Franklin's Notes


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. or, two sets are the same iff they have the same elements.
2. Null set. or, there is a set with no elements.
3. Pairs. or, for any two sets, there exists a set which has precisely those two sets as its elements.
4. Unions. or, given a set of sets, we may form a set which is the union of all sets in that set.
5. Power set. or, we may form the set of all subsets of a given set.
6. Infinity. or, there exist infinite sets.
7. Regularity. or, every nonempty set contains an element which is disjoint from it.
8. Subsets. 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. 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:

model-theory

set-theory

axioms

back to home page