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Zermelo-Fraenkel axioms (Axiom)

Ernst Zermelo and Abraham Fraenkel proposed the following axioms as a foundation for what is now called Zermelo-Fraenkel set theory, or ZF. If this set of axioms are accepted along with the Axiom of Choice, it is often denoted ZFC.

  • Equality of sets: If $ X$ and $ Y$ are sets, and $ x \in X$ iff $ x \in Y$, then $ X = Y$.
  • Pair set: If $ X$ and $ Y$ are sets, then there is a set $ Z$ containing only $ X$ and $ Y$.
  • Union over a set: If $ X$ is a set, then there exists a set that contains every element of each $ x \in X$.
  • Axiom of power set: If $ X$ is a set, then there exists a set $ \mathcal{P}(x)$ with the property that $ Y \in \mathcal{P}(x)$ iff any element $ y \in Y$ is also in $ X$.
  • Replacement axiom: Let $ F(x,y)$ be some formula. If, for all $ x$, there is exactly one $ y$ such that $ F(x,y)$ is true, then for any set $ A$ there exists a set $ B$ with the property that $ b \in B$ iff there exists some $ a \in A$ such that $ F(a,b)$ is true.
  • Regularity axiom: Let $ F(x)$ be some formula. If there is some $ x$ that makes $ F(x)$ true, then there is a set $ Y$ such that $ F(Y)$ is true, but for no $ y \in Y$ is $ F(y)$ true.
  • Existence of an infinite set: There exists a non-empty set $ X$ with the property that, for any $ x \in X$, there is some $ y \in X$ such that $ x \subseteq y$ but $ x \neq y$.
  • Separation axiom: If $ X$ is a set and $ P$ is a condition on sets, there exists a set $ Y$ whose members are precisely the members of $ X$ satisfying $ P$.



"Zermelo-Fraenkel axioms" is owned by mathcam. [ full author list (2) | owner history (1) ]
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See Also: axiom of choice, Russell's paradox, von Neumann ordinal, axiom, continuum hypothesis, generalized continuum hypothesis, set theory, von Neumann-Bernays-Gödel set theory, set

Other names:  Zermelo-Fraenkel set theory, ZFC, ZF
Keywords:  set theory
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Cross-references: infinite set, formula, property, contains, iff, equality, axiom of choice, axioms
There are 40 references to this entry.

This is version 14 of Zermelo-Fraenkel axioms, born on 2001-10-18, modified 2006-10-25.
Object id is 317, canonical name is ZermeloFraenkelAxioms.
Accessed 24176 times total.

Classification:
AMS MSC: 03E99 (Mathematical logic and foundations :: Set theory :: Miscellaneous)
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The ZF Axioms of set theory by quantum on 2007-12-07 01:16:59
I looked at some of these ZF axioms of set theory to know what a set is, and they mention property. I clicked on property, it goes to function. I clicked on functions , it goes to relations. I clicked to relations, it goes to cartesian products. I clicked on cartesian products, it goes to ordered pairs. I clicked on ordered pairs, it goes to back to sets.
lol.
[ reply | up ]
effective procedure to determine sethood? by jac on 2004-11-19 10:36:00



I'm trying to write some code that will test an object for
sethood. I'm not sure where to start with this. I could,
for example, assume that every object was a set. Seems sort
of silly, but it is definitely simple. On the other hand, I
suppose I could in theory work up a program that would encode
these axioms (or some other set theoretic axioms) that test
for sethood in a nontrivial way. Does anyone have pointers
for anything like this that is out there already?
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