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1) Is EFA stronger than recursive algorithm? (This can be in term of proof theoretic ordinal, or whatsoever - to rephrase the question, are all problems that can be solved(and halt) by recursive algorithm expressible in EFA?

2) Is Friedman's grand conjecture proven?

3) Is it proven that there cannot be an algorithm that receives the proof that is not expressed in EFA form as an input and outputs a proof in EFA (for problems that have proofs expressible in EFA)?

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What is an EFA? –  filmor Oct 19 '12 at 10:16
    
Elementary function arithmetic –  W12 Oct 19 '12 at 10:22
    
any comment???? –  W12 Oct 20 '12 at 23:22
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Parts of the question make no sense and parts are ambiguous because "recursive algorithm" has different meanings in different contexts. The following might be relevant though: There are algorithms for which it is true but not provable in EFA that the algorithm halts on all inputs. In fact, there are algorithmically computable (total) functions $\mathbb N\to\mathbb N$ that grow asymptotically faster than any function given by an algorithm for which EFA can prove that it halts on all inputs. Furthermore, the same is true with far stronger theories, like ZFC, in place of EFA.

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