Code (set theory): Difference between revisions

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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}.

{{refimprove|date=June 2024}}

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

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

is a set

==See also==

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

*[[Hereditarily countable set]]

such that there is an [[isomorphism]] between <math>(\omega,E)</math> and <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>.

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.

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)\mapsto(n^2+2nk+k^2+n+3k)/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. Therefore, <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.

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

==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>

}}

{{math-stub}}

[[Category:Set theory]]

[[Category:Inner model theory]]

{{mathsettheory-stub}}