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Choice principles in constructive and classical set theories
 POHLERS (EDS.): PROCEEDINGS OF THE LOGIC COLLOQUIUM 2002
, 2002
"... The objective of this paper is to assay several forms of the axiom of choice that have been deemed constructive. In addition to their deductive relationships, the paper will be concerned with metamathematical properties effected by these choice principles and also with some of their classical models ..."
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The objective of this paper is to assay several forms of the axiom of choice that have been deemed constructive. In addition to their deductive relationships, the paper will be concerned with metamathematical properties effected by these choice principles and also with some of their classical models.
Versions of normality and some weak forms of the axiom of choice. preprint
, 1996
"... Abstract. We investigate the set theoretical strength of some properties of normality, including Urysohn’s Lemma, TietzeUrysohn Extension Theorem, normality of disjoint unions of normal spaces, and normality of Fσ subsets of normal spaces. Introduction. The notion of a normal topological space has ..."
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Abstract. We investigate the set theoretical strength of some properties of normality, including Urysohn’s Lemma, TietzeUrysohn Extension Theorem, normality of disjoint unions of normal spaces, and normality of Fσ subsets of normal spaces. Introduction. The notion of a normal topological space has been of interest to topologists for many years. (See, for example, [2], [4], [7], [12], [13], [18], and [20].) In [4] it has been shown that there exists a close connection between properties of normality and the axiom of choice. In particular, in [4], van Douwen established that the proposition: “The disjoint union of a family X = {Xn: n ∈ ω} of orderable topological spaces such that each Xn
Metamathematical Properties of Intuitionistic Set Theories with Choice Principles
"... This paper is concerned with metamathematical properties of intuitionistic set theories with choice principles. It is proved that the disjunction property, the numerical existence property, Church’s rule, and several other metamathematical properties hold true for Constructive ZermeloFraenkel Set T ..."
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This paper is concerned with metamathematical properties of intuitionistic set theories with choice principles. It is proved that the disjunction property, the numerical existence property, Church’s rule, and several other metamathematical properties hold true for Constructive ZermeloFraenkel Set Theory and full Intuitionistic ZermeloFraenkel augmented by any combination of the principles of Countable Choice, Dependent Choices and the Presentation Axiom. Also Markov’s principle may be added. Moreover, these properties hold effectively. For instance from a proof of a statement ∀n ∈ ω ∃m ∈ ω ϕ(n, m) one can effectively construct an index e of a recursive function such that ∀n ∈ ω ϕ(n, {e}(n)) is provable. Thus we have an explicit method of witness and program extraction from proofs involving choice principles. As for the proof technique, this paper is a continuation of [32]. [32] introduced a selfvalidating semantics for CZF that combines realizability for extensional set theory and truth.
ZORN’S LEMMA AND SOME APPLICATIONS
"... Zorn’s lemma is a result in set theory which appears in proofs of some nonconstructive existence theorems throughout mathematics. We will state Zorn’s lemma below and use it in later sections to prove some results in linear algebra, ring theory, and group theory. In an appendix, we will give an app ..."
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Zorn’s lemma is a result in set theory which appears in proofs of some nonconstructive existence theorems throughout mathematics. We will state Zorn’s lemma below and use it in later sections to prove some results in linear algebra, ring theory, and group theory. In an appendix, we will give an application to metric spaces. The statement of Zorn’s lemma
KINNAWAGNER SELECTION PRINCIPLES, AXIOMS OF CHOICE AND MULTIPLE CHOICE
, 1995
"... of the KinnaWagner Selection Principle (KW), the Axiom of Choice (AC), and the Axiom of Multiple Choice (MC). The forms of these statements that we shall use are given below. AC: For every set x of of nonempty sets there is a function f (choice function) ..."
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of the KinnaWagner Selection Principle (KW), the Axiom of Choice (AC), and the Axiom of Multiple Choice (MC). The forms of these statements that we shall use are given below. AC: For every set x of of nonempty sets there is a function f (choice function)
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, 803
"... Abstract. We work in settheory without choice ZF. Denoting by AC(N) the countable axiom of choice, we show in ZF+AC(N) that the closed unit ball of a uniformly convex Banach space is compact in the convex topology (an alternative to the weak topology in ZF). We prove that this ball is (closely) con ..."
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Abstract. We work in settheory without choice ZF. Denoting by AC(N) the countable axiom of choice, we show in ZF+AC(N) that the closed unit ball of a uniformly convex Banach space is compact in the convex topology (an alternative to the weak topology in ZF). We prove that this ball is (closely) convexcompact in the convex topology. Given a set I, a real number p ≥ 1 (resp. p = 0), and some closed subset F of [0, 1] I which is a bounded subset of ℓ p (I), we show that AC(N) (resp. DC, the axiom of Dependent Choices) implies the compactness of F.