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Talk:Axiom of Choice (view source)
Revision as of 23:38, November 27, 2008
, 23:38, November 27, 2008→My edits: And an explanation of why Foxtrot was apparently wrong to ban AndyJM.
The Axiom of Choice is not equivalent to the existence of non-Lebesgue measurable subsets of the real line and so I have removed this claim from the article. Intuitively one would not expect these to be equivalent. The existence of a non-Lebesgue measurable subset of the real line only says something about cardinals up to and including the continuum. However the Axiom of Choice makes a statement about all cardinals (in fact it is typically invoked when describing cardinals). [[User:AndyJM|AndyJM]] 14:27, 27 November 2008 (EST)
The Axiom of Choice is not equivalent to the existence of non-Lebesgue measurable subsets of the real line and so I have removed this claim from the article. Intuitively one would not expect these to be equivalent. The existence of a non-Lebesgue measurable subset of the real line only says something about cardinals up to and including the continuum. However the Axiom of Choice makes a statement about all cardinals (in fact it is typically invoked when describing cardinals). [[User:AndyJM|AndyJM]] 14:27, 27 November 2008 (EST)
:I got curious and actually looked into this. The Axiom of Choice does imply the existence of non-Lebesgue measurable subsets of the real line. However, the existence of non-Lebesgue measurable subsets of the real line doesn't imply the Axiom of Choice. I found a paper called [http://matwbn.icm.edu.pl/ksiazki/fm/fm138/fm13812.pdf "The Hahn-Banach theorem implies the existence of a non-Lebesgue measurable set"] and will quote the abstract for those who don't want to download a 500+kb document:
{{cquote|'''Abstract.''' In this paper we show that the Axioms of Zermelo-Fraenkel set theory together with the Hahn-Banach theorem imply the existence of a non-Lebesgue measurable set. Our construction does not make any use of the Axiom of Choice.}}
:So the other direction of the claim is apparently wrong, and AndyJM was right to remove it. He was blocked for "removing information from articles" by the same user who also re-inserted the apparently wrong claim. This is exactly why I oppose this "Block first" approach: How can we improve this encyclopedia when we can get blocked and banned by whoever ''thinks'' we're wrong? I suggest an unblock. --[[User:AlanS|AlanS]] 18:38, 27 November 2008 (EST)