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For question in Proof Theory, where "proofs" themselves are the object of mathematical investigation. It is not to be used to request a proof of some result.

2 votes
Accepted

Is adding all sentences true of terms in skolemized theory conservative?

So I can mark this answered (when I can tmw) I'm posting Emil Jeřábek's comment as an answer but they deserve all the credit. Suppose that $(\forall x)\phi(x)$ is in $T' - T_S$. That means we've pro …
Peter Gerdes's user avatar
  • 3,029
2 votes
1 answer
153 views

Is adding all sentences true of terms in skolemized theory conservative?

Suppose I have a (incomplete) theory $T$ (e.g. PA) which I skolemize to get a theory $T_S$ in the expanded language. I now build $T'$ by adding to $T_S$ any sentence $(\forall x)\phi(x)$ where I can …
Peter Gerdes's user avatar
  • 3,029
4 votes
1 answer
409 views

Model of PA with false $\Sigma^0_1$ sentence but no false Con sentence?

This is probably a really basic result that I'm forgetting but if $M \models \text{PA}$ and $M \models \phi$ for some $\Sigma^0_1$ sentence $\phi$ such that $\mathbb{N} \models \lnot \phi$ does it fol …
Peter Gerdes's user avatar
  • 3,029
2 votes
0 answers
249 views

A formal definition of a useful theorem?

Sorry if this feels a bit squishy, but I'm wondering if there is any published work trying to give a fully formal definition of the notion of a useful theorem. I mean, in mathematics we all know that …
Peter Gerdes's user avatar
  • 3,029