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{{short description|Concept in set theory}}

In [[set theory]], a '''code''' for a set x <math>\in H_{\aleph_1}</math> is a set E <math>\subset</math> &omega;&times;&omega; such that there is an [[isomorphism]] between (&omega;,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 &omega;&times;&omega; and (n,E) instead of (&omega;,E).

{{refimprove|date=June 2024}}

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

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

is a set

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 &omega; 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.

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

In [[set theory]], a '''code''' for a set x <math>\in H_{\aleph_1}</math> is a set E <math>\subset</math> &omega;&times;&omega; such that there is an [[isomorphism]] between <math>(&\omega;,E)</math> and (X,<math>(X,\in)</math>) where <math>X</math> is the [[transitive set|transitive closure]] of <math>\{x\}</math>.{{r|mitchell}} If <math>X</math> is finite (with cardinality <math>n</math>), then use <math>n&\times; n</math> instead of &<math>\omega;&\times;&\omega;</math> and <math>(n,E)</math> instead of <math>(&\omega;,E)</math>.

So codes are a way of mapping <math>H_{\aleph_1}</math> into the [[powerset]] of &omega;&times;&omega;. Using a pairing function on &omega; (such as (n,k) goes to (n²+2&middot;n&middot;k+k²+n+3&middot;k)/2), we can map the powerset of &omega;&times;&omega; into the powerset of &omega;. And we can map the powerset of &omega; 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>.

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 <math>X</math>, then one knows what <math>x</math> is. (We use the transitive closure of <math>\{x\}</math> rather than of <math>x</math> itself to avoid confusing the elements of <math>x</math> with elements of its elements or whatever.) A code includes that information identifying <math>x</math> and also information about the particular injection from <math>X</math> into &<math>\omega;</math> which was used to create <math>E</math>. The extra information about the injection is non-essential, so there are many codes for the same set which are equally useful.

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

So codes are a way of mapping <math>H_{\aleph_1}</math> into the [[powerset]] of &<math>\omega;&\times;&\omega;</math>. Using a [[pairing function]] on &<math>\omega;</math> (such as <math>(n,k) goes to \mapsto(n^{^2}+2&middot;n&middot;k2nk+k^{^2}+n+3&middot;k3k)/2)</math>, we can map the powerset of &<math>\omega;&\times;&\omega;</math> into the powerset of &<math>\omega;</math>. And we can map the powerset of &<math>\omega;</math> 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. AndTherefore, <math>H_{\aleph_1} \subset L(R)</math>, where [[L(R)|{{math|''L''(''R'')}}]] is the smallest transitive inner model of ZF containing all the ordinals and all the reals.

==See also==

*[[Hereditarily countable set]]

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

*[[L(R)]]

==References==

{{reflist|refs=

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

<ref name=mitchell>{{citation

| last = Mitchell | first = William J.

| arxiv = math/9210202

| doi = 10.2307/2586656

| issue = 4

| journal = The Journal of Symbolic Logic

| jstor = 2586656

| mr = 1665735

| pages = 1393–1398

| title = The complexity of the core model

| volume = 63

| year = 1998}}</ref>

}}

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[[Category:Set theory]]

[[Category:Inner model theory]]

{{mathsettheory-stub}}