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Code (set theory)

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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 .[1] If is finite (with cardinality ), 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. (We use the transitive closure of rather than of itself to avoid confusing the elements of with elements of its elements or whatever.) 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.

References

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  1. ^ Mitchell, William J. (1998), "The complexity of the core model", The Journal of Symbolic Logic, 63 (4): 1393–1398, arXiv:math/9210202, doi:10.2307/2586656, JSTOR 2586656, MR 1665735