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Axiom of Choice, quotes *June 1, 2008*

*Posted by Alexandre Borovik in Uncategorized.*

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- The Axiom of Choice is obviously true, the Well-ordering theorem obviously false, and who can tell about Zorn’s lemma ?” ::— Jerry Bona -(This is a joke: although the axiom of choice, the well-ordering principle, and Zorn’s lemma are all mathematically equivalent, most mathematicians find the axiom of choice to be intuitive, the well-ordering principle to be counterintuitive, and Zorn’s lemma to be too complex for any intuition.)
- “The Axiom of Choice is necessary to select a set from an infinite number of socks, but not an infinite number of shoes.” ::— Bertrand Russell -(The observation here is that one can define a function to select from an infinite number of pairs of shoes by stating for example, to choose the left shoe. Without the axiom of choice, one cannot assert that such a function exists for pairs of socks, because left and right socks are (presumably) identical to each other.)
- “The axiom gets its name not because mathematicians prefer it to other axioms.” ::— A. K. Dewdney -( This quote comes from the famous April Fool’s Day article in the computer recreations column of the Scientific American , April 1989.)

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Tarski told me the following story. He

tried to publish his theorem ( |X|=|X*X| -> AxiomOfChoice ) in the

Comptes Rendus Acad. Sci. Paris but Fréchet and

Lebesgue refused to present it. Fréchet wrote that

an implication between two well known propositions

is not a new result. Lebesgue wrote that an

implication between two false propositions is of no

interest. And Tarski said that after this misadventure

he never tried to publish in the Comptes Rendus.

There is an interesting pespective in The Principles of Mathematics Revisited

by Jaakko Hintikka – 1996, saying that the axiom of choice is fine, but we have to pay attention to the nature of the choice functions. Saying that it just exists in some metaphysical sense is not saying much.