# Talk:Code (set theory)

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Field:  Foundations, logic, and set theory

## A definition easier to follow

There is probably nothing wrong in the article as it stands, but I want to change the beginning of it anyway. The reason is that I don't like when (for most people) nontrivial concepts are introduced in a single sentence and, in addition, the notation is introduced in nested "where-clauses" and "such-that-clauses"

I'll replace

In set theory, a code for a set
x ${\displaystyle \in H_{\aleph _{1}},}$
the notation standing for the hereditarily countable sets,
is a set
E ${\displaystyle \subset }$ ω×ω
such that there is an isomorphism between (ω,E) and (X,${\displaystyle \in }$) where X is the transitive closure of {x}.

with this

In set theory a code of a set is defined as follows. Let ${\displaystyle x}$ be a hereditarily countable set, and let ${\displaystyle X}$ be the transitive closure of ${\displaystyle \{x\}}$. Let as usual ${\displaystyle \omega }$ denote the set of natural numbers and let ${\displaystyle \aleph _{1}}$ be the first uncountable cardinal number. In this notation we have ${\displaystyle x\in H_{\aleph _{1}}}$. Also recall that ${\displaystyle (X,\in )}$ denotes the relation of belonging in ${\displaystyle X}$.
A code for ${\displaystyle x}$ is any set ${\displaystyle E}$ satisfying the following two properties:
1.) ${\displaystyle E}$ ${\displaystyle \subset }$ ω×ω
2.) There is an isomorphism between ${\displaystyle (\omega ,E)}$ and ${\displaystyle (X,\in )}$.

if there are no objections.

YohanN7 (talk) 23:13, 6 June 2008 (UTC)