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:
A Silver machine is an eligible structure of the form which satisfies the following conditions:
Condensation principle. If then there is an such that .
Finiteness principle. For each there is a finite set such that for any set we have
Skolem property. If is *definable from the set , then ; moreover there is an ordinal , uniformly definable from , such that .