Comment on The Pascal's Wager Fallacy Fallacy

Tue, 15 Jan 2013 11:47:06 GMT

The set of all definable objects is countable, for obvious reasons.

With this definition, your diagonal trick actually doesn't work (which is good, because otherwise we'd have a paradox): Definability isn't a notion expressible in the theory itself, only in the metatheory. Hence if you attempt to "define" something in terms of the set of all definable numbers, the result is only, uh, "metadefinable".

We could similarly argue that the definable objects should be thought of as "meta-countable" rather than countable, right? The reals-implied-by-a-theory would always be uncountable-in-the-theory. (I'm tempted to imagine a world in which this ended the argument between constructivists and classicists, but realistically, one side or the other would end up feeling uneasy about such a compromise... more likely, both.)

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