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cousin_it comments on Logical Pinpointing - Less Wrong

62 Post author: Eliezer_Yudkowsky 02 November 2012 03:33PM

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Comment author: cousin_it 23 January 2018 10:33:15AM *  0 points [-]

How come we never see anything physical that behaves like any of of the non-standard models of first order PA?

Qiaochu's answer: because PA isn't unique. There are other (stronger/weaker) axiomatizations of natural numbers that would lead to other nonstandard models. I don't think that answer works, because we don't see nonstandard models of these other theories either.

wedrifid's answer: because PA was designed to talk about natural numbers, not other things in reality that humans can tell apart from natural numbers.

My answer: because PA was designed to talk about natural numbers, and we provably did a good job. PA has many models, but only one computable model. Since reality seems to be computable, we don't expect to see nonstandard models of PA in reality. (Though that leaves the mystery of whether/why reality is computable.)