# Did Gödel's First Incompleteness Theorem ever led to number theoretical insights?

Gödel's First Incompleteness Theorem is based on the construction of a formula - a so-called Gödel formula - stating its own unprovability. Has ever such a formula led to novel insights in number theory (beside arithmetics' incompleteness)?

Intuitively, a Gödel formula would be an excellent candidate as an additional axiom which, even though not making arithmetics' axiomatization complete, would most likely result in novel and interesting theorems.

To the best of my knowledge, this path has so far not been followed. Am I right or were number theoretical results gained from exploiting a Gödel formula?

• I think a more general question would be: what are examples of true statements about natural number that are not deducible from Peano Arithmetic but which have been used in number theory? I am convinced that there are ways to find such statements without passing by Gödel's First Incompleteness Theorem (for example: Paris–Harrington theorem). It would be hard to imagine for me that the statement coming from Gödel's Theorem would be of special importance among such... – Alexey Mar 10 '19 at 12:58
• I do not think that Goedel's results are useful to detect additional aspects in number theory, but since Goedel we know that famous open conjectures could be unsolvable. An important example of a known unsolvable problem is to determine whether a diophantine equation has an integer solution. – Peter Mar 10 '19 at 13:16
• I think you answered your own question by invoking the 10th problem. There is been significant insight gained from work on generalizations of the DPRM theorem. – Andrés E. Caicedo Mar 10 '19 at 13:43
• @Carl Yes, that is true. It would be great to see something more direct, I think the question has some interesting potential. – Andrés E. Caicedo Mar 10 '19 at 14:06
• @Andrés E. Caicedo: I am doubtful there are many examples, if there are any, for the reasons I sketched below. – Carl Mummert Mar 10 '19 at 14:06