Code (set theory) explained

In set theory, a code for a hereditarily countable set

x\in

H
	\aleph₁

is a set

E\subset\omega x \omega

such that there is an isomorphism between

(\omega,E)

and

(X,\in)

where

is the transitive closure of

\{x\}

. If

is finite (with cardinality

), then use

n x n

instead of

\omega x \omega

and

(n,E)

instead of

(\omega,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

, then one knows what

is. (We use the transitive closure of

\{x\}

rather than of

itself to avoid confusing the elements of

with elements of its elements or whatever.) A code includes that information identifying

and also information about the particular injection from

into

\omega

which was used to create

. 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

H
	\aleph₁

into the powerset of

\omega x \omega

. Using a pairing function on

\omega

such as

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

, we can map the powerset of

\omega x \omega

into the powerset of

\omega

. And we can map the powerset of

\omega

into the Cantor set, a subset of the real numbers. So statements about

H
	\aleph₁

can be converted into statements about the reals. Therefore,

H
	\aleph₁

\subsetL(R)

, where is the smallest transitive inner model of ZF containing all the ordinals and all the reals.

Codes are useful in constructing mice.