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Codes are useful in constructing mice. |
If X is finite (with cardinality n), then use (n,E) instead of (ω,E). |
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In [[set theory]], a '''code''' for a set x <math>\in H_{\aleph_1}</math> is a set E <math>\subset</math> ω×ω such that there is an [[isomorphism]] between (ω,E) and (X,<math>\in</math>) where X is the [[transitive set|transitive closure]] of {x}. If X is finite (with cardinality n), then use n×n instead of ω×ω and (n,E) instead of (ω,E).
So codes are a way of mapping <math>H_{\aleph_1}</math> into the [[powerset]] of ω×ω. Using a pairing function on ω (such as (n,k) goes to (n<sup>2</sup>+2·n·k+k<sup>2</sup>+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 number]]s. So statements about <math>H_{\aleph_1}</math> can be converted into statements about the reals. And <math>H_{\aleph_1} \subset L(R)</math>.
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