Code (set theory)

In set theory, a code for a hereditarily countable set
is a set
such that there is an isomorphism between and where is the transitive closure of. If is finite, then use instead of and instead of.
According to the axiom of extensionality, the identity of a set is determined by its elements. And since those elements are also sets, their identities are determined by their elements, etc.. So if one knows the element relation restricted to, then one knows what is. A code includes that information identifying and also information about the particular injection from into which was used to create. The extra information about the injection is non-essential, so there are many codes for the same set which are equally useful.
So codes are a way of mapping into the powerset of. Using a pairing function on such as, we can map the powerset of into the powerset of. And we can map the powerset of into the Cantor set, a subset of the real numbers. So statements about can be converted into statements about the reals. Therefore,, where [L(R)|] is the smallest transitive inner model of ZF containing all the ordinals and all the reals.
Codes are useful in constructing mice.