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shminux comments on Consistent extrapolated beliefs about math? - Less Wrong Discussion
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Do you conflate the mathematical meaning of "existence" with the colloquial one? As others mentioned, what happens if you replace the colloquial "exist" with something like "is accurate" or "is useful"? After all, integers are a mathematical model, not a physical object, and models are ultimately judged by their usefulness and accuracy. For example, the objects in the naive set theory may not exist in the mathematical sense, but they still appear in this model (of limited but non-zero usefulness and accuracy).
My post is dealing with the question whether human beliefs about the integers can be covered by some specific formal theory, or all formal theories are inadequate for that purpose. I've seen many people argue for the latter point of view, in the post I try to argue for the former.
Ah, I see. I misunderstood your point. You don't worry about your current beliefs, you worry about the extrapolated ones. How do you see this represented? An algorithm, which, when fed a theorem, outputs a proof/disproof/proof of undecidability? Or something that spits all possible true statements, unprompted, never shutting up?
Yeah. Judging from the comments, it looks like most people misunderstood the point, though the point is right there in the title :-( I think the result of extrapolation should be an algorithm that spits out all true statements, or an algorithm that checks proofs. These are equivalent, and known as "effectively generated" or "recursively axiomatizable" theories. It can't be an algorithm that answers provable/disprovable/independent, because that's impossible even for PA, and I'd like to think that I believe in more than just PA.