What's the big deal with Gödel's second incompleteness theorem?

Question

Edit: My question is specifically about Gödel's second incompleteness theorem. I get the significance of his first incompleteness theorem, which is of course completely amazing.

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According to the Wikipedia entry on Gödel's second incompleteness theorem, "the broadly accepted natural language statement of the theorem is" as follows.

For any formal effectively generated theory T including basic arithmetical truths and also certain truths about formal provability, if T includes a statement of its own consistency then T is inconsistent.

I accept that the above theorem is mathematically useful (say, for proving the inconsistency of a system). But why is it philosophically interesting? Assume it was false. Suppose there was a theory that was consistent, and also included the basic arithmetical truths that Gödel's theorem speaks of, and could also prove its own consistency. For concreteness, suppose ZFC could do this.

Well, so what? It's clearly circular.

Now I'm not doubting the mathematical usefulness of Gödel's theorem. But my question is, on a philosophical level, what's all the commotion about?

Answer 1

You observe, correctly, that just because a formal system S "asserts" its own consistency — by means of a proof which, in a meta-language M, is isomorphic to a proof of consistency of S — does not mean that you should therefore trust S to be consistent. Any inconsistent system which is rich enough to admit Gödel numbering (or an equivalent technique), and which has an explosive implication (so that everything follows from a falsehood), is able to prove its own consistency; although it would be interesting to know whether or not it allows you to derive "consistency claims" without passing through blatant contradictions of the form A & ¬A to do so.

The big deal with Gödel's Second Incompleteness theorem is that the only formal systems which can "prove" their own consistency via encoding in Peano Arithmetic (or an equivalent system) — and which is also able to prove that addition and multiplication are total functions — are in fact inconsistent. Even if we knew from first principles that we could not rely on internally proven consistency claims, there is the ironic twist that such "proofs of consistency" are in fact proofs for precisely the opposite.

What this really means is that consistency is a bit of a chimeral property of a formal system to have. We are denied even the conceit of self-verifiability in totalizing formal systems. You can of course prove that a formal system S is consistent in another formal system M — but then why should you accept that M is consistent? Proving it so in another system M' is just pushing the problem away a further step. The consequence is that consistency of a formal system is an unavoidably negative property: a failure to be able to exhibit a contradiction, in which case you can never be sure if it is really consistent, or if you just haven't realized how to produce a contradiction in the system.

In the end, Gödel's Second Incompleteness theorem says that unless (like Gödel himself) you believe that humans somehow have a sort of occult-ish access to timeless Platonic truth, mathematics is subject to the same epistemological limitations as the natural sciences, in which formal systems play the role of theories and the discovery of inconsistencies play the role of falsification.

Answer 2

The other answers miss something important. The real impact of the Second Theorem isn't in the limitations it places on a theory's proving its own consistency. The key point is this. If a nice arithmetical theory T can't even prove itself to be consistent, it certainly can't prove the consistency of a richer theory T+ which extends T (since proving the richer theory is consistent, proves a cut-down part of it is consistent). Hence the fact that an arithmetic theory like PA can't prove its own consistency means we can't use 'safe' reasoning of the kind we can encode in ordinary arithmetic to prove that other more 'risky' mathematical theories are in good shape.

For example, we can't use unproblematic arithmetical reasoning to convince ourselves of the consistency of set theory (with its postulation of a universe of wildly infinite sets).

And that is a very interesting result, for it seems to sabotage what is called Hilbert's Programme, which is precisely the project of trying to defend the wilder reaches of infinitistic mathematics by giving consistency proofs which use only 'safe' methods. (For a great deal more about this, see my Gödel book!)

Answer 3

An additional gloss to Peter Smiths answer that a weaker system cannot prove a stronger system consistency, is that a system incomparable to another may prove it consistent.

For example, in Gentzen's proof of the consistency of PA (Peano Arithmatic) he uses PRA (Primitive Recursive Arithmatic) and quantifier free transinfinite induction. It is not stronger than first order arithmetic (it can't prove induction), nor is it weaker (it can prove consistency of PA which PA can't).

I know that after learning about Gödel's incompleteness theorem, it came as a surprise to me that there could be a proof of PA's consistency.

Answer 4

Here's what I wrote on circularity, but didn't post (I am really just agreeing with what Michael Hardy said in more detail.. this should be a comment but it's too long):

I think it's a really good point that proving a mathematical theory (like PA or ZFC) consistent using the theory itself is circular. If the theory did happen to be inconsistent, it could prove its own consistency! So the mere existence of a consistency proof doesn't tell you anything: The real value in a consistency proof is telling you how strong an induction principle (or how big an infinite ordinal) you need to show that there isn't a proof of false: For example we need $\omega^2$ for the simple typed lambda calculus (which corresponds to a very simple constructive propositional logic), $\epsilon_0$ for PA.

Answer 5

It's not circular!

If the mere fact that one can prove consistency of a system within the system is cited as evidence that it's consistent, that is circular reasoning. But such a proof gives you more than that: you know the details of the specific proof. In particular, you know which of the axioms of the system it relies on. If you use the first three of the system's axioms to prove consistency of the whole system of 20 axioms (including those first three), and therefore conclude that the whole system is consistent because you were already confident of the consistency of the first three, then you are not reasoning circularly. But Gödels incompleteness result tells you that in certain kinds of cases, that cannot be done

Answer 6

Gödel showed that truth and provability don't necessarily coincide. In a certain sense, he also showed that not every statement is necessarily either meaningless or either true or false. Already Aristotle envisioned that this might be the case for statements about the future.

The question whether mathematics is consistent is interesting, but so is the question: "Are mathematical models fundamentally different from the real world around us, or is it possible to approximate by mathematical models the important aspects of the world around us which we want to investigate?"

Once we believed that the world around us is deterministic, and that mathematics itself is "even more" deterministic. We learned later that the world around us is not as deterministic as we thought, and Gödel's incompleteness theorems taught us that mathematics is also less deterministic than we expected.

I once asked myself: "How to model non-existence (or not-yet existence) mathematically?". If I recall correctly, I was thinking about questions of measurability at that time. However, the point remains that a consistent and finitistic mathematic would probably be too limited to reflect the infinite, unbounded and inconsistent real world sufficiently accurate.