Saturday, May 16, 2009

Pro-Choice vs Pro-Life

Surely, Facebook gives us a choice here:

Facebook: I support the right to choose one element from each set in a collection
Do you believe that an infinite product of nonempty sets should be nonempty? Do you feel that non-measurable subsets of the reals should exist? Or games of perfect information with no winning strategy for either player? Do you believe that every set should have a well-ordering, or that any poset in which every chain has an upper bound is entitled to a maximal element? Do nonzero rings have the right to a maximal ideal? Is every vector space entitled to a basis? Should fields have algebraic closures? Should products of compact topological spaces be compact, and countable unions of countable sets be countable? Do you want to be able to cut a sphere up into a finite number of pieces and reassemble them, with only rigid motions, into a sphere twice as large?

It's all possible if you're pro-axiom-of-choice!


Facebook: The Axiom of Life (aka Negation of Axiom of Choice)
Do you have nightmares of being split apart, reassembled, and finding two of yourself? Do you wonder just how large is a non-measurable set, roughly speaking? Then this is the group for you, advocating the negation of AC (or at least replacement by the axiom of dependent choice if you actually need to prove a theorem for some reason).

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