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Some diophantine forms of gödel's theorem:

Dyson, Verena H. ; Jones, James P. ; Shepherdson, John C.
Archiv für mathematische Logik und Grundlagenforschung.  22 (1980)  1-2 - p. 51-60 , 1980
Link: https://doi.org/10.1007/bf02318026