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Sniffnoy comments on Gödel and Bayes: quick question - Less Wrong
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Comments (36)
You cannot get number theory out of the first-order theory of the reals; how would you write a first-order predicate to tell you whether or not a real number was an integer?
I don't see what least upper bounds have to do with anything. Again, Gödel's theorem deals with systems that can model number theory (i.e. the integers, the whole numbers); it has nothing to do with the real numbers.
The Wikipedia article and this shorter account both say that some form of Gödel's incompleteness theorem applies to second-order logic. I asked about the limits of the first-order approach to the reals because it looks like we'd need to use that if we want to stop the theorem from applying.
That approach still seems odd, but I can sort of see how you could do probability that way. I'll edit the OP to reflect my real question as soon as I feel up to it.