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Freeman Dyson has claimed that

$\not \exists m,n Reversed(2^n) = m ^ 5 $

(where Reversed(l) just is the reverse of the digits of l in base 10), is probably an example of an unproveable truth (source), and that even if it's not, there are many similar examples, some of which will be examples. As a heuristic argument, he says that a proof would have to rely on some pattern in the digits of powers of two, but those seem to be random.

Is this heuristic argument reasonable? I was surprised because I had thought that a search for undecidable arithmetic statements in Peano Arithmetic (let alone ZFC or something stronger) was rather challenging. But maybe that's only for (a) provably unprovable statements or (b) interesting statements. I'd tend to assume Dyson knows what he's talking about here, but am still curious.

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In this area, as in many others, guesses are easier than theorems. There are now a few more or less "natural" statements about the integers that have been shown not provable in Peano Arithmetic (modulo the usual consistency assumptions). Look for the Paris-Harrington Theorem. – André Nicolas Jun 10 '12 at 20:35
I suspect that Dyson doesn't know whether the statement is true or not (though in a sense he thinks it is very probably true) but thinks that there is no obvious way of proving it and so says it is probably not provable. – Henry Jun 10 '12 at 20:35
It's clearly a heuristic argument. Logically, you can't do any better, since we're talking about absolutely unprovable true statements, not just ones unprovable in a particular system. My question is whether it's a good heuristic argument. – user802500 Jun 10 '12 at 20:40
André, what's interesting about this is that Dyson's claim seems to be that it is probably undecidable in any reasonable system we might come up with. In contrast, the version of the Ramsey Theorem involved in the Paris Harrington theorem is decidable, just not in PA. – user802500 Jun 10 '12 at 20:43

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