A geometric form of the axiom of choice
|Abstract||Consider the following well-known result from the theory of normed linear spaces (, p. 80, 4(b)): (g) the unit ball of the (continuous) dual of a normed linear space over the reals has an extreme point. The standard proof of (~) uses the axiom of choice (AG); thus the implication AC~(w) can be proved in set theory. In this paper we show that this implication can be reversed, so that (*) is actually eq7I2valent to the axiom of choice. From this we derive various corollaries, for example: the conjunction of the Boolean prime ideal theorem and the Krein-Milman theorem implies the axiom of choice, and the Krein-Milman theorem is not derivable from the Boolean prime ideal theorem.|
|Keywords||No keywords specified (fix it)|
|External links||This entry has no external links. Add one.|
|Through your library||Only published papers are available at libraries|
Similar books and articles
Paul E. Howard (1973). Limitations on the Fraenkel-Mostowski Method of Independence Proofs. Journal of Symbolic Logic 38 (3):416-422.
G. P. Monro (1983). On Generic Extensions Without the Axiom of Choice. Journal of Symbolic Logic 48 (1):39-52.
Mitchell Spector (1988). Ultrapowers Without the Axiom of Choice. Journal of Symbolic Logic 53 (4):1208-1219.
Paul E. Howard, Arthur L. Rubin & Jean E. Rubin (1978). Independence Results for Class Forms of the Axiom of Choice. Journal of Symbolic Logic 43 (4):673-684.
Vivian Charles Walsh (1967). On the Significance of Choice Sets with Incompatibilities. Philosophy of Science 34 (3):243-250.
Mitchell Spector (1991). Extended Ultrapowers and the Vopěnka-Hrbáček Theorem Without Choice. Journal of Symbolic Logic 56 (2):592-607.
Paul Howard & Jean E. Rubin (1995). The Axiom of Choice for Well-Ordered Families and for Families of Well- Orderable Sets. Journal of Symbolic Logic 60 (4):1115-1117.
Lorenz Halbeisen & Saharon Shelah (2001). Relations Between Some Cardinals in the Absence of the Axiom of Choice. Bulletin of Symbolic Logic 7 (2):237-261.
Harvey Friedman (2000). Does Mathematics Need New Axioms? The Bulletin of Symbolic Logic 6 (4):401 - 446.
David W. Miller (2007). Some Restricted Lindenbaum Theorems Equivalent to the Axiom of Choice. Logica Universalis 1 (1).
John L. Bell, The Axiom of Choice. Stanford Encyclopedia of Philosophy.
Olivier Esser (2000). Inconsistency of the Axiom of Choice with the Positive Theory GPK+ ∞. Journal of Symbolic Logic 65 (4):1911 - 1916.
Added to index2010-12-22
Total downloads10 ( #106,150 of 548,969 )
Recent downloads (6 months)1 ( #63,511 of 548,969 )
How can I increase my downloads?