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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 …
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 …
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 …
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 …