# A question about set theory and Frege logic

Does there exist a very weak axiomatic theory of arithmetic-weaker than (but possibly a sub-theory of) Robinson's theory Q-which can be interpreted in the first order fragment of Frege logic? If so, what are the axioms of such a theory? By "the first order fragment of Frege logic" I mean the system discussed on pages 251-252 of the book "One Hundred years of Russell's Paradox" (edited by Godehard Link).

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I don't have a copy of this book. Is the list of axioms too long to post here? –  François G. Dorais May 3 '10 at 21:54
How weak will you allow? For example, the empty theory? After all, Q is already extremely weak (the weakest known theory to support the incompleteness theorem). –  Joel David Hamkins May 3 '10 at 23:08
Presburger arithmetic is weaker in some sense than Q, but I don't know if one can interpret Frege's FOL in it. See math.stackexchange.com/questions/4107/… –  Carlo Von Schnitzel Oct 10 '10 at 7:52
My question is actually the reverse of this. Can one interpret Presburger Arithmetic in Frege's FOL? –  Garabed Gulbenkian Jun 3 '11 at 20:29