family of 2-element subsets of the powerset 𝓟(ℝ) of ℝ has a choice function". 5. The Countable Axiom of Choice restricted to non-empty finite sets, ($CAC_{fin}$), is, in ZF, strictly weaker than the statement "for every infinite set X, $2^{X}$ is countably compact".
3
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On the set-theoretic strength of the n-compactness of generalized Cantor cubes

64%
Howard P., Tachtsis E.
Fundamenta Mathematicae
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2016
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tom 234
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nr 3
241-252
EN
We investigate, in set theory without the Axiom of Choice 𝖠𝖢, the set-theoretic strength of the statement Q(n): For every infinite set X, the Tychonoff product $2^{X}$, where 2 = {0,1} has the discrete topology, is n-compact, where n = 2,3,4,5 (definitions are given in Section 1). We establish the following results: (1) For n = 3,4,5, Q(n) is, in 𝖹𝖥 (Zermelo-Fraenkel set theory minus 𝖠𝖢), equivalent to the Boolean Prime Ideal Theorem 𝖡𝖯𝖨, whereas (2) Q(2) is strictly weaker than 𝖡𝖯𝖨 in 𝖹𝖥𝖠 set theory (Zermelo-Fraenkel set theory with the Axiom of Extensionality weakened in order to allow atoms). This settles the open problem in Tachtsis (2012) on the relation of Q(n), n = 2,3,4,5, to 𝖡𝖯𝖨.
4
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On rigid relation principles in set theory without the axiom of choice

64%
Howard P., Tachtsis E.
Fundamenta Mathematicae
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2016
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tom 232
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nr 3
199-226
EN
We study the deductive strength of the following statements: 𝖱𝖱: every set has a rigid binary relation, 𝖧𝖱𝖱: every set has a hereditarily rigid binary relation, 𝖲𝖱𝖱: every set has a strongly rigid binary relation, in set theory without the Axiom of Choice. 𝖱𝖱 was recently formulated by J. D. Hamkins and J. Palumbo, and 𝖲𝖱𝖱 is a classical (non-trivial) 𝖹𝖥𝖢-result by P. Vopěnka, A. Pultr and Z. Hedrlín.
5
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On a Certain Notion of Finite and a Finiteness Class in Set Theory without Choice

51%
Herrlich H., Howard P., Tachtsis E.
Bulletin of the Polish Academy of Sciences. Mathematics
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2015
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tom 63
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nr 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.
6
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Remarks on the Stone Spaces of the Integers and the Reals without AC

51%
Herrlich H., Keremedis K., Tachtsis E.
Bulletin of the Polish Academy of Sciences. Mathematics
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2011
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tom 59
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nr 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 $2^{𝓟(X)}$, where 2 is 2 = {0,1} 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.
7
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On the Compactness and Countable Compactness of $2^{ℝ}$ in ZF

51%
Keremedis K., Felouzis E., Tachtsis E.
Bulletin of the Polish Academy of Sciences. Mathematics
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2007
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tom 55
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nr 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 "$2^{ℝ}$ is countably compact" and "$2^{ℝ}$ is compact"
8
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Countable Compact Scattered T₂ Spaces and Weak Forms of AC

51%
Keremedis K., Felouzis E., Tachtsis E.
Bulletin of the Polish Academy of Sciences. Mathematics
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2006
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tom 54
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nr 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.
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