Zermelo-Fraenkel Axioms

The Zermelo-Fraenkel axioms are the basis for Zermelo-Fraenkel Set Theory. In the following, $\exists$ stands for Exists, $\in$ for ``is an element of,'' $\forall$ for For All, $\Rightarrow$ for Implies, $\neg$ for Not (Negation), $\wedge$ for And, $\vee$ for Or, $\rightleftharpoons$ for ``is Equivalent to,'' and ${\mathcal S}$ denotes the union $y$ of all the sets that are the elements of $x$.

1. Existence of the empty set: $\exists x\forall u\neg(u\in x)$.

2. Extensionality axiom: $\forall x\forall y(\forall u(u\in x\rightleftharpoons u\in y)\to x=y)$.

3. Unordered pair axiom: $\forall x\forall y\exists z\forall u(u\in z\rightleftharpoons u=x\vee u=y)$.

4. Union (or ``sum-set'') axiom: $\forall x\exists y\forall u(u\in y\rightleftharpoons\exists v(u\in v\wedge v\in x))$.

5. Subset axiom: $\forall x\exists y\forall u(u\in y\rightleftharpoons\forall v(v\in u\to v\in x))$.

6. Replacement axiom: For any set-theoretic formula $A(u,v)$,

$$\forall u\forall v\forall w(A(u,v)\wedge A(u,w)\to v=w)\to\f... ...all v(v\in y\rightleftharpoons\exists u(u\in x\wedge A(u,v))).$$

7. Regularity axiom: For any set-theoretic formula $A(u)$, $\exists x A(x)\to\exists x(A(x)\wedge\neg\exists y(A(y)\wedge y\in x))$.

8. Axiom of Choice:

$\forall x[\forall u(u\in x\to\exists v(v\in u))$
$\wedge\forall u\forall v((u\in x\wedge v\in x\wedge\neg u=v)$
$\to\neg\exists w(w\in u\wedge w\in v))\to\exists y \{y\subset{\mathcal S}(x)$
$\wedge\forall u(u\in x\to\exists z(z\in u\wedge z\in y\wedge\forall w(w\in u\wedge w\in y\to w=z)))\}]$

9. Infinity axiom: $\exists x(\exists u(u\in x)\wedge\forall u(u\in x\to\exists v(v\in x\wedge u\subset v\wedge\neg v=u)))$.

If Axiom 6 is replaced by

6'. Axiom of subsets: for any set-theoretic formula $A(u)$, $\forall x\exists y\forall u(u\in y\rightleftharpoons u\in x \wedge A(u))$,

which can be deduced from Axiom 6, then the set theory is called Zermelo Set Theory instead of Zermelo-Fraenkel Set Theory.

Abian (1969) proved Consistency and independence of four of the Zermelo-Fraenkel axioms.

See also Zermelo-Fraenkel Set Theory

References

Abian, A. ``On the Independence of Set Theoretical Axioms.'' Amer. Math. Monthly 76, 787-790, 1969.

Iyanaga, S. and Kawada, Y. (Eds.). ``Zermelo-Fraenkel Set Theory.'' §35B in Encyclopedic Dictionary of Mathematics, Vol. 1. Cambridge, MA: MIT Press, pp. 134-135, 1980.