The Axiom of Choice

Abstract

The Axiom of Choice was first enunciated by Zermelo. The standard formulation is as follows.

To choose one sock from each of infinitely many pairs of socks requires the Axiom of Choice, but for shoes the Axiom is not needed.—Bertrand Russell

The Axiom of Choice is obviously true; the well-ordering principle is obviously false; and who can tell about Zorn’s lemma?—Jerry Bona

The test of every religious, political, or educational system, is the man which it forms. If a system injures the intelligence it is bad. If it injures the character it is vicious, if it injures the conscience it is criminal.—Henri Frédéric Amiel

Every person thinks his own intellect perfect, and his own child handsome.—Sa’di

The Axiom of Choice allows you to take things out of sets that you never should have been allowed to put in there in the first place.—anon.

A choice function exists in constructive mathematics, because a choice is implied by the very meaning of existence.—Errett Bishop

Thought is subversive and revolutionary, destructive and terrible; thought is merciless to privilege, established institutions, and comfortable habit. Thought looks into the pit of hell and is not afraid. Thought is great and swift and free, the light of the world, and the chief glory of man.—Bertrand Russell

It is no exaggeration to say that a straightforward realistic approach to mathematics has yet to be tried. It is time to make the attempt.—Errett Bishop