Apologies for the mis-sent post. Somehow the To: line must have changed just before I hit send, due to some overhelpfulness of gmail. Some math (or at least philosophy of math) content in recompense... On Thu, May 28, 2009 at 11:57 AM, Mike Speciner <ms@alum.mit.edu> wrote:

Intuitively, if I'm prisoner j, the probability that the chosen equivalence class representative agrees with reality starting at or before my position (j) is 0, and so my chance of being correct is still 1/2.

But equally intuitively, if there were 50 who got their guess right, and 10 who got it wrong, your chance of being correct would be 1/6. And in this case, the number of people who get it right is more than 1000 times as great as the number who get it wrong, so your chance of getting it right is > 99.9%. As one of the commenters on the web page points out, the reason these two intuitions don't match up is that the intuition that says they should get the same result is based on using Fubini's theorem on the probability measure, and this doesn't work because we're dealing with non-measurable sets here. I guess the reason this paradox bothers me (and the author of the web page) more than the Banach-Tarski paradox is that we are comfortable with non-measureable sets in the domain where measure=volume, but we have an intuition that everything has a probability, so non-measurable sets in a probability measure seem wrong. One could ask what the expected value is for how far along the sequence I have to look to match with any of the chosen representatives, and I'd guess that if that expected value exists it would have to be infinity. Oh, and I agree with Jim about axiomatizing, rather than constructing, the reals. All the construction does is show that if integers and set theory are consistent, than so is the theory of the reals. But I'm much more confident of the existence of the reals than I am of the existence of a lack of contradiction in ZF, so what does this really buy me? To put it another way, if someone *did* find a contradiction in ZF, no-one would stop using the reals; they would just look for a new "foundation", while still assuming that when one was found, it would allow the construction of a set of real numbers that had the properties we know and love. Andy

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