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Jaakko Hintikka - One of the best experts on this subject based on the ideXlab platform.
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Is the Axiom of Choice a Logical or Set‐Theoretical Principle?
Dialectica, 2005Co-Authors: Jaakko HintikkaAbstract:A generalization of the Axioms of Choice says that all the Skolem functions of a true first-order sentence exist. This generalization can be implemented on the first-order level by generalizing the rule of existential instantiation into a rule of functional instantiation. If this generalization is carried out in first-order Axiomatic set theory (FAST), it is seen that in any model of FAST, there are sentences S which are true but whose Skolem functions do not exist. Since this existence is what the truth of S means in a combinational (model-theoretical) sense, in any model of FAST there are sentences which are set-theoretical true but false in the normal sense of the word. This shows that the assumptions on which the Axiom of Choice rests cannot be fully implemented in FAST. The Axiom of Choice is not a set-theoretical principle.
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is the Axiom of Choice a logical or set theoretical principle
Dialectica, 2005Co-Authors: Jaakko HintikkaAbstract:A generalization of the Axioms of Choice says that all the Skolem functions of a true first-order sentence exist. This generalization can be implemented on the first-order level by generalizing the rule of existential instantiation into a rule of functional instantiation. If this generalization is carried out in first-order Axiomatic set theory (FAST), it is seen that in any model of FAST, there are sentences S which are true but whose Skolem functions do not exist. Since this existence is what the truth of S means in a combinational (model-theoretical) sense, in any model of FAST there are sentences which are set-theoretical true but false in the normal sense of the word. This shows that the assumptions on which the Axiom of Choice rests cannot be fully implemented in FAST. The Axiom of Choice is not a set-theoretical principle.
Michael Rathjen - One of the best experts on this subject based on the ideXlab platform.
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Power Kripke–Platek set theory and the Axiom of Choice
Journal of Logic and Computation, 2020Co-Authors: Michael RathjenAbstract:It is shown that adding the Axiom of Choice to Power Kripke-Platek set theory, KP(P), does not increase its proof-theoretic strength.
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power kripke platek set theory and the Axiom of Choice
Journal of Logic and Computation, 2020Co-Authors: Michael RathjenAbstract:It is shown that adding the Axiom of Choice to Power Kripke-Platek set theory, KP(P), does not increase its proof-theoretic strength.
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Power Kripke-Platek set theory and the Axiom of Choice
arXiv: Logic, 2018Co-Authors: Michael RathjenAbstract:Whilst Power Kripke-Platek set theory, KPP, shares many properties with ordinary Kripke-Platek set theory, KP, in several ways it behaves quite differently from KP. This is perhaps most strikingly demonstrated by a result, due to Mathias, to the effect that adding the Axiom of constructibility to KPP gives rise to a much stronger theory, whereas in the case of KP the constructible hierarchy provides an inner model, so that KP and KP+V=L have the same strength. This paper will be concerned with the relationship between KPP and KPP plus the Axiom of Choice or even the global Axiom of Choice, GAC. Since L is the standard vehicle to furnish a model in which this Axiom holds, the usual argument for demonstrating that the addition of AC or GAC to KPP does not increase proof-theoretic strength does not apply in any obvious way. Among other tools, the paper uses techniques from ordinal analysis to show that KPP+GAC has the same strength as KPP, thereby answering a question of Mathias. Moreover, it is shown that KPP+GAC is conservative over KPP for Pi-1-4 statements of analysis. The method of ordinal analysis for theories with power set was developed in an earlier paper. The technique allows one to compute witnessing information from infinitary proofs, providing bounds for the transfinite iterations of the power set operation that are provable in a theory. As the theory KPP+GAC provides a very useful tool for defining models and realizability models of other theories that are hard to construct without access to a uniform selection mechanism, it is desirable to determine its exact proof-theoretic strength. This knowledge can for instance be used to determine the strength of Feferman's operational set theory with power set operation as well as constructive Zermelo-Fraenkel set theory with the Axiom of Choice.
Kyriakos Keremedis - One of the best experts on this subject based on the ideXlab platform.
