Originally Posted by

**bram28** Thanks! I think I got it: I think I was trying to show that there is no *sound* (i.e. true under the standard interpretation) and recursive theory that is complete for arithmetic, but Godel's Incompleteness Theorem states that there is no *consistent* (i.e. you never have P and ~P as elements of the theory) and recursive theory that is complete for arithmetic. Indeed, in my last paragraph of the original post I used 'consistent' when I should have used 'sound'. It still leaves me wondering though: isn't soundness the more 'interesting' property? I mean, I have always interpreted the Godel results as there not being any way to axiomatize arithmetic, i.e. that is was impossible to have a finite (and, more generally, recursive) set of first-order logic axioms for arithmetic from which all arithmetical truths can be derived, and for that, isn't it enough to say that there is no sound and recursive theory that is complete? And wouldn't that be easier to show? (again, because you wouldn't need to go into notions like representability_in_a_theory or definability_in_a_theory?)