Set of all true first-order statements about set Today a TA claimed that "set of all true first-order statements about sets" is actually a set. After searching around on the net, all I can find is this question on this site, which is about completely different things. So, can anyone give me a proof that this is a set?
EDIT: Or if this is definitely not a set, can someone give me a proof?
EDIT: if you don't believe in the existence of a world we live in so that the notion of truth is meaningful, the question can be rephrase as follow: let $M$ be a model of ZFC, and for each statements (in the first-order language for ZFC) that is true about $M$, we can compute the Godel encoding of such statements in $M$ (where the computation is carried out according to $M$). Now is there an element of $M$ such that (according to $M$) it contains exactly those Godel encoding as above?
EDIT: Let me clarify this, since some people are confused about the crux of the problem and confuse this with a basic question. We are looking for a set with some properties about its elements. Specifically, a set $T$ such that $\sigma\in T$ if and only if $\sigma$ is a true statement. Generally, specifying properties of elements are not sufficient to guarantee the existence of a set; for example the "set" $S$ of all natural number $x$ such that $x\notin S$ does not exist. However, in standard set theory, the axiom schema of restricted comprehension does in fact guarantee the existence of a set $S$ of all elements $a\in A$ that satisfy a first-order property $\phi(x)$. So the obvious attempt to solve this problem is to find a first-order formula $TRUE(\tau)$ that pick out all the true statement. But by Tarski's undefinability of truth, no such formula exist. So if this can be proven to be a set, you need to do something clever. In fact, you might need some sort of proof by contradiction otherwise you will run into the undefinability of truth still. But my feeling is that this one is not a set, but I can't prove it either.
Thank you.
 A: Some models of ZFC contain their own theories as a set, and some do not.
If there is an inaccessible cardinal $\kappa$, for example, then an easy Löwenheim-Skolem argument shows that there are many ordinals $\lambda$ where $V_\lambda\prec V_\kappa$, an elementary substructure. Thus, the theory of $V_\kappa$ is the same as the theory of $V_\lambda$, which is an element of $V_\kappa$, since $V_\lambda$ is a set structure in $V_\kappa$, which can form its theory. So $V_\kappa$ is a model of ZFC containing its own theory as an element.
Meanwhile, if there is a transitive model of ZFC, then there is a least such model, the Shephardson-Cohen model, the least $L_\alpha$ that is a model of ZFC. This model is pointwise definable, meaning that every element of it is definable without parameters. But no pointwise-definable model of ZFC can have its own theory as an element, (similar to Tarski's theorem on the non-definability of truth) since if it did this theory would be definable and we would be able to use the fixed-point theorem to find a sentence asserting it's own falsity, which is a contradiction. Alternatively, if a model is pointwise definable, then the theory of the model contains the definitions of all the various ordinals of the model, and the theory tells you how to order them, and so the model would realize that every ordinal is countable, contrary to ZFC. Thus, one can see that no $\omega$-standard Paris model (a model in which every ordinal is definable without parameters) can contain its own theory as an element.
The previous examples are both involving transitive models of ZFC, which is the best kind of case, for we have an agreement between the $\omega$ of the model and in the meta-theory. But let me point out that the concept of "containing your own theory" is problematic for models that are not $\omega$-standard, and one would have to say more about what was meant for this case. The problem is that the theory of a model is defined externally to the model, and consists of entirely standard sentences in the meta-theory. But no $\omega$-nonstandard can have such a set inside it. One might ask for a set in the model whose standard part is the theory, and this is possible, even for models of ZFC that think $\neg$Con(ZFC). The reason is that every model of ZFC has an elementary extension $\bar M$, such that any desired set of natural numbers is coded as the standard part of a set in the model $\bar M$. And so every model of ZFC has an elementary extension to a model $\bar M$ of ZFC, such that the theory of $\bar M$ is coded as an element of $\bar M$. 
A: Basically the issue is that the claim is ill-defined. Truth is always relative to a structure. You can ask for truth of some sentence in some structure, but never absolute truth of any sort. So if the claim was "There is a set of all first-order sentences of set theory that are true in $M$, where $M$ is a model of ZFC." then ZFC itself cannot prove this claim, and hence he is essentially wrong.
There is a way to axiomatize truth without structures, but it is certainly not what the TA you quote is doing, and even then any theory of truth that has sufficient arithmetic cannot have a computable deductive system. I can make my claims in this paragraph precise on request.

Now that you've added the second "EDIT" section, your question here is clear and is answered by JDH's answer. I just want to emphasize that the answer is "It depends on $M$." (which is precisely why I said the TA's claim is wrong), since if it did not depend on $M$ then the existence of such a set would be either provable or disprovable in ZFC itself. If it is provable, then ZFC proves Con(ZFC), and so ZFC is inconsistent. If it is disprovable, then ZFC proves the non-existence of a model of ZFC that has the full power-set of the naturals, which set theorists who subscribe to ZFC do not believe. Either way, the TA's claim cannot be proven as far as is known today.
A: To talk about "all true sentences" you need not merely a model. You also need to specify a theory out of which you are going to pick out true sentences.  Without such a step it is easy specify a counterexample to what you report as your TA's claim.  Thus, consider the collection of statements "$X$ is a set" where $X$ ranges over all sets. This collection is not a set.
