I'm seriously confused about absoluteness.
A formula in the language of a theory $T$ is absolute for $T$ structures if its truth value is the same in all standard transitive models of $T$ (this may not be the most general def, but lets go with it here - it's the def. Barwise uses in Admissible Sets and Structures, which gave rise to this question).
There's a theorem of Feferman Kreisel to the effect of:
if $\phi$ is absolute for $T$, then $\phi$ is $\Delta^T_1$
so far so good. But sentences are formulas too, and some of those will be theorems and hence true in all standard transitive models; but surely the above is not guaranteed for all theorems? e.g. it is a theorem that $\forall x \exists y [y = TC(x)]$, but there's no way of writing this to meet the above?
I'm sure I am deeply confused about something very simple. Please help!
Thanks,
Chris
If $T$ proves a sentence $\phi$ then $T$ proves $\phi$ is equivalent to every other sentence provable in $T$, including $(\exists x)[x=x]$ and $(\forall x)[x=x]$. We're not "rewriting" the formula in a strong sense, we're just finding a provably equivalent formula of a certain form.
The real application of absoluteness is to sentences and formulas that are not theorems of $T$.