Code (set theory)

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

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

Codes are useful in constructing mice.

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.