We work in ZFC. Let denote the class of infinite cardinal numbers such that (see von Neumann hierarchy) is a -elementary submodel of the universe , i.e. such that the embedding preserves and reflects the truth of -formulas. In particular:
is if and only if it is infinite, and
is if and only if it is uncountable and coincides with the hereditarily -sized sets , i.e. those whose transitive closure has cardinality .
If and both are in , we say is --extendible if there is an elementary embedding with critical point , such that , , and .
We say is -extendible if it is --extendible for all with .