/agree neil

forever is a long time. im thinking natural numbers had a pretty ancient origin in that several species exhibit ’self-awareness’ and are thus cognisant of 0, 1, and at least 2.

too bad plato isnt still around to argue with.

You’re asked “Does Godel’s Theorem matter on a day-to-day basis to practicing mathematicians?”, to which you answer “no.” The answer is actually “yes”, at least unless you’re artificially restricting the class of ‘working mathematicians.”

This is exactly why I qualified my statement thus:

(Of course this is an overgeneralization; some mathematicians have built distinguished careers on worrying about what’s provable from various sets of axioms. But they are a small minority.)

The hidden assumption here appears to be essentially Platonic: that there is a single, ideal Truth to which we can refer.

As an assumption, an axiom, this seems perfectly reasonable, but it is an assumption, and a somewhat controversial one.

It’s better, I think, not to use the word “axiom” here, since the word is already being used in this discussion to refer to the axioms of some formal system like Peano arithmetic, and what you’ve got in mind is something else altogether. Let’s stick with “assumption”.

it’s perfectly possible to have one formal system with models M_0 and M_1 in which statements that are referred to as “true” in one system aren’t considered “true” in the other.

Sure. But what I’m talking about here is truth within the standard model of the natural numbers.

Now you might say that the existence of the standard model is an assumption, and so it is, in exactly the same sense that the existence of the external world or of conscious minds other than your own is an assumption. But, just like those assumptions, it’s one that we all learned by age six that we can’t possibly live without.

So to summarize, when I say “true”, I mean “true about the natural numbers”, and when I say “the natural numbers”, I mean the numbers 0,1,2,3, et cetera. No first order logical system can formalize what “et cetera” means here, but every six year old knows what it means perfectly well. It’s that six year old understanding I’m appealing to.

(PS: This is as good a place as any to mention that you don’t need Godel to know that no first-order system can distinguish a unique model of the natural numbers; Lowenheim-Skolem is enough for that.)

The answer is actually “yes”, at least unless you’re artificially restricting the class of ‘working mathematicians.”

Now, do people doing analysis usually care? Nope, hardly ever, just like they don’t usually really care if you can get to the standard axioms at the front of Apostol from Whitehead and Russell. But it matters dramatically, for example, to logicians; it’s a central result — along with its morphisms to Turing’s Theorem and Chaitin’s algorithmic complexity — in theory of computer science, and the metamathematical tool of the “diagonalization” method gets practically beaten to death in a good CS theory course.

Thus, for example, we have a sentence “2+3=5″; we have a formal interpretation of that as “the third successor of the second successor of 0 is the same construct as the fifth successor of 0″ via Peano; and we can show that correspondence seems to match what we find to be “true”.

What it doesn’t do is tell us anything about what is “true”, and in fact we can easily find examples of systems in which what is “true” in one system isn’t necessarily “true” in another.

The hidden assumption here appears to be essentially Platonic: that there is a single, ideal Truth to which we can refer.

As an assumption, an axiom, this seems perfectly reasonable, but it is an assumption, and a somewhat controversial one. Certainly it’s one you can’t use model theory to support, since we know (cf. Robinson’s nonstandard analysis versus standard analysis) that it’s perfectly possible to have one formal system with models M_0 and M_1 in which statements that are referred to as “true” in one system aren’t considered “true” in the other.

Let “NN” be the natural numbers defined by some set of axioms.

I have no idea what this means.

We can BELIEVE in some properties of NN.

We can KNOW some properties of “NN”.

We can BELIEVE that NN = “NN”

But we can’t KNOW anything about NN.

After the uprising on June 17th,
The Secretary of the Writers Union
Had leaflets distributed in the Stalinallee
Upon which was to be read that the people
Had forfeited the confidence of the government
And could only reclaim it
Through redoubled efforts. Would it not be easier
Still for the government
To dissolve the people
And elect another?

—Bertolt Brecht, “The Solution,” Buckow Elegies No. 9 (S.H. transl)

There is no math paper demonstrating the truth of the consistency of ZFC, except for those that derive it from axioms in a larger system, as above.

I might believe that S6 has no complex structure, but that does not make it a mathematical truth unless someone finds a proof. So you may have an unproved belief if you want, but it is certainly the case that mathematicians care deeply about having a finite proof from the axioms.