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Nov 27, 2010 at 23:17 comment added Henry Towsner I believe the theories I mentioned qualify. They don't contain quantifiers, so all statements, including intermediate steps in a proof, are quantifier free.
Nov 27, 2010 at 23:16 history edited Henry Towsner CC BY-SA 2.5
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Nov 27, 2010 at 22:11 comment added Lucas K. Thanks for the answer and it helps me further, but I am actually looking what happens if you really limit the logic. So, any intermediate result of a theorem must be $\Pi_2$
Nov 24, 2010 at 1:27 comment added Henry Towsner I should add that, in general, a huge portion of proof theory is precisely concerned with the behavior of $\Pi^0_2$ sentences.
Nov 24, 2010 at 1:25 history answered Henry Towsner CC BY-SA 2.5