Gödel's theorem vs unprovable mathematical results

Question

As an answer to this question, Peter Smith wrote:

Indeed, it is a fairly gross misunderstanding of what Gödel's theorem says to summarize it as asserting that "there exist mathematical results that cannot be proven"

It made me realize that I don't understand the difference between Gödel's first incompleteness theorem and the assertion "there exist mathematical results that cannot be proven."

Since ZFC is a fixed set of axioms, aren't there mathematical sentences (e.g., the continuum hypothesis) which can neither be proved nor disproved, as long as we stick with ZFC? And isn't this what Gödel's first incompleteness theorem predicts?

There's a big difference between "can't be proven as long as we stick with ZFC" and "can't be proven". — Andreas Blass, May 14 '19 at 23:58
As long as we stick with ZFC. But we can add more axioms and prove more results. — Asaf Karagila, May 14 '19 at 23:58
Godel himself famously thought there were no absolutely undecidable questions. (In fact my recommendation would be to google "absolutely undecidable proposition" and skim the various papers that come up.) — spaceisdarkgreen, May 15 '19 at 00:07

score 4 · Accepted Answer · answered May 15 '19 at 02:55

You're understanding Godel correctly - you're misunderstanding the misunderstanding. :P The following will just reaffirm things you already know, but readers may find it useful:

The issue is in understanding what "cannot be proven" means. All too frequently someone will make the implicit error of assuming that this is said with respect to every appropriate axiom system. This amounts to a quantifier mix-up:

"For every appropriate axiom system there is an statement undecidable in that system"

(which is correct) becomes

"There is a statement undecidable in any appropriate axiom system"

(which ... isn't).

More generally, one should never say "prove" without specifying an axiom system (or acknowledging that there's some handwaving going on). For example, somewhere on this site is at least one question about GIT which goes roughly: "GIT1 proves that ZFC is incomplete, which means ZFC is consistent, but doesn't that contradict GIT2?" The issue of course is that the first "proves" is with respect to a system including the hypothesis that ZFC is consistent, while GIT2 would only apply if we had been using ZFC itself.

Gödel's theorem vs unprovable mathematical results

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