# axiom schema of separation

Let $\varphi (u,p)$ be a formula^{}. For any $X$ and $p$, there exists a set $Y=\{u\in X:\varphi (u,p)\}$.

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 $\varphi $. In symbols, it reads:

$$\forall X\forall p\exists Y\forall u(u\in Y\leftrightarrow u\in X\wedge \varphi (u,p)).$$ |

By Extensionality, the set $Y$ is unique.

The Axiom Schema of Separation implies that $\varphi $ may depend on more than one parameter $p$.

We may show by induction^{} that if $\varphi (u,{p}_{1},\mathrm{\dots},{p}_{n})$ is a formula, then

$$\forall X\forall {p}_{1}\mathrm{\cdots}\forall {p}_{n}\exists Y\forall u(u\in Y\leftrightarrow u\in X\wedge \varphi (u,{p}_{1},\mathrm{\dots},{p}_{n}))$$ |

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 $\mathbf{C}=\{u:\varphi (u,{p}_{1},\mathrm{\dots},{p}_{n})\}$. 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 $\mathrm{\varnothing}=\{x\in X:x\ne x\}$ 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

$${V}^{\prime}=\{x\in V:x\notin x\}$$ |

is a set and we have reached a Russell paradox.

Title | axiom schema of separation |
---|---|

Canonical name | AxiomSchemaOfSeparation |

Date of creation | 2013-03-22 13:42:46 |

Last modified on | 2013-03-22 13:42:46 |

Owner | Sabean (2546) |

Last modified by | Sabean (2546) |

Numerical id | 18 |

Author | Sabean (2546) |

Entry type | Axiom |

Classification | msc 03E30 |

Synonym | separation schema |

Synonym | separation |