Notions of compactness for special subsets of ℝ I and some weak forms of the axiom of choice

Journal of Symbolic Logic 75 (1):255-268 (2010)
  Copy   BIBTEX


We work in set-theory without choice ZF. A set is Countable if it is finite or equipotent with ${\Bbb N}$ . Given a closed subset F of [0, 1] I which is a bounded subset of $\ell ^{1}(I)$ (resp. such that $F\subseteq c_{0}(I)$ ), we show that the countable axiom of choice for finite sets, (resp. the countable axiom of choice AC N ) implies that F is compact. This enhances previous results where AC N (resp. the axiom of Dependent Choices) was required. If I is linearly orderable (for example $I={\Bbb R}$ ), then, in ZF, the closed unit ball of the Hilbert space $\ell ^{2}(I)$ is (Loeb-)compact in the weak topology. However, the weak compactness of the closed unit ball of $\ell ^{2}(\scr{P}({\Bbb R}))$ is not provable in ZF



    Upload a copy of this work     Papers currently archived: 89,311

External links

Setup an account with your affiliations in order to access resources via your University's proxy server

Through your library

Similar books and articles

On generic extensions without the axiom of choice.G. P. Monro - 1983 - Journal of Symbolic Logic 48 (1):39-52.
A note on some weak forms of the axiom of choice.Gary P. Shannon - 1991 - Notre Dame Journal of Formal Logic 33 (1):144-147.
Dependent Choices and Weak Compactness.Christian Delhommé & Marianne Morillon - 1999 - Notre Dame Journal of Formal Logic 40 (4):568-573.
Sequential compactness and the axiom of choice.Norbert Brunner - 1983 - Notre Dame Journal of Formal Logic 24 (1):89-92.
The isomorphism property versus the special model axiom.Renling Jin - 1992 - Journal of Symbolic Logic 57 (3):975-987.
The axiom of choice and combinatory logic.Andrea Cantini - 2003 - Journal of Symbolic Logic 68 (4):1091-1108.


Added to PP

70 (#208,785)

6 months
5 (#237,114)

Historical graph of downloads
How can I increase my downloads?

References found in this work

The Axiom of Choice.Gershon Sageev - 1976 - Journal of Symbolic Logic 41 (4):784-785.
Products of compact spaces in the least permutation model.Norbert Brunner - 1985 - Mathematical Logic Quarterly 31 (25‐28):441-448.
Products of Compact Spaces in the Least Permutation Model.Norbert Brunner - 1985 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 31 (25-28):441-448.
The Compactness of 2^R and the Axiom of Choice.Kyriakos Keremedis - 2000 - Mathematical Logic Quarterly 46 (4):569-571.

Add more references