In the Wikipedia article [Diophantine set](https://en.wikipedia.org/wiki/Diophantine_set) there is a section entitled "Further applications" in regards to Matiyasevich's theorem and it states:



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> Matiyasevich's theorem has since been used to prove that many problems from calculus and differential equations are unsolvable.
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> One can also derive the following stronger form of Gödel's first incompleteness theorem from Matiyasevich's result:
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and then states the theorem. However, this section contains no specific citations. Could somebody provide me with references of this proof worthy of citation? Specifically, I mean the proof regarding the first incompleteness theorem. Granted, it is easy to see how Matiyasevich's theorem can prove Gödel's first incompleteness theorem, but I am looking for (possible the first) published paper with this result to be able to reference it.



Matiyasevich's theorem and Gödel's first incompleteness theorem
If you are writing a research paper on this, the usual practice in mathematics is that you do not need a reference for such an obvious result which is known to everyone in the field already. For things like this, there is often no paper that claims credit for the result - one of the roles of books is to summarize these "well known" results that nobody would take credit for in a paper. 


I suspect that the result was "first known" even before Matiyasevich proved the final part of the MDRP theorem, in the sense that everyone realized that a negative answer to Hilbert's 10th problem would lead immediately to this kind of result. 


However, Martin Davis does mention this result as Theorem 7.7 of ''Hilbert's 10th problem is unsolvable'', *The American Mathematical Monthly*, 80(3), 1973, pp. 233-269. He provides what he calls an "informal heuristic" argument, which is the standard argument sketched by Carlo Beenakker in another answer here.


