Definition:Zermelo-Fraenkel Set Theory with Axiom of Choice

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Definition

Zermelo-Fraenkel Set Theory with the Axiom of Choice is a system of axiomatic set theory upon which the whole of (at least conventional) mathematics can be based.

Its basis consists of a system of Aristotelian logic, appropriately axiomatised, together with the Zermelo-Fraenkel axioms of Set Theory and the (controversial) Axiom of Choice.

These are as follows:


The Axiom of Extension

Let $A$ and $B$ be sets.

The Axiom of Extension states that:

$A$ and $B$ are equal

if and only if:

they contain the same elements.


That is, if and only if:

every element of $A$ is also an element of $B$

and:

every element of $B$ is also an element of $A$.


This can be formulated as follows:

$\forall x: \paren {x \in A \iff x \in B} \iff A = B$


The Axiom of the Empty Set

$\exists x: \forall y \in x: y \ne y$


The Axiom of Pairing

For any two sets, there exists a set to which only those two sets are elements:

$\forall a: \forall b: \exists c: \forall z: \paren {z = a \lor z = b \iff z \in c}$


The Axiom of Specification

For any well-formed formula $\map P y$, we introduce the axiom:

$\forall z: \exists x: \forall y: \paren {y \in x \iff \paren {y \in z \land \map P y} }$

where each of $x$, $y$ and $z$ range over arbitrary sets.


The Axiom of Unions

For every set of sets $A$, there exists a set $x$ (the union set) that contains all and only those elements that belong to at least one of the sets in the $A$:

$\forall A: \exists x: \forall y: \paren {y \in x \iff \exists z: \paren {z \in A \land y \in z} }$


The Axiom of Powers

For every set, there exists a set of sets whose elements are all the subsets of the given set.

$\forall x: \exists y: \paren {\forall z: \paren {z \in y \iff \forall w: \paren {w \in z \implies w \in x} } }$


The Axiom of Infinity

There exists a set containing:

$(1): \quad$ a set with no elements
$(2): \quad$ the successor of each of its elements.

That is:

$\exists x: \paren {\paren {\exists y: y \in x \land \forall z: \neg \paren {z \in y} } \land \forall u: u \in x \implies u^+ \in x}$


The Axiom of Replacement

For every mapping $f$ and subset $S$ of the domain of $f$, there exists a set containing the image $f \sqbrk S$.

More formally, let us express this as follows:


Let $\map P {x, z}$ be a propositional function, which determines a mapping.

That is, we have:

$\forall x: \exists ! y : \map P {x, y}$.

Then we state as an axiom:

$\forall A: \exists B: \forall y: \paren {y \in B \iff \exists x \in A : \map P {x, y} }$


The Axiom of Foundation

For all non-empty sets, there is an element of the set that shares no element with the set.

That is:

$\forall S: \paren {\paren {\exists x: x \in S} \implies \exists y \in S: \forall z \in S: \neg \paren {z \in y} }$

The antecedent states that $S$ is not empty.


The Axiom of Choice

For every set of non-empty sets, it is possible to provide a mechanism for choosing one element of each element of the set.

$\ds \forall s: \paren {\O \notin s \implies \exists \paren {f: s \to \bigcup s}: \forall t \in s: \map f t \in t}$

That is, one can always create a choice function for selecting one element from each element of the set.


Also known as

Zermelo-Fraenkel Set Theory with the Axiom of Choice is popularly seen abbreviated as ZFC.


Also see


Source of Name

This entry was named for Ernst Friedrich Ferdinand Zermelo and Abraham Halevi Fraenkel.


Historical Note

ZFC is generally accepted by mathematicians as a "reasonably good foundation" of mathematics.

We are far from claiming superiority of ZFC over alternative foundations of mathematics. For whatever reason, it won the competition. It does a decent job; so let us stick to it. It should be pointed out though that, to the best of our knowledge, none of the competitors to ZFC resolves the question of truth or falsity of CH, SH, MA, $\diamondsuit$, or of any other statement whose independence of ZFC has been established by the method of forcing.
-- 1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics: Introduction


Sources

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