AxiomSchemaOfSeparation 2003-06-24 19:14:06 2003-06-25 06:58:37 Axiom % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Let $\phi(u, p)$ be a formula. For any $X$ and $p$, there exists a set $Y = \{ u \in X : \phi(u, p) \}$. The Axiom Schema of Separation is an axiom schema of Zermelo-Fraenkel set theory. Note that it represents infinitely many individual axioms, one for each formula $\phi$. In symbols, it reads: \[ \forall X \forall p \exists Y \forall u(u \in Y \leftrightarrow u \in X \land \phi(u, p)). \] By Extensionality, the set $Y$ is unique. The Axiom Schema of Separation implies that $\phi$ may depend on more than one parameter $p$. We may show by induction that if $\phi(u, p_1, \ldots, p_n)$ is a formula, then \[ \forall X \forall p_1 \cdots \forall p_n \exists Y \forall u(u \in Y \leftrightarrow u \in X \land \phi(u, p_1, \ldots, p_n)) \] holds, using the Axiom Schema of Separation and the Axiom of Pairing. Another consequence of the Axiom Schema of Separation is that a subclass of any set is a set. To see this, let $\mathbf{C}$ be the class $\mathbf{C} = \{ u : \phi(u, p_1, \ldots, p_n) \}$. Then \[ \forall X \exists Y (\mathbf{C} \cap X = Y) \] holds, which means that the intersection of $\mathbf{C}$ with any set is a set. Therefore, in particular, the intersection of two sets $X \cap Y = \{ x \in X : x \in Y \}$ is a set. Furthermore the difference of two sets $X - Y = \{ x \in X : x \notin Y \}$ is a set and, provided there exists at least one set, which is guaranteed by the Axiom of Infinity, the empty set is a set. For if $X$ is a set, then $\emptyset = \{ x \in X : x \neq x \}$ is a set. Moreover, if $\mathbf{C}$ is a nonempty class, then $\bigcap \mathbf{C}$ is a set, by Separation. $\bigcap \mathbf{C}$ is a subset of every $X \in \mathbf{C}$. Lastly, we may use Separation to show that the class of all sets, $V$, is not a set, i.e., $V$ is a proper class. For example, suppose $V$ is a set. Then by Separation \[ V' = \{ x \in V : x \notin x \} \] is a set and we have reached a Russell paradox.