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dankane comments on Second-Order Logic: The Controversy - Less Wrong
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Eliezer - why is it that you seem to adamant that our physical reality is better modeled by the standard natural numbers rather than some supernatural model? If we try to see whether a physical implementation of a Turing machine provides an inconsistency in PA, then of course it won't because it will run for fewer than Ackermann(20) iterations, and all number theories say that it won't halt by then. If we instead lived in something that looked like the game of life and we did expect it to last infinitely long why exactly would we expect time steps to be indexed by some particular model of the natural numbers when we haven't yet seen more than finitely many of them (and those finitely many are predicted in all models). Claiming that we don't expect the Turing machine to ever halt because it hasn't yet seems a little like saying that all times I've seen come before 2014, and therefore I expect that all times ever will have this property.
Actually, here's maybe a better way of saying what I'm trying to get at here: What evidence would actually convince you that we lived in a universe given by a non-standard model of the natural numbers?
Before you say "I run a computer program which returns a proof of 0=1 in PA", think about it for a while. Ignoring the fact that in practice, you would probably suspect that the proof is wrong, might you instead take this as evidence that PA is actually inconsistent. After all, Godel tells us that we can't really be sure about that. In fact, if you did live in a nonstandard model of PA, and found a nonstandard proof of 0=1, wouldn't it feel from the inside like you found an inconsistency in PA rather than that you found a nonstandard number?