Editing Diophantine set (section)

==Further applications==

Matiyasevich's theorem has since been used to prove that many problems from [[calculus]] and [[differential equation]]s are unsolvable.

One can also derive the following stronger form of [[Gödel's first incompleteness theorem]] from Matiyasevich's result:

:''Corresponding to any given consistent axiomatization of number theory,<ref>More precisely, given a [[arithmetical hierarchy#The arithmetical hierarchy of formulas|<math>\Sigma^0_1</math>-formula]] representing the set of [[Gödel number]]s of [[sentence (mathematical logic)|sentences]] that recursively axiomatize a [[consistency|consistent]] [[theory (mathematical logic)|theory]] extending [[Robinson arithmetic]].</ref> one can explicitly construct a Diophantine equation that has no solutions, but such that this fact cannot be proved within the given axiomatization.''

According to the [[incompleteness theorem]]s, a powerful-enough consistent axiomatic theory is incomplete, meaning the truth of some of its propositions cannot be established within its formalism. The statement above says that this incompleteness must include the solvability of a diophantine equation, assuming that the theory in question is a number theory.