# Nisan comments on Exponentiation goes wrong first - Less Wrong

10 14 December 2010 04:13AM

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Comment author: 15 December 2010 03:14:52AM *  0 points [-]

There is just no self-consistent assignment of probabilities possible.

Really? Do you have an example? I'd be interested in seeing one.

(What do you mean here by "self-consistent"?)

Comment author: 15 December 2010 03:53:22AM *  0 points [-]

Really? Do you have an example? I'd be interested in seeing one.

Making this statement more precise is difficult. To make things easy I'm going to assume that probabilities live between 0 and 1 and that our general system of axioms is first order R with first order Q.

Now, our probability assignments are consistent iff they don't lead to a contradiction. So, assign our probabilities, then whatever you do, I note that by Robinson's theorems, I can describe PA using our axioms (this isn't precisely true but is close enough to be true for our purposes.) I then invoke the contradiction we have in PA. No matter what initial probability assignment you choose I can do this.

Note that this argument might not work if we just have first order reals as our axiomatic system because that's not sufficient to define PA in R, assuming that PA is consistent (to see this note that we have a decision procedure for first order reals and so if we could we'd have a way of deciding claims in PA.). I don't think there's any way to import contradictions in PA into first order R and I suspect this can't be done in general. (I suspect that I'm missing some technical details here so take this with a handful of salt.)