An ordinal is *definable from a class of ordinals X if and only if there is a formula and such that is the unique ordinal for which where for all we define to be the name for within .
A structure is eligible if and only if:
- .
- < is the ordering on On restricted to X.
- is a partial function from to X, for some integer k(i).
If is an eligible structure then is defined to be as before but with all occurrences of X replaced with .
Let be two eligible structures which have the same function k. Then we say if and we have: