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Revision difference : axiom schema of separation |
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| Let $\phi(u, p)$ be a formula. For any $X$ and $p$, there exists a set $Y = \{ u \in X : \phi(u, p) \}$. |
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: |
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)). |
\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. |
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 on $n$, |
The Axiom Schema of Separation implies that $\phi$ may depend on more than one parameter $p$. We may show, by induction on $n$, that if $\phi(u, p_1, \ldots, p_n)$ is a formula, then |
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\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)) |
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holds, using the Axiom Schema of Separation and the Axiom of Pairing. |
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