Consequences of the Axiom of Choice

Howard, Paul E.; Rubin, Jean E.

doi:10.1090/surv/059

Cited by 282 publications

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“…Thus, if the principle WO ℵ ω holds true, then D ω is well orderable, hence X is well orderable, and we may conclude that M(Loeb,hLoeb) is a theorem of ZF+WO ℵ ω . Now, in Fraenkel-Mostowski permutation models for ZF 0 set theory (= ZF minus the axiom of regularity), selective and Loeb metric spaces are always well orderable since the axiom PW (hence the weaker WO ℵ ω ) is true in each such model; see [5].…”

Section: Introduction and Some Known Resultsmentioning

confidence: 99%

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Weak axioms of choice for metric spaces

Keremedis

Tachtsis

2005

Proc. Amer. Math. Soc.

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Abstract. In the framework of ZF, i.e., Zermelo-Fraenkel set theory without the axiom of choice AC, we show that if the family of all non-empty, closed subsets of a metric space (X, d) has a choice function, then so does the family of all non-empty, open subsets of X. In addition, we establish that the converse is not provable in ZF.We also show that the statement "every subspace of the real line R with the standard topology has a choice function for its family of all closed, non-empty subsets" is equivalent to the weak choice form "every continuum sized family of non-empty subsets of reals has a choice function". Notation and terminologyIn the following, each of the statements "Form x" has been considered in [5], where all known implications between these forms are given in Table 1; see http:// www.math.purdue.edu/˜jer /Papers/conseq.html. Definition 1.(1) Let (X, T ) be a topological space. (1a) X is said to be a Loeb space iff the family of all non-empty, closed subsets of X has a choice function. (1b) X is said to be selective iff the family of all non-empty, open subsets of X has a choice function. (1c) X is called second countable if it has a countable base for T . (1d) X is called separable if it has a countable dense subset.

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Section: Introduction and Some Known Resultsmentioning

confidence: 99%

“…In the following, each of the statements "Form x" has been considered in [5], where all known implications between these forms are given in Table 1; see http:// www.math.purdue.edu/˜jer /Papers/conseq.html.…”

Section: Notation and Terminologymentioning

confidence: 99%

Weak axioms of choice for metric spaces

Keremedis

Tachtsis

2005

Proc. Amer. Math. Soc.

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“…Reverse mathematics was prefigured by some results in set theory, such as the classical theorem that states that the axiom of choice, well-ordering principle of Zermelo, maximal chain priciple of Hausdorff, Zorn's lemma [38], and statements of the vector basis theorem [39] and Tychonov product theorem [40] are equivalent over ZF set theory (Howard and Rubin, 1998) [41]. The goal of reverse mathematics, however, is to study ordinary theorems of mathematics rather than possible axioms for set theory.…”

Section: Projective Mathematics Vs Reverse Mathematics Vs Classicalmentioning

confidence: 99%

Unconventional Algorithms: Complementarity of Axiomatics and Construction

Dodig-Crnković

Burgin

2012

Entropy

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Abstract:In this paper, we analyze axiomatic and constructive issues of unconventional computations from a methodological and philosophical point of view. We explain how the new models of algorithms and unconventional computations change the algorithmic universe, making it open and allowing increased flexibility and expressive power that augment creativity. At the same time, the greater power of new types of algorithms also results in the greater complexity of the algorithmic universe, transforming it into the algorithmic multiverse and demanding new tools for its study. That is why we analyze new powerful tools brought forth by local mathematics, local logics, logical varieties and the axiomatic theory of algorithms, automata and computation. We demonstrate how these new tools allow efficient navigation in the algorithmic multiverse. Further work includes study of natural computation by unconventional algorithms and constructive approaches.

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“…For example, we have the following short list (see [12], [13], and [19] for much longer lists): In this paper we are especially interested in (TT). Recall that a collection A of subsets of a set X has finite character if for every A ⊆ X, A ∈ A if and only if every finite subset of A is in A.…”

Section: Introductionmentioning

confidence: 99%

“…There are other such non-constructive principles that are perhaps not so well known as those given above. Here is a short list (see [12], form 14, for a much longer list; additional references are [2], [3], [13], [18], and [20]). The logical status of these ideas is not trivial.…”

Section: Introductionmentioning

confidence: 99%

Restricted versions of the Tukey-Teichmüller theorem that are equivalent to the Boolean prime ideal theorem

Hodel

2004

Arch. Math. Logic

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Dedicated to W.W. Comfort on the occasion of his seventieth birthday. AbstractWe formulate a restricted version of the Tukey-Teichmüller Theorem that we denote by (rTT). We then prove that (rTT) and (BPI) are equivalent in ZF and that (rTT) applies rather naturally to several equivalent forms of (BPI): Alexander Subbase Theorem, Stone Representation Theorem, Model Existence and Compactness Theorems for propositional and first-order logic. We also give two variations of (rTT) that we denote by (rTT) + and (rTT) ++ ; each is equivalent to (rTT) in ZF. The variation (rTT) ++ applies rather naturally to various Selection Lemmas due to Cowen, Engeler, and Rado. IntroductionLet ZF denote Zermelo-Fraenkel set theory. It is well known that the axiom of choice (AC) is independent of ZF. When we add AC to ZF we obtain the most widely used axiom system for classical mathematics, namely ZFC = ZF + AC.It is also well known that there are many theorems of ZFC that are in fact equivalent forms of AC. For example, we have the following short list (see [12], [13], and [19] for much longer lists): In this paper we are especially interested in (TT). Recall that a collection A of subsets of a set X has finite character if for every A ⊆ X, A ∈ A if and only if every finite subset of A is in A. Note that ∅ ∈ A (assuming that A is non-empty). There are two versions of the Tukey-Teichmüller Theorem; however, in ZF they are equivalent to each other and also to AC.TT (weak form) Let X be a set and let A be a non-empty collection of subsets of X with finite character. Then A has a maximal element (with respect to ⊆).TT (strong form) Let X be a set and let A be a non-empty collection of subsets of X with finite character. Then for each A ∈ A, there exists B ∈ A such that A ⊆ B and B is maximal.The Tukey-Teichmüller Theorem seems tailor-made for certain applications. For example, it can be used to prove that every vector space has a basis by simply noting that the property of being linearly independent has finite character. Another nice application is a proof of the Alexander Subbase Theorem (see [14] or [7]).The axiom of choice and its equivalent forms are non-constructive principles; they assert the existence of a set without giving instructions on how to construct the set. There are other such non-constructive principles that are perhaps not so well known as those given above. Here is a short list (see The logical status of these ideas is not trivial. The easy part is:The hard part states:• AC cannot be proved from ZF + BPI (Halpern and Lévy [10]);• BPI cannot be proved from ZF (S. Feferman [8]).In section 2 we formulate a version of the Tukey-Teichmüller Theorem (TT) that is equivalent to (BPI) rather than to (AC). We then use this new 2 non-constructive axiom, denoted (rTT) (for restricted Tukey-Teichmüller), to give direct proofs of the Alexander Subbase Theorem and the Stone Representation Theorem. In section 3 we show directly that (rTT) and (BPI) are equivalent in ZF. Finally, in section 4 we use (rTT) to derive various theorems...

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Consequences of the Axiom of Choice

Cited by 282 publications

References 0 publications

Weak axioms of choice for metric spaces

Weak axioms of choice for metric spaces

Unconventional Algorithms: Complementarity of Axiomatics and Construction

Restricted versions of the Tukey-Teichmüller theorem that are equivalent to the Boolean prime ideal theorem

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