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Unions and the Axiom of Choice
Mathematical Logic Quarterly, 2008Co-Authors: Omar De La Cruz, Eric J. Hall, Paul E. Howard, Kyriakos Keremedis, Jean E. RubinAbstract:We study statements about countable and well-ordered unions and their relation to each other and to countable and well-ordered forms of the Axiom of Choice. Using WO as an abbreviation for “well-orderable”, here are two typical results: The assertion that every WO family of countable sets has a WO union does not imply that every countable family of WO sets has a WO union; the Axiom of Choice for WO families of WO sets does not imply that the countable union of countable sets is WO. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
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Properties of the real line and weak forms of the Axiom of Choice
MLQ, 2005Co-Authors: Omar De La Cruz, Eric J. Hall, Paul E. Howard, Kyriakos Keremedis, Eleftherios TachtsisAbstract:We investigate, within the framework of Zermelo-Fraenkel set theory ZF, the interrelations between weak forms of the Axiom of Choice AC restricted to sets of reals. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
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Metric spaces and the Axiom of Choice
MLQ, 2003Co-Authors: Omar De La Cruz, Eric J. Hall, Paul E. Howard, Kyriakos Keremedis, Jean E. RubinAbstract:We study conditions for a topological space to be metrizable, properties of metriz- able spaces, and the role the Axiom of Choice plays in these matters.
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Disasters in topology without the Axiom of Choice
Archive for Mathematical Logic, 2001Co-Authors: Kyriakos KeremedisAbstract:We show that some well known theorems in topology may not be true without the Axiom of Choice.
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Some Weak Forms of the Axiom of Choice Restricted to the Real Line
MLQ, 2001Co-Authors: Kyriakos Keremedis, Eleftherios TachtsisAbstract:It is shown that AC(ℝ), the Axiom of Choice for families of non-empty subsets of the real line ℝ, does not imply the statement PW(ℝ), the powerset of ℝ can be well ordered. It is also shown that (1) the statement “the set of all denumerable subsets of ℝ has size 2” is strictly weaker than AC(ℝ) and (2) each of the statements (i) “if every member of an infinite set of cardinality 2 has power 2, then the union has power 2” and (ii) “ℵ(2) ≠ ℵω” (ℵ(2) is Hartogs' aleph, the least ℵ not ≤ 2), is strictly weaker than the full Axiom of Choice AC.
Philipp Kleppmann - One of the best experts on this subject based on the ideXlab platform.
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Nielsen‐Schreier and the Axiom of Choice
Mathematical Logic Quarterly, 2015Co-Authors: Philipp KleppmannAbstract:The Nielsen-Schreier theorem asserts that subgroups of free groups are free. In the first section we show that this theorem does not follow from the Linear Ordering Principle, thus strengthening the fact that it implies the Axiom of Choice for families of finite sets. In the second section, we show that a stronger variant of the Nielsen-Schreier theorem implies the Axiom of Choice.
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nielsen schreier and the Axiom of Choice
Mathematical Logic Quarterly, 2015Co-Authors: Philipp KleppmannAbstract:The Nielsen-Schreier theorem asserts that subgroups of free groups are free. In the first section we show that this theorem does not follow from the Linear Ordering Principle, thus strengthening the fact that it implies the Axiom of Choice for families of finite sets. In the second section, we show that a stronger variant of the Nielsen-Schreier theorem implies the Axiom of Choice.
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Nielsen-Schreier implies the finite Axiom of Choice
arXiv: Logic, 2015Co-Authors: Philipp KleppmannAbstract:We present a new proof that the statement 'every subgroup of a free group is free' implies the Axiom of Choice for finite sets.
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Generating sets of free groups and the Axiom of Choice
Mathematical Logic Quarterly, 2014Co-Authors: Philipp KleppmannAbstract:Write F(X) for the free group generated by X. We show that the statement for infinite sets is equivalent to the Axiom of Choice, whereas the statement is strictly weaker than the Axiom of Choice.
Eleftherios Tachtsis - One of the best experts on this subject based on the ideXlab platform.
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Properties of the real line and weak forms of the Axiom of Choice
MLQ, 2005Co-Authors: Omar De La Cruz, Eric J. Hall, Paul E. Howard, Kyriakos Keremedis, Eleftherios TachtsisAbstract:We investigate, within the framework of Zermelo-Fraenkel set theory ZF, the interrelations between weak forms of the Axiom of Choice AC restricted to sets of reals. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
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Some Weak Forms of the Axiom of Choice Restricted to the Real Line
MLQ, 2001Co-Authors: Kyriakos Keremedis, Eleftherios TachtsisAbstract:It is shown that AC(ℝ), the Axiom of Choice for families of non-empty subsets of the real line ℝ, does not imply the statement PW(ℝ), the powerset of ℝ can be well ordered. It is also shown that (1) the statement “the set of all denumerable subsets of ℝ has size 2” is strictly weaker than AC(ℝ) and (2) each of the statements (i) “if every member of an infinite set of cardinality 2 has power 2, then the union has power 2” and (ii) “ℵ(2) ≠ ℵω” (ℵ(2) is Hartogs' aleph, the least ℵ not ≤ 2), is strictly weaker than the full Axiom of Choice AC.