Code (set theory)

In set theory, a code for a set x ${\displaystyle \in H_{\aleph _{1}}}$ is a set E ${\displaystyle \subset }$ ω×ω such that there is an isomorphism between (ω,E) and (X,${\displaystyle \in }$) where X is the transitive closure of {x}.

So codes are a way of mapping ${\displaystyle H_{\aleph _{1}}}$ into the powerset of ω×ω. Using a pairing function on ω (such as (n,k) goes to (n²+2n·k+k²+n+3·k)/2), we can map the powerset of ω×ω into the powerset of ω. And using, say, continued fractions, we can map the powerset of ω into the real numbers. So statements about ${\displaystyle H_{\aleph _{1}}}$ can be converted into statements about the reals. And ${\displaystyle H_{\aleph _{1}}\subset L(R)}$.

See also

• Hereditarily countable set
• L(R)

References

• William J. Mitchell,"The Complexity of the Core Model","Journal of Symbolic Logic",Vol.63,No.4,December 1998,page 1393.