Axiom of choice

From Academic Kids

In mathematics, the axiom of choice is an axiom of set theory. It was formulated in 1904 by Ernst Zermelo. While it was originally controversial, it is now accepted and used casually by most mathematicians. However, there are still schools of mathematical thought, primarily within set theory, which either reject the axiom of choice, or even investigate consequences of its negation.

The axiom of choice is typically abbreviated AC, or C as a suffix.

Contents

Statement

The axiom of choice states: Template:Axiom Stated more formally: Template:Axiom Another formulation of the axiom of choice states: Template:Axiom

Usage

Until the late 19th century, the axiom of choice was often used implicitly. For example, after having established that the set S contains only non-empty sets, a mathematician might have said "let F(X) be one of the members of X for all X in S." In general, it is impossible to prove that F exists without the axiom of choice, but this seems to have gone unnoticed until Zermelo.

Not every situation requires the axiom of choice. For finite sets X, the axiom of choice follows from the other axioms of set theory. It is equivalent to saying that if we have several (finite number of) boxes, each containing at least one item, then we can choose exactly one item from each box. Clearly we can do this: We start at the first box, choose an item; go to the second box, choose an item; and so on. There are only finitely many boxes, so eventually we stop. This gives us an explicit choice function. It takes the first box to the first element we chose, the second box to the second element, and so on.

For certain infinite sets X, it is also possible to avoid the axiom of choice. For example, suppose that we have some sets of natural numbers. Every set of natural numbers has a least element, so to construct our choice function we choose the least element of each set. This gives us a definite choice of an element from each set, that is, we can write down an explicit expression that tells us what value our choice function takes. Indeed, any time it is possible to make such an explicit choice, the axiom of choice is unnecessary.

The difficulty appears when there is no natural choice of elements from each set. If we can't make explicit choices, how do we know that our set exists? For example, suppose that X is the set of all non-empty subsets of the real numbers. First we might try to proceed as if X were finite. If we try to choose an element from each set, then, because X is infinite, we will never stop, and consequently, we will never be able to produce a choice function for all of X. So that won't work. Next we might try to choose the least element from each set, as if X were subsets of the natural numbers. But some subsets of the real numbers don't have least elements, for example, { x | x > 0}. So that won't work, either.

The reason why we could choose subsets of the natural numbers is because the natural numbers are well-ordered: Every subset of the natural numbers has a unique least element. Perhaps if we were clever we might say, "Even though the usual ordering of the real numbers does not work, it may be possible to find a different ordering of the real numbers which is a well-ordering. Then our choice function can choose the least element of every set under our unusual ordering." The problem then becomes constructing such an ordering, and it turns out that every set can be well-ordered if and only if the axiom of choice is true.

Proofs involving the axiom of choice are always nonconstructive: They produce an object, but it is impossible to say exactly what that object is. Consequently, while the axiom of choice asserts that there is a well-ordering of the real numbers, it doesn't tell you what that ordering is. Yet the reason why we chose above to well-order the real numbers was so that for each set in X, we could explicitly choose an element of that set. If we can't write down the well-ordering we are using, then our choice is not very explicit! This is one of the reasons why some mathematicians dislike the axiom of choice. For example, constructivists believe that all existence proofs should be totally explicit; you should be able to construct anything that exists. They reject the axiom of choice because it asserts the existence of an object without telling you what it is.

Independence of AC

By work of Kurt Gdel and Paul Cohen, the axiom of choice is logically independent of the other axioms of Zermelo-Fraenkel set theory (ZF). This means that neither it nor its negation can be proven to be true in ZF. Consequently, assuming the axiom of choice, or its negation, or neither will never lead to a contradiction that you couldn't get otherwise; in other words, the axiom of choice on its own does not cause mathematics to fall apart. Because of this, one argument given in favor of the axiom of choice is that it is convenient: It doesn't hurt, and it's easier to use the axiom of choice than not.

One reason that some mathematicians dislike the axiom of choice is that it implies the existence of some bizarre counter-intuitive objects. An example of this is the Banach-Tarski paradox which says in effect that it is possible to "carve up" the 3-dimensional solid unit ball into finitely many pieces and, using only rotation and translation, reassemble the pieces into two balls each with the same volume as the original. Note that the proof, like all proofs involving the axiom of choice, is an existence proof only: it does not tell us how to carve up the unit sphere to make this happen, it simply tells us that it can be done.

On the other hand, the negation of the axiom of choice is also bizarre. For example, the statement that for any two sets S and T, either the cardinality of S is less than or equal to the cardinality of T or the cardinality of T is less than or equal to the cardinality of S is equivalent to the axiom of choice. Put differently, if the axiom of choice is false, then there are sets S and T of incomparable size: Neither can be mapped in a one-to-one fashion onto a subset of the other.

