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cousin_it comments on The Pascal's Wager Fallacy Fallacy - Less Wrong
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There's some weirdnesses down that route - for example, it turns out that you can't distinguish zero from nonzero, so the step function is actually uncomputable.
My contrarian claim is that everyone could live with the nameable numbers - that is, the numbers that can be pointed out using a finite number of books to describe them. People who really strongly care about the uncountability of the reals have a hard time coming up with a concrete example of what they'd miss.
Of course, whether a number is definable or not depends on the surrounding theory. Stick to first-order theory of the reals and only algebraic numbers will be definable! Definable in ZF? Or what?
EDIT Apr 30: Oops! Obviously definability depends only on the ambient language, not the actual ambient theory... no difference here between ZF and ZFC...