Hilberts tenth problem, Godels incompleteness, Halting problem, a unifying perspective

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 the insolvability of Hilberts tenth problem, Godels incompleteness theorem, and Turings halting problem. By slightly strengthening the property $P$, we get Tarskis definability theorem, namely that truth is not first order definable. The property $P$ together with a Cantors diagonalization process emphasizes that all the above theorems are a variation on a theme, that of self reference and diagonalization combined. We relate our results to self referential paradoxes, including a formalisation of the Liar paradox, and fixed point theorems. We also discuss the property $P$ for arbitrary rings. We give a survey on Hilberts tenth problem for quadratic rings and for the rationals pointing the way to ongoing research in main stream mathematics involving recursion theory, definability in model theory, algebraic geometry and number theory.