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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: Schema of Separation Axioms (Restricted Principle of Comprehension)

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

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

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, and if we abbreviate the aliteration $x_1,\ldots,x_n$ by $\overset{n}{x}$, the axiom states: For every predicate of the form $p(z,\overset{n}{x})$ the following axiom holds: For all $\overset{n}{x}$ and all sets $x$ there is a set $y$ containing exactly those elements $z$ of $x$ fulfilling $p(z,\overset{n}{x})$: $$\forall \overset{n}{x}(\forall X(\exists Y(\forall z(z\in Y\Rightarrow z\in X\wedge p(z,\overset{n}{x}))))).$$

The schema justifies the set-builder notation, since it ensures the existence (and with the axiom of extensionality the uniqueness) of a subset of elements of a set $x$ fulfilling some given property $p(z,\overset{n}{x})$:
$$\forall \overset{n}{x} (\forall X(\exists Y(Y=\{z\in X\mid p(z,\overset{n}{x})\}))).$$

| | | | | created: 2014-06-09 13:16:44 | modified: 2020-11-22 13:29:49 | 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

Edit or AddNotationAxiomatic Method

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Bibliography (further reading)

[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