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?
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?