Douglas_Knight comments on No one knows what Peano arithmetic doesn't know - Less Wrong

17 Post author: cousin_it 16 December 2011 09:36PM

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Comment author: Douglas_Knight 17 December 2011 01:23:48AM 0 points [-]

For you final question, I find it hard to imagine a notion of "formal system" that does not just mean "language." Then an RE formal system is the same as an RE language. If you insist on the terminology of proofs, you can take the words of the language as axioms and not have any rules of inference.

Comment author: cousin_it 17 December 2011 02:06:14AM 0 points [-]

For your final question, I find it hard to imagine a notion of "formal system" that does not just mean "language."

How about http://en.wikipedia.org/wiki/First-order_logic#Syntax ?

Comment author: Douglas_Knight 17 December 2011 02:17:13AM *  0 points [-]

So you agree that "provability oracle for an RE formal system" is the same as "membership oracle for an RE language" and your question is trivial?

ETA: No, first order languages does restrict the set of languages. But I object to this usage. "Formal systems" should include more general systems.

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