TITLE: The set of Godel numbers of true sentences.
QUESTION [15 upvotes]: Tarski's Theorem on the undefinability of truth gives me a bit of a headache, and as a beginner I am still trying to grapple with its consequences. Here's a question.
Let $T$ be the set of Godel numbers of sentences $\sigma$ such that $V \models \sigma$. 
Now, I know that by Tarski's theorem, there is no formula that defines $T$ - that is to say, there is no first order formula $\tau (x)$ such that for each $n$, $n \in T \leftrightarrow V \models \tau (n)$. 
I can live with that, because obviously some sets are not definable. (Indeed, most are not definable.) But must $T$ exist somewhere in $V$, even if it not definable? The argument is not so obvious to me. Or will this depend on background assumptions? (Whether $V=L$, or whether this or that large cardinal exists, and so on ...)

REPLY [7 votes]: Allow me to complement the other answers by mentioning that Kelly Morse set theory KM proves the existence of a satisfaction predicate for first-order set-theoretic truth, and consequently in Kelly-Morse set theory you can prove the existence of your truth sets as desired.
A satisfaction predicate is a class $T$ of pairs $\langle \varphi,a\rangle$, which obeys the following Tarskian conditions:

(atomic truth) $T$ contains $\langle \varphi,\vec a\rangle$ for atomic $\varphi$ just in case $\varphi(\vec a)$ holds.
(Boolean combinations) $T$ contains $\langle \varphi\wedge\psi,\vec a\rangle$ just in case it also contains $\langle \varphi,\vec a\rangle$ and $\langle \psi,\vec a\rangle$, and similarly, it contains $\langle \neg\varphi,\vec a\rangle$ just in case it doesn't contain $\langle \varphi,\vec a\rangle$. 
(quantifiers) $T$ contains $\langle \exists x\varphi,\vec a\rangle$ if and only if there is some $b$ such that $T$ contains $\langle\varphi,(b,\vec a)\rangle$, where we place $b$ in the right position to be interpreted by variable $x$. 

A partial satisfaction predicate, in contrast, satisfies these requirements for all formulas up to a specified syntactic complexity. It is not difficult to prove that if $T$ is a partial satisfaction class for $\Sigma_n$ formulas, then one may define from it a partial satisfaction class for $\Sigma_{n+1}$ formulas, simply by applying one more step of the inductive definition of truth. Furthermore, one may prove by induction that if there is a partial satisfaction class for $\Sigma_n$ formulas, then it is unique. 
It follows from those observations that one may prove in KM by induction on the natural numbers that for every natural number $n$ there is a partial satisfaction class for $\Sigma_n$ formulas, and furthermore, that these partial satisfaction classes agree with each other, cohering into a full satisfaction class $T$ that works with all formulas.
In particular, in Kelly-Morse, you have your desired truth sets about first-order truth.
(Note that the induction that partial satisfaction classes exist is a second order claim, and although it can be proved by induction in KM, this part of the argument cannot be carried out in ZFC.)