Naive Set Theory by Paul R. Halmos

Author:Paul R. Halmos
Language: eng
Format: epub, pdf
Publisher: INscribe Digital
Published: 2017-04-17T04:00:00+00:00

SECTION 15

THE AXIOM OF CHOICE

For the deepest results about partially ordered sets we need a new set-theoretic tool; we interrupt the development of the theory of order long enough to pick up that tool.

We begin by observing that a set is either empty or it is not, and, if it is not, then, by the definition of the empty set, there is an element in it. This remark can be generalized. If X and Y are sets, and if one of them is empty, then the Cartesian product X × Y is empty. If neither X nor Y is empty, then there is an element x in X, and there is an element y in Y; it follows that the ordered pair (x, y) belongs to the Cartesian product X × Y, so that X × Y is not empty. The preceding remarks constitute the cases n = 1 and n = 2 of the following assertion: if {Xi} is a finite sequence of sets, for i in n, say, then a necessary and sufficient condition that their Cartesian product be empty is that at least one of them be empty. The assertion is easy to prove by induction on n. (The case n = 0 leads to a slippery argument about the empty function; the uninterested reader may start his induction at 1 instead of 0.)

The generalization to infinite families of the non-trivial part of the assertion in the preceding paragraph (necessity) is the following important principle of set theory.

Axiom of choice. The Cartesian product of a non-empty family of nonempty sets is non-empty.

In other words: if {Xi} is a family of non-empty sets indexed by a nonempty set I, then there exists a family {xi}, i ∈ I, such that xi ∈ Xi for each i in I.

Suppose that is a non-empty collection of non-empty sets. We may regard as a family, or, to say it better, we can convert into an indexed set, just by using the collection itself in the role of the index set and using the identity mapping on in the role of the indexing. The axiom of choice then says that the Cartesian product of the sets of has at least one element. An element of such a Cartesian product is, by definition, a function (family, indexed set) whose domain is the index set (in this case ) and whose value at each index belongs to the set bearing that index. Conclusion: there exists a function f with domain such that if A ∈ , then f(A) ∈ A. This conclusion applies, in particular, in case ∈ is the collection of all non-empty subsets of a non-empty set X. The assertion in that case is that there exists a function f with domain (X) – {∅} such that if A is in that domain, then f(A) ∈ A. In intuitive language the function f can be described as a simultaneous choice of an element from each of many sets; this is the reason for the name of the axiom.

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