Choice principles for special subsets of the real line

Keremedis, Kyriakos; Tachtsis, Eleftherios

doi:10.1002/malq.200310048

Cited by 3 publications

(3 citation statements)

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“…Truss [29, Theorem 3]. From the above considerations, it follows that neither (#) is provable in

\mathsf {ZF}

.Finally, we point out that it is an open problem whether or not

\mathsf {AC}(\mathsf {P}\mathbb {R})

is equivalent to

\mathsf {AC}(\mathbb {R})

\mathsf {ZF}

; for a partial (non‐trivial) solution to this open problem, the reader is referred to Keremedis and Tachtsis [18, § 5]. Let (##) stand for the following weaker form of the Hewitt–Marczewski–Pondiczery Theorem:(##) A non‐empty product of

\le 2^{\aleph _{0}}

separable Hausdorff spaces, each with at least two points, is separable.We have the following implications:To see that (a) is true, fix a continuum sized family

\mathcal {A}=\lbrace A_{i}:i\in \mathbb {R}\rbrace

of non‐empty subsets of

\mathbb {R}

.…”

Section: Resultsmentioning

confidence: 99%

“…Finally, we point out that it is an open problem whether or not

\mathsf {AC}(\mathsf {P}\mathbb {R})

is equivalent to

\mathsf {AC}(\mathbb {R})

\mathsf {ZF}

; for a partial (non‐trivial) solution to this open problem, the reader is referred to Keremedis and Tachtsis [18, § 5].…”

Section: Resultsmentioning

confidence: 99%

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On Hausdorff operators in ZF$\mathsf {ZF}$

Keremedis

Tachtsis

2023

Mathematical Logic Qtrly

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A Hausdorff space is called effectively Hausdorff if there exists a function F—called a Hausdorff operator—such that, for every with , , where U and V are disjoint open neighborhoods of x and y, respectively. Among other results, we establish the following in , i.e., in Zermelo–Fraenkel set theory without the Axiom of Choice (): is equivalent to “For every set X, the Cantor cube is effectively Hausdorff”. This enhances the result of Howard, Keremedis, Rubin and Rubin [13] that is equivalent to “Hausdorff spaces are effectively Hausdorff” in . The Boolean Prime Ideal Theorem and the statement “For every infinite set X, the Stone space of the Boolean algebra is effectively Hausdorff” are mutually independent. In particular, the latter statement is not provable in . The Axiom of Choice for non‐empty subsets of () is equivalent to each of “Separable Hausdorff spaces are effectively Hausdorff” and “The Cantor cube is effectively Hausdorff”. The Principle of Dependent Choices in conjunction with the Axiom of Choice for continuum sized families of non‐empty subsets of does not imply the axiom of choice for partitions of . The latter independence result fills the gap in information in Howard and Rubin's book “Consequences of the Axiom of Choice”. The axiom of countable choice for non‐empty subsets of is equivalent to each of “Denumerable Hausdorff spaces are effectively Hausdorff”, “Denumerable T3 spaces are completely normal” and “Denumerable Tychonoff spaces are Urysohn”.

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“…Truss [29, Theorem 3]. From the above considerations, it follows that neither (#) is provable in

\mathsf {ZF}

.Finally, we point out that it is an open problem whether or not

\mathsf {AC}(\mathsf {P}\mathbb {R})

is equivalent to

\mathsf {AC}(\mathbb {R})

\mathsf {ZF}

\le 2^{\aleph _{0}}

separable Hausdorff spaces, each with at least two points, is separable.We have the following implications:To see that (a) is true, fix a continuum sized family

\mathcal {A}=\lbrace A_{i}:i\in \mathbb {R}\rbrace

of non‐empty subsets of

\mathbb {R}

.…”

Section: Resultsmentioning

confidence: 99%

“…Finally, we point out that it is an open problem whether or not

\mathsf {AC}(\mathsf {P}\mathbb {R})

is equivalent to

\mathsf {AC}(\mathbb {R})

\mathsf {ZF}

; for a partial (non‐trivial) solution to this open problem, the reader is referred to Keremedis and Tachtsis [18, § 5].…”

Section: Resultsmentioning

confidence: 99%

On Hausdorff operators in ZF$\mathsf {ZF}$

Keremedis

Tachtsis

2023

Mathematical Logic Qtrly

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“…(12) In [21] it is shown that Form 212 does not imply either of AC(R), Form 203, and in [10] it is shown that Form 212 does not imply Form 368. (13) In [10] we showed that Form 368 is true in M1. Since (14) Tarski [19] has shown that Form 368 implies Form 170.…”

mentioning

confidence: 99%

Properties of the real line and weak forms of the Axiom of Choice

Cruz

Hall

Howard

et al. 2005

MLQ - Math. Log. Quart.

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2005 Summer Meeting of the Association for Symbolic Logic. Logic Colloquium '05

Wainer¹

2006

Bull. symb. log.

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and the Sigalas Wine Co. Ltd. We thank J. Moore, Uncountable linear orders. A. Nies, Algebras with finite descriptions. C. Parsons, Paul Bernays' later philosophy of mathematics. H. Schwichtenberg, Minimal logic for computable functionals. M. Sheard, A transactional approach to the logic of truth. S. Tupailo, Monotone inductive definitions and consistency of New Foundations. K. Weihrauch, Computable analysis. J. Zapletal, Forcing idealized. The proceedings of Logic Colloquium '05, edited by C. Dimitracopoulos, L. Newelski, D. Normann, and J. Steel, will contain many of the invited lectures and will be published in the ASL series Lecture Notes in Logic. More information about the meeting can be found at the conference webpage http:// www.math.uoa.gr/lc2005/.

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Choice principles for special subsets of the real line

Abstract: We study the role the axiom of choice plays in the existence of some special subsets of R and its power set ℘(R).

Cited by 3 publications

References 10 publications

On Hausdorff operators in ZF$\mathsf {ZF}$

On Hausdorff operators in ZF$\mathsf {ZF}$

Properties of the real line and weak forms of the Axiom of Choice

2005 Summer Meeting of the Association for Symbolic Logic. Logic Colloquium '05

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