[FOM] Sigma-1 Completeness

[Richard Heck]

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

-- 
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Richard G Heck, Jr
Professor of Philosophy
Brown University
http://frege.brown.edu/heck/
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