A third possibility is to prove theorems using neither the axiom of choice nor its negation, a tactic often preferred in constructive mathematics. Such statements will be true in any model of Zermelo-Fraenkel set theory, regardless of the truth or falsity of the axiom of choice in that particular model. This renders any claim that relies on either the axiom of choice or its negation undecidable. For example, under such an assumption, the Banach-Tarski paradox is neither true nor false: It is impossible to construct a decomposition of the unit ball which can be reassembled into two unit balls, and it is also impossible to prove that it can't be done. However, the Banach-Tarski paradox can be rephrased as a statement about models of ZF by saying, "In any model of ZF in which AC is true, the Banach-Tarski paradox is true." Similarly, all the statements listed below under Results requiring AC are undecidable in ZF, but since each is provable in any model of ZFC, there are models of ZF in which each statement is true.

Weaker versions of choice

There are several weaker statements which are not equivalent to the axiom of choice, but which are closely related. One simple one is the axiom of countable choice, which states that a choice function exists for any countable set X. This usually suffices when trying to make statements about the real numbers, for example, because the rational numbers, which are countable, form a dense subset of the reals. See also the Boolean prime ideal theorem and the axiom of dependent choice.

Results requiring AC

One of the most interesting aspects of the axiom of choice is the large number of places in mathematics that it shows up. There are also a remarkable number of important statements that, assuming the axioms of ZF but neither AC nor ¬AC, are equivalent to the axiom of choice. The most important among them are Zorn's lemma and the well-ordering theorem: every set can be well-ordered. In fact, Zermelo initially introduced the axiom of choice in order to formalize his proof of the well-ordering principle. Here are some statements that require the axiom of choice or a weakened version of it:

  • Set theory
    • Any union of countably many countable sets is itself countable.
    • If the set A is infinite, then there exists an injection from the natural numbers N to A.
    • If the set A is infinite, then A and A×A have the same cardinality.
    • If two sets are given, then they either have the same cardinality, or one has a smaller cardinality than the other.
    • The product of any nonempty family of nonempty sets is nonempty.

Results requiring ¬AC

There are models of Zermelo-Fraenkel set theory in which the axiom of choice is false. We will abbreviate "Zermelo-Fraenkel set theory plus the negation of the axiom of choice" by ZF¬C. For certain models of ZF¬C, it is possible to prove the negation of some standard facts. We do not claim that the following statements are true in all models of ZF¬C, only that there exists a model of ZF¬C in which the statement is true. Note that any model of ZF¬C is also a model of ZF, so for each of the following statements, there exists a model of ZF in which that statement is true.

  • There exists a model of ZF¬C in which there is a function f from the real numbers to the real numbers such that f is not continuous at a, but for any sequence {xn} converging to a, limn f(xn)=f(a).
  • There exists a model of ZF¬C in which real numbers are a countable union of countable sets.
  • There exists a model of ZF¬C in which there is a field with no algebraic closure.
  • There exists a model of ZF¬C in which there is a vector space with no basis.
  • There exists a model of ZF¬C in which there is a vector space with two bases of different cardinalities.

For proofs, see Thomas Jech, The Axiom of Choice, American Elsevier Pub. Co., New York, 1973.

Quotes

The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?
— Jerry Bona
This is a joke that although the axiom of choice, the well-ordering principle, and Zorn's lemma are mathematically equivalent, most mathematicians find the axiom of choice to be intuitive, the well-ordering principle to be counterintuitive, and Zorn's lemma to be too complex for any intuition.
The Axiom of Choice is necessary to select a set from an infinite number of socks, but not an infinite number of shoes.
— Bertrand Russell
The observation here is that one can define a function to select from an infinite number of pairs of shoes by stating for example, to choose the left shoe. Without the axiom of choice, one cannot assert that such a function exists for pairs of socks, because left and right socks are (presumably) identical to each other.
The axiom gets its name not because mathematicians prefer it to other axioms.
— A. K. Dewdney
From the famous April Fool's Day article in the computer recreations column of the Scientific American, April 1989.

External links

  • A leisurely introduction to the axiom, popular consequences, and further links are found at Eric Schechter's homepage (http://www.math.vanderbilt.edu/~schectex/ccc/choice.html).
  • There are many people still doing work on the axiom of choice and its consequences. If you are interested in more, look up Paul Howard at EMU (http://www.emunix.emich.edu/~phoward/).

da:Udvalgsaksiomet de:Auswahlaxiom fr:Axiome du choix it:Assioma della scelta ko:선택공리 nl:Keuzeaxioma ja:選択公理 pl:Aksjomat wyboru ru:Аксиома выбора sv:Urvalsaxiomet

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