Stable ordinals appear in several areas of higher recursion
theory. Sacks and Simpson used them to distinguish cases in their positive
solution to Post's Problem in $\alpha$-recursion theory. A more recent example
is Welch's use of them to characterise the running times of infinite time
Turing machines. In this talk I'll show that any real
(indeed any set or class of ordinals) is generic relative to
its "stability predicate" and then develop an analogy
between stability and strength in large cardinal set theory.
The main result is that there are models of set theory in
which the two notions coincide. This has the consequence
that the universe of sets is generic over a definable
"$L$-like" inner model with only very modest large cardinals.