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NoSuchPlace comments on Consistent extrapolated beliefs about math? - Less Wrong
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That is true, however you don't appear to be asserting the consistency of your beliefs, you are asserting the consistency of a particular subset of your beliefs which does not contain the assertion of its consistency. This is not in conflict with Gödel's incompleteness theorem which implies that no theory may consistently assert its own consistency. Gödel's incompleteness theorem does not forbid proofs of consistency by more powerful theories: for example there are proofs of the consistency of Peano arithmetic
Yeah, that's a fair point. If I believed the sentence "my beliefs about the integers are consistent", it would be a pretty complicated sentence about the integers, containing an encoding of itself by the diagonal lemma. Maybe you're right that I don't actually believe that, not even intuitively. I just believe a bunch of other sentences, and believe that they are consistent. That would agree with the conclusion of the post, that my beliefs about the integers (both actual and extrapolated) can be covered by some specific formal theory.
I think we can make this more precise. Second-order arithmetic theoretically covers what I believe about the natural numbers. We then need a theory of sets or collections to interpret it. Gaifman, in his discussion of non-standard models, suggests that we go meta instead, and say that whenever we learn a new formula we expand the principle of induction to include this formula. In both cases, the open-ended part comes less from arithmetic than from some larger framework.
So: I don't think my larger 'theory' is necessarily self-consistent. I do think there exists some consistent intermediate theory which includes the consistency of second-order PA, and the uniqueness of its model up to isomorphism.
If we define phi+1 as the union of the statement(s) phi and the assertion of phi's consistency, someone might possibly doubt the consistency of PA+omega, since the concept "omega" need not appear in first-order PA. I find myself hesitating slightly about PA+the Church-Kleene ordinal, although ultimately I think many large cardinals are consistent with(in) set theory. And yes, I think at this point we should start assigning probabilities to mathematical statements.