[FOM] Sigma-1 Completeness
rgheck at brown.edu
Fri Aug 3 15:22:08 EDT 2007
Q, and even PA, is \Sigma_1 complete. My question concerns what theories
know about their own \Sigma_1 completeness. We can distinguish two
versions of this question:
(i) A theory T might be able to show, for EACH \Sigma_1 sentence A,
that, if A, then T proves A.
(ii) A theory T might be able to show that, for EVERY \Sigma_1 sentence
A, if A, then T proves A.
What is the weakest theory T with each property---if such there is?
I'm sure this is well-known to many of you, but I can't seem to find the
answer. I'm guessing that maybe the answer to (ii) is I\Sigma_1.
Richard G Heck, Jr
Professor of Philosophy
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