# Arithmetisation of metamathematics in a general setting

### 's Introduction and Conundrum

This paper by Feferman seems to provide the answer, that it works if the theory you start with is as strong as PA. Now I can't be sure about that, because I'm just not that good a logician. I havn't actually read the paper properly, and if I had I still wouldn't have fully understood it.

It looks like its a corollary of:

 Theorem 6.2 let A= and S be axiom systems, where P S. Suppose that a numerates A in S. Then A is interpretable in S + {Con(a)}.

Where P is the peano axioms.

There is the problem that this is just about first order theories and so I have to guess whether there is any analogous result for logics in general.

### 's second conjecture

(the first one was that its possible to steal proof theoretic strength)
 This is how I really got to be writing these notes. I was reading this paper by Hartry Field ([Field97]) in which he dismisses a number of Gödelian arguments against extreme anti-objectivism concerning arithmetic. It seemed to me that Gödelian arguments, even if they work, are overkill. For me, first port of call (in defence of the objectivity of truth in arithmetic) is just arguing that the natural numbers and the successor operation form a well understood and uniquely defined structure. However, if you want something more, my next port of call would be w-consistency. Clearly an w-inconsistent extension of PA is not worth considering (i.e. its not really about the natural numbers any more, you've thrown away the intended interpretation). So the question arises, how much does this buy you? And I conjectured "everything" (i.e. true arithmetic is the only w-consistent completion of Peano arithmetic). And then I decided that this same result of Feferman (Theorem 6.2) could be used to prove it. I think I was wrong, but that made me include the reference to the paper, and then these notes. The conjecture and the incorrect "proof" of it using Theorem 6.2. is here.