  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 Hilbert's tenth problem, G\"odel's incompleteness theorem, and
Turing's halting problem. By slightly strengthening the property $P$, we get
Tarski's definability theorem, namely that truth is not first order definable.
The property $P$ together with a "Cantor's 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 Hilbert's 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.
