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We formulate a property $P$ on a class of relations on the natural numbers,and formulate a general theorem on $P$, from which we get as corollaries theinsolvability of Hilbert's tenth problem, G\"odel's incompleteness theorem, andTuring's halting problem. By slightly strengthening the property $P$, we getTarski's definability theorem, namely that truth is not first order definable.The property $P$ together with a "Cantor's diagonalization" process emphasizesthat all the above theorems are a variation on a theme, that of self referenceand diagonalization combined. We relate our results to self referentialparadoxes, including a formalisation of the Liar paradox, and fixed pointtheorems. We also discuss the property $P$ for arbitrary rings. We give asurvey on Hilbert's tenth problem for quadratic rings and for the rationalspointing the way to ongoing research in main stream mathematics involvingrecursion theory, definability in model theory, algebraic geometry and numbertheory.

Hilbert's tenth problem, G\"odel's incompleteness, Halting problem, a unifying perspective