Gödel's Incompleteness Theorem — meta-reasoning “loophole”?

Gödel's Theorem says that I can construct a mathematical statement like "f(x1,x2,...,x_n)=0 has no integer solution", where it is impossible (in a certain system of axioms) to formally prove that it's true, and also impossible to formally prove that it's false.

I have often heard a school of thought that goes like "Well, in reality, we know that this particular statement is true. Why? Because if f has an integer solution, then it would obviously be possible to prove that f has an integer solution. (Plug it in and check it.) Yet I just constructed f in an elaborate way to ensure that no such formal proof exists."

Although this isn't a formal axiomatic proof that the statement is true, it is still a "proof" using meta-reasoning. (Is that a fair description?)

If this is correct so far, is there some generalization of Gödel's Theorem that says "There are statements which cannot be "proven" true, nor "proven" false, nor "proven" formally undecidable, even if the word "proven" is taken more broadly to allow any kind of "meta-reasoning" that mathematicians are capable of?

Makes my head spin to think about it. Thanks in advance!

• Whenever you say "in reality", "meta-reasoning" or "what mathematicians are capable of", you seem to mean "outside of the system about which Gödel's theorem says something". I wonder if this outside system has a name besides "common sense". – Daniel S. Sep 3 '15 at 17:31
• Once we have called this system "common sense", we can consider whether the statement "This statement cannot be proved using common sense" is true. Going meta just changes this to the liar paradox: en.wikipedia.org/wiki/Liar_paradox – gmatht Aug 13 '17 at 13:38

Yes, for most reasonable meanings of the words, there are statements that can neither be proved nor disproved nor proved to be independent.

However, "any kind of meta-reasoning that mathematicians are capable of" is slightly too fuzzy to work as a reasonable meaning here. Some mathematicians are capable of reasonings that other mathematicians consider faulty, for example.

Generally Gödel-like results always work particular proof systems, which are presumed to be "sane" in the sense that:

• Any valid proof or disproof can be encoded as a sequence of bytes.

• (Consistency): there is nothing that has both a proof and a disproof.

• One can write a computer program that reads a sequence of bytes and then tells us whether it encodes a valid proof or disproof -- and if so, what it proves/disproves.

The last of these conditions fail if you try to make "provability" mean something non-operational such as "whatever will convince most real-life mathematicians". And it happens to be an essential technical part of the arguments.

One way to see there must be a sentence thar is neither provable nor disprovable nor provably independent is to consider all sentences of the form "$P$ halts" for all programs $P$ in your favorite programming language. If they are all either provable or disprovable, you could decide the halting problem by searching simultaneously for a proof and a disproof. So some of then (we don't know exactly which) must be neither provable nor disprovable. But if all of those could be proved independent, then we could solve the halting problem by searching simultaneously for a proof or a disproof or a proof of independendce. (If we know that the program's halting sentence is independent, then it cannot actually halt, because everything that halts does so provably).

The integer solution might be too big to simply "plug in and check it": "most" models of Peano arithmetic have elements that are simply too big to have a decimal representation in the formal language you're using.

More precisely, a nonstandard model of Peano arithmetic contains the standard model of the natural numbers, along with extra integers that are larger than every standard integer. But only the standard integers can be expressed in decimal notation using the formal language of arithmetic.

And depending on your philosophical beliefs, even the standard model of the natural numbers contains numbers that are too big to ever write down in decimal, even in principle.

Because we know or are assuming something that the original theory doesn't know: that the theory is consistient!

Namely, $f$ could have an integer solution. It would just turn out that the original theory was inconsistent. If we didn't know that original theory was consistent, this would be an actual possibility, and your proof would fall through.

In general, to prove a theory is consistient, we need a different theory.