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cousin_it comments on What's a "natural number"? - Less Wrong
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Comments (17)
komponisto, did you leave my question unanswered because you don't know the answer, or because you thought the question stupid and decided to bail out? If you can dissolve my confusion, please do.
Sorry! I didn't have an answer immediately, but thought I might come up with one after a day or two. Unfortunately, by that time, I had forgotten about the question!
Anyway, the way I'd approach it is to ask what is wrong, from our point of view, with a given nonstandard theory.
Actually, I just thought of something while writing this comment. Take your example of adding a "PA is inconsistent" axiom to PA. Yes, we could add such an axiom, but why bother? What use do we get from this new system that we didn't already get from PA? If the answer is "nothing", then we can invoke a simplicity criterion. On the other hand, if there is some situation where this system is actually convenient, then there is indeed nothing "wrong" with it, and we wouldn't want an AI to think that there was.
(Edit: I'll try to make sure I reply more quickly next time.)