Code (set theory)

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In set theory, a code for a hereditarily countable set

${\displaystyle x\in H_{\aleph _{1}}\,}$

is a set

${\displaystyle E\subset \omega \times \omega }$

such that there is an isomorphism between ${\displaystyle (\omega ,E)}$ and ${\displaystyle (X,\in )}$ where ${\displaystyle X}$ is the transitive closure of ${\displaystyle \{x\}}$.^[1] If ${\displaystyle X}$ is finite (with cardinality ${\displaystyle n}$), then use ${\displaystyle n\times n}$ instead of ${\displaystyle \omega \times \omega }$ and ${\displaystyle (n,E)}$ instead of ${\displaystyle (\omega ,E)}$.

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 ${\displaystyle X}$, then one knows what ${\displaystyle x}$ is. (We use the transitive closure of ${\displaystyle \{x\}}$ rather than of ${\displaystyle x}$ itself to avoid confusing the elements of ${\displaystyle x}$ with elements of its elements or whatever.) A code includes that information identifying ${\displaystyle x}$ and also information about the particular injection from ${\displaystyle X}$ into ${\displaystyle \omega }$ which was used to create ${\displaystyle E}$. 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 ${\displaystyle H_{\aleph _{1}}}$ into the powerset of ${\displaystyle \omega \times \omega }$. Using a pairing function on ${\displaystyle \omega }$ such as ${\displaystyle (n,k)\mapsto (n^{2}+2nk+k^{2}+n+3k)/2}$, we can map the powerset of ${\displaystyle \omega \times \omega }$ into the powerset of ${\displaystyle \omega }$. And we can map the powerset of ${\displaystyle \omega }$ into the Cantor set, a subset of the real numbers. So statements about ${\displaystyle H_{\aleph _{1}}}$ can be converted into statements about the reals. Therefore, ${\displaystyle H_{\aleph _{1}}\subset L(R)}$, 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.

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