Welcome guest
You're not logged in.
300 users online, thereof 0 logged in

If you recall the historical development of set theory, the general principle of comprehension led to contradictions. Zermelo restricted this principle and limited a general comprehension to the following axiom of separation:

## Axiom: Axiom of Separation (Restricted Principle of Comprehension)

If $$p(z)$$ is a definite property1, then for all sets $$X$$ there is a subset $$Y$$ consisting of those elements $$x$$, for which $$p(x)$$ is satisfied. Formally, this axiom can be written as

$\forall X~\exists Y~\forall z~(z\in Y \Leftrightarrow z\in X\wedge p(z)).$

1 Albert Skolem proposed to state “definite property” more precisely by replacing $p(z)$ by an atomic formula in predicate logic $p(z,X_1,\ldots,X_n).$ This makes the axiom in fact a whole schema for infinitely many axioms, in which the placeholder $p(z,X_1,\ldots,X_n)$ stands for an arbitrary, $n+1$-ary logical formula, in which $z$ is a free variable. With this specification, the axiom reads

$\forall X_1,\ldots,X_n \forall X~\exists Y~\forall z~(z\in Y \Leftrightarrow z\in X\wedge p(z,X_1,\ldots,X_n)).$

| | | | | created: 2014-06-09 13:16:44 | modified: 2018-03-23 00:04:21 | by: bookofproofs | references: [656], [983]

(none)