Code (set theory): Difference between revisions

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In [[set theory]], a '''code''' for a set[[hereditarily countable set]]

:x <math>x \in H_{\aleph_1} \,</math>

is a set

:x <math>\in H_{\aleph_1},</math>

:<math>E \subset \omega \times \omega</math>

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''&times;''n'' instead of ω&times;ω and (''n'',''E'') instead of (ω,''E'').

the notation standing for the [[hereditarily countable set]]s,

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 ω 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.

is a set

So codes are a way of mapping <math>H_{\aleph_1}</math> into the [[powerset]] of ω&times;ω. Using a [[pairing function]] on ω (such as (''n'',''k'') goes to (''n''²+2·''n''·''k''+''k''²+''n''+3·''k'')/2), we can map the powerset of ω&times;ω into the powerset of ω. And we can map the powerset of ω into the [[Cantor set]], a subset of the [[real number]]s. So statements about <math>H_{\aleph_1}</math> can be converted into statements about the reals. AndConsequently, <math>H_{\aleph_1} \subset L(R) \,.</math>.

:E <math>\subset</math> ω&times;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&times;n instead of ω&times;ω and (n,E) instead of (ω,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 ω 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 <math>H_{\aleph_1}</math> into the [[powerset]] of ω&times;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&times;ω into the powerset of ω. And we can map the powerset of ω into the [[Cantor set]], a subset of 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>.

Codes are useful in constructing [[mouse (set theory)|mice]].

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[[Category:Set theory]]

[[Category:Inner model theory]]

{{settheory-stub}}