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 property^{1}, 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]

## 1.**Corollary**: Justification of the Set-Builder Notation

## 2.**Corollary**: Justification of Subsets and Supersets

## 3.**Corollary**: Equality of Sets

## 4.**Explanation**: How the Axiom of Separation Avoids Russel's Paradox

## 5.**Corollary**: There is no set of all sets

## 6.**Corollary**: Justification of the Set Intersection

## 7.**Corollary**: Justification of the Set Difference

## 8.**Corollary**: Set Difference and Set Complement are the Same Concepts

[983] **Ebbinghaus, H.-D.**: “Einführung in die Mengenlehre”, BI Wisschenschaftsverlag, 1994, 3

[656] **Hoffmann, Dirk W.**: “Grenzen der Mathematik – Eine Reise durch die Kerngebiete der mathematischen Logik”, Spektrum Akademischer Verlag, 2011

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