Zermelo-Frankel Axioms

The following are the axioms of Zermelo-Frankel set theory, stated in first-order logic with the language $\mathscr{L}={\in}$:

1. Extensionality. $$(\forall xy)(x\equiv y\leftrightarrow (\forall z)(z\in x \leftrightarrow z\in y))$$ or, two sets are the same iff they have the same elements.
2. Null set. $$(\exists x\forall y)(\neg y\in x)$$ or, there is a set with no elements.
3. Pairs. $$(\forall xy\exists z\forall u)(u\in z\leftrightarrow (u\equiv x\lor u\equiv y))$$ or, for any two sets, there exists a set which has precisely those two sets as its elements.
4. Unions. $$(\forall x\exists y\forall z)(z\in y \leftrightarrow (\exists w)(z\in w\land w\in x))$$ or, given a set of sets, we may form a set which is the union of all sets in that set.
5. Power set. $$(\forall x\exists y\forall z)(z\in y\leftrightarrow (\forall w)(w\in z\rightarrow w\in x))$$ or, we may form the set of all subsets of a given set.
6. Infinity. $$(\exists x)((\exists y)(y\in x)\land (\forall y)(y\in x\rightarrow (\exists z)(y\in z\land z\in x)))$$ or, there exist infinite sets.
7. Regularity. $$(\forall x)((\exists y)(y\in x)\rightarrow (\exists y)(y\in x\land \neg(\exists z)(z\in y\land z\in x)))$$ or, every nonempty set contains an element which is disjoint from it.
8. Subsets. $$(\forall x\exists y\forall z)(z\in y\leftrightarrow (z\in x\land\varphi (zu_1...u_n)))$$ where $\varphi$ is a formula in which $y$ does not occur - or, for any set $x$, we may form the set ${z\in x:\varphi(zu_1...u_n)}$. This is actually a scheme of countably many axioms (one for each first-order formula $\varphi$ with at least one free variable in which the variable $y$ does not occur).
9. Collection. $$(\forall x \exists y)\varphi(xyv_1...v_n)\rightarrow (\forall A\exists B\forall x\exists y)(x\in A\implies (y\in B\land \varphi(xyv_1...v_n)))$$ where $\varphi$ is a formula in which $y$ does not occur. Intuitively, if $f$ is a definable class function, then the image of any domain set $A$ under $f$ falls inside some other set $B$.

Here are some notable theories comprised of these axioms:

• Zermelo set theory consists of axioms $(1)$ through $(8)$.

• Zermelo-Frankel set theory consists of axioms $(1)$ through $(9)$.

• ZFC set theory consists of axioms $(1)$ through $(8)$ plus the Axiom of Choice in the form $$\begin{align}(\forall z)\big(&(\forall w\exists x)(w\in z\rightarrow x\in w)\\ &\rightarrow (\exists w\forall x\exists! y)(x\in z\rightarrow y\in x\land y\in w)\big)\end{align}$$ Note that the above statement is in the form of the General Principle of Choice, which can be shown equivalent to AC.

• Zermelo-Frankel set theory without foundation/regularity, denoted $\mathrm{ZF}^{-}$, consists of Zermelo-Frankel set theory without axiom $(7)$. It suffices to complete many of the constructions usually done in ZF, but it allows for nonstandard models containing atoms, or sets which are their own unique element. Theories of these models are sometimes called ZF with atoms or ZFA.

• Internal set theory or IST is a consistent supertheory of ZFC that contains three new powerful axioms.

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