Code (set theory)

In set theory, a code for a hereditarily countable set : x \in H_{\aleph_1} \, is a set : E \subset \omega \times \omega such that there is an isomorphism between (\omega,E) and (X,\in) where X is the transitive closure of \{x\} . If X is finite (with cardinality n ), then use n\times n instead of \omega\times\omega and (n,E) instead of (\omega,E) . 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 X , then one knows what x is. (We use the transitive closure of \{x\} rather than of x itself to avoid confusing the elements of x with elements of its elements or whatever.) A code includes that information identifying x and also information about the particular injection from X into \omega which was used to create E . 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 H_{\aleph_1} into the powerset of \omega\times\omega . Using a pairing function on \omega such as (n,k)\mapsto(n^2+2nk+k^2+n+3k)/2 , we can map the powerset of \omega\times\omega into the powerset of \omega . And we can map the powerset of \omega into the Cantor set, a subset of the real numbers.…

In set theory, a code for a hereditarily countable set x \in H_{\aleph_1} \, is a set E \subset \omega \times \omega such that there is an isomorphism between (\omega,E) and (X,\in) where X is the transitive closure of \{x\} . If X is finite (with cardinality n ), then use n\times n instead of \omega\times\omega and (n,E) instead of (\omega,E) . 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 X , then one knows what x is. (We use the transitive closure of \{x\} rather than of x itself to avoid confusing the elements of x with elements of its elements or whatever.) A code includes that information identifying x and also information about the particular injection from X into \omega which was used to create E . 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 H_{\aleph_1} into the powerset of \omega\times\omega . Using a pairing function on \omega such as (n,k)\mapsto(n^2+2nk+k^2+n+3k)/2 , we can map the powerset of \omega\times\omega into the powerset of \omega . And we can map the powerset of \omega into the Cantor set, a subset of the real numbers. So statements about H_{\aleph_1} can be converted into statements about the reals. Therefore, H_{\aleph_1} \subset L(R) , where 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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