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Axiom of Choice

The Axiom of Choice is not part of ZF. It is however widely accepted and critical to some proofs. The combination of this axiom and the others in ZF is called ZFC.

The axiom states that for any collection of non empty sets $C$ there exists a choice function $f$ that can select an element from every member of $C$. In other words for every $e\in C$ $f(e)\in e$.


\begin{displaymath}\forall C \exists f \forall e [ (e \in C \wedge e \ne \emptyset) \rightarrow f(e) \in e]\end{displaymath}

A mathematically complete statement of the above requires a definition in the language of set theory of function. A function is a set of ordered pairs where the first element is in the domain of the function and the second element is in the range of the function. Each pair maps an element of the domain uniquely into an element of the range. Thus each first element must occur only once as in the set that defines the function.

Gödel proved that one could construct a model for the axioms of ZF using the constructible sets. Essentially these are the sets one can build up by applying the axioms of ZF. In this model the axiom of choice is true. However Paul Cohen constructed models of ZF in which the Axiom of Choice was false making it clear that this axiom cannot be derived from the other axioms.

It is a strange axiom since it would seem to be obvious. If one has a collection of sets then one should be able to choose one member from each set. But in general there is no way to do this using the axioms of ZF. It is one example of the strange nature of the infinite in formal mathematics. The real numbers derived from the power set allow one to search over all reals. This leads to many other strange questions and another postulate sometimes needed for theorems that is not derivable from the other axioms. This is the Continuum Hypothesis discussed in Section 5.7. Unlike the axiom of choice the Continuum Hypothesis is not generally accepted as true.

Figure 6.4: The axioms of ZFC set theory
\begin{figure} % latex2html id marker 782 \sf \begin{enumerate} \item Axiom of e... ...tyset) \rightarrow f(e) \in e]\end{displaymath}\par \end{enumerate} \end{figure}

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