"Practical" Implications of Godel's Incompleteness Theorems?

Question

Godel showed that there are "Godelian" sentences within sufficiently powerful axiomatic systems (Principia Mathematica and the like). I interpret such Godelian sentences as the number theoretic equivalent of the Liar's paradox, where self reference leads to "weirdness". Besides the ability to construct such "weird" sentences, does Godel's Incompleteness Theorems have any impact on possibly preventing proofs for things like the Goldbach conjecture (or proving statements of such nature that don't seem to have anything to do with self reference)? I have heard people claim such consequences, and I don't understand how Godelian sentences have anything to do with possibly preventing proofs for such number theoretic conjectures.

Answer 1

First of all, the MRDP theorem showed that independence already exists at a very basic level: given any "appropriate" theory $T$ there is a Diophantine equation $\mathcal{E}_T$ which has no solutions but whose unsolvability cannot be proved in $T$. Moreover, the process $T\leadsto\mathcal{E}_T$ is completely constructive; there are no "shenanigans" here at all. Of course the Diophantine equations which emerge from this process are extremely complicated, but this shows that we can't escape Godel by only paying attention to "simple" sentences.

Having said that, here's an example of how Godel's incompleteness theorem can be used to prove an unprovability result around a non-logic-y sentence:

As a consequence of (the original proof of) the first incompleteness theorem we get the second incompleteness theorem: that no "appropriate" formal system can prove its own consistency. It turns out that consistency statements can be consequences of initially-innocuous principles. For example, a variant of Ramsey's theorem turns out to imply the consistency of $\mathsf{PA}$, and so - by applying Godel - must be unprovable in $\mathsf{PA}$ itself (unless $\mathsf{PA}$ is inconsistent of course)!