TITLE: Can we prove set theory is consistent?
QUESTION [34 upvotes]: Disclaimer
Of course not, I'm aware of Gödel's second incompleteness theorem. Still there is something which does not persuade me, maybe it's just that I've taken my logic class too long ago. On the other hand, it may turn out I'm just confused. :-)
Background
I will be talking about models of set theory; these are sets on their own, so a confusion can arise, since the symbol $\in$, viewed as "set belonging" in the usual sense, may have a different meaning from the symbol $\in$ of the theory. So, to avoid confusion, I will speak about levels.
On the first level is the set theory mathematicians use all day. This has axioms, but is not a theory in the usual sense of logic. Indeed, to speak about logic we already need sets (to define alphabets and so on). In this naif set theory we develop logic, in particular the notions of theory and model. We call this theory Set1.
On the second level is the formalized set theory; this is a theory in the sense of logic. We just copy the axioms of the naif set theory and take the (formal) theory which has these strings of symbols as axioms. We call this theory Set2.
Now Gödel's result tells us that if Set2 is consistent, it cannot prove its own consistence. Well, here we need to be a bit more precise. The claim as stated is obvious, since Set2 cannot prove anything about the sets in the first level. It does not even know that they exist.
So we repeat the process that carried from Set1 to Set2: we define in Set2 the usual notions of logic (alphabets, theories, models...) and use these to define another theory Set3.
A correct statement of Gödel's result is, I think, that

if Set2 is consistent, then it cannot prove the consistence of Set3.

The problem
Ok, so we have a clear statement which seems to be completely provable in Set1, and indeed it is. This doesn't tell us, however that

if Set1 is consistent, then it cannot prove the consistence of Set2.

So I'm left with the doubt that what one can do "from the outside" may be a bit more than what one can do in the formalized theory. Compare this with Gödel's first incompleteness theorem, where one has a statement which is true for natural numbers (and we can prove it from the outside) but which is not provable in PA.
So the question is:

is there any reason to believe that Set1 cannot prove the consistence of Set2? Or I'm just confused and what I said does not make sense?

Of course one could just argue that Set1, not being formalized, is not amenable to mathematical investigation; the best model we have is Set2, so we should trust that we can always "shift our theorems one level". But this argument does not convince me: indeed Gödel's first incompleteness theorem shows that we have situations where the theorem in the formalized theory are strictly less then what we can see from the outside.
Final comment
In a certain sense, it is far from intuitive that set theory should have a model. Because models are required to be sets, and sets are so small...
Of course I know about universes, and how one can use them to "embed" the theory of classes inside set theory, so sets may be bigger than I think. But then again, existence of universes is not provable from the usual axioms of set theory.

REPLY [51 votes]: Your question certainly makes sense and it is a point that I feel is too often glossed over in textbooks.
Let me rephrase your question.  Goedel's second theorem says that, assuming that a certain formal system (ZFC, say) has a certain property that we call "consistency," then there is no formal proof in ZFC of a certain string, commonly denoted by "Con(ZFC)."  Fine.  But why on earth should this theorem say anything about whether the consistency of ZFC can be proved mathematically?  The theorem is just a theorem about abstract strings of symbols, not about what human beings can and cannot do.  The string denoted "Con(ZFC)" is commonly taken to "say" that "ZFC is consistent," but what is the justification for doing so?  A string is just a string, and doesn't "say" anything.  If we choose to think of the string as "meaning" something then that's our business, but surely that kind of human social activity is not something we can prove mathematical theorems about?
The answer is that, underlying the usual discussions of Goedel's second theorem, there is the following Key Assumption: If someone were to come up with a mathematical proof of the consistency of ZFC, then by mimicking that proof, we could produce a formal proof of Con(ZFC) from the axioms of ZFC.  The Key Assumption is crucial.  Without it, we cannot make the leap from Goedel's second theorem to a meta-mathematical statement about the (im)possibility of proving the consistency of mathematics.  And note that the Key Assumption is not a purely mathematical one; it cannot be, because it is a statement linking something that is not purely mathematical (namely, mathematical proof, which is a product of human activity) and something that is purely mathematical (namely, ZFC and theorems of ZFC).  Therefore the Key Assumption is not susceptible to mathematical proof, and the reasons we have for accepting it must be in part philosophical.
So what reasons do we have for accepting the Key Assumption?  The chief reason is that long experience has taught us that all mathematical proofs that mathematicians come up with can indeed be mimicked by formal proofs in ZFC.  This may seem obvious to us today, but it is not at all a trivial statement.  Prior to the set-theoretic revolution, it was by no means obvious that all the diverse areas of mathematics could be formulated in a single common language (i.e., set theory) and deduced from a short list of axioms.  It is only through the hard work of those working in the foundations of mathematics that we now take for granted that for any precise mathematical statement we want to make, there exists a formal sentence $S$ in the first-order language of set theory with the property that any mathematically acceptable proof of the original mathematical statement can be mimicked to produce a formal proof of $S$ from the axioms of ZFC.  And if you had any lingering doubts about whether this formal mimicry existed only in theory and not in practice, then in recent years, the advent of formal theorem-proving software such as Mizar, HOL Light, Coq, Isabelle, etc., should have swept away such doubts by demonstrating concretely that large areas of mathematics can be mimicked formally in practice, and not just in theory.
Finally, let me mention that although I believe it is very reasonable to accept the Key Assumption, it is possible to reject it.  Perhaps most notably, the philosopher Michael Detlefsen has challenged the standard claim that the string Con(ZFC) properly mimics the statement "ZFC is consistent" in the sense of the Key Assumption, and has suggested that Hilbert's program to prove the consistency of mathematics is not yet dead.  I believe that Detlefsen is simply mistaken and that there is nothing unsatisfactory about the standard string Con(ZFC), but he is at least correct that there is something to be checked here, and it is not a purely mathematical point but a partially philosophical one.