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From: alopez-o@neumann.uwaterloo.ca (Alex Lopez-Ortiz)
Subject: sci.math FAQ: Relevance of AC
Summary: Part 26 of many, New version,
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THE AXIOM OF CHOICE
There are several equivalent formulations:
* The Cartesian product of nonempty sets is nonempty, even if the
product is of an infinite family of sets.
* Given any set S of mutually disjoint nonempty sets, there is a set
C containing a single member from each element of S . C can thus
be thought of as the result of ``choosing" a representative from
each set in S . Hence the name.
Relevance of the Axiom of Choice
THE AXIOM OF CHOICE
There are many equivalent statements of the Axiom of Choice. The
following version gave rise to its name:
For any set X there is a function f , with domain X\(0) , so that
f(x) is a member of x for every nonempty x in X .
Such an f is called a ``choice function" on X . [Note that X\ (0)
means X with the empty set removed. Also note that in Zermelo-Fraenkel
set theory all mathematical objects are sets so each member of X is
itself a set.]
The Axiom of Choice (AC) is one of the most discussed axioms of
mathematics, perhaps second only to Euclid's parallel postulate. The
axioms of set theory provide a foundation for modern mathematics in
the same way that Euclid's five postulates provided a foundation for
Euclidean geometry, and the questions surrounding AC are the same as
the questions that surrounded Euclid's Parallel Postulate:
1. Can it be derived from the other axioms?
2. Is it consistent with the other axioms?
3. Should we accept it as an axiom?
For many sets, including any finite set, the first six axioms of set
theory (abbreviated ZF) are enough to guarantee the existence of a
choice function but there do exist sets for which AC is required to
show the existence of a choice function. The existence of such sets
was proved in 1963 by Paul Cohen. This means that AC cannot be derived
from the other six axioms; in other words ``AC is independent of ZF."
This answers question [1] posed above.
The question of whether AC is consistent with the other axioms
(question [2] above) was answered by Goedel in 1938. Goedel showed
that if the other axioms are consistent then AC is consistent with
them. This is a ``relative consistency" proof which is the best we can
hope for because of Goedel's Second Incompleteness Theorem.
The third question, ``Should we accept it as an axiom?", moves us into
the realm of philosophy. Today there are three major schools of
thought concerning the use of AC:
1. Accept it as an axiom and use it without hesitation.
2. Accept it as an axiom but use it only when you cannot find a proof
without it.
3. AC is unacceptable.
Most mathematicians today belong to school A. Mathematicians who are
in school B are usually there because of a belief in Occam's Razor
(use as few assumptions as possible when explaining something) or an
interest in metamathematics. There are a growing number of people
moving to school C, especially computer scientists who work on
automated reasoning using constructive type theories.
Underlying the schools of thought about the use of AC are views about
truth and the nature of mathematical objects. Three major views are
platonism, constructivism, and formalism.
Platonism
A platonist believes that mathematical objects exist independent of
the human mind, and a mathematical statement, such as AC, is
objectively either true or false. A platonist accepts AC only if it is
objectively true, and probably falls into school A or C depending on
her belief. If she isn't sure about AC's truth then she may be in
school B so that once she finds out the truth about AC she will know
which theorems are true.
Constructivism
A constructivist believes that the only acceptable mathematical
objects are ones that can be constructed by the human mind, and the
only acceptable proofs are constructive proofs. Since AC gives no
method for constructing a choice set constructivists belong to school
C.
Formalism
A formalist believes that mathematics is strictly symbol manipulation
and any consistent theory is reasonable to study. For a formalist the
notion of truth is confined to the context of mathematical models,
e.g., a formalist would say "The parallel postulate is false in
Riemannian geometry." but she wouldn't say "The parallel postulate is
false." A formalist will probably not allign herself with any school.
She will comfortably switch between A, B, and C depending on her
current interests.
So: Should you accept the Axiom of Choice? Here are some arguments for
and against it.
Against
* It's not as simple, aesthetically pleasing, and intuitive as the
other axioms.
* It is equivalent to many statements which are not intuitive such
as "Every set can be well ordered." How, for example, would you
well order the reals?
* With it you can derive non-intuitive results, such as the
existence of a discontinuous additive function, the existence of a
non-measurable set of reals, and the Banach-Tarski Paradox (see
the next section of the sci.math FAQ).
* It is nonconstructive - it conjures up a set without providing any
sort of procedure for its construction.
For
The acceptance of AC is based on the belief that our intuition about
finite sets can be extended to infinite sets. The main argument for
accepting it is that it is useful. Many important, intuitively
plausible theorems are equivalent to it or depend on it. For example
these statements are equivalent to AC:
* Every vector space has a basis.
* Trichotomy of Cardinals: For any cardinals k and l , either k < l
or k = l or k > l .
* Tychonoff's Theorem: The product of compact spaces is compact in
the product topology.
* Zorn's Lemma: Every nonempty partially ordered set P in which each
chain has an upper bound in P has a maximal element.
And these statements depend on AC (i.e., they cannot be proved in ZF
without AC):
* The union of countably many countable sets is countable.
* Every infinite set has a denumerable subset.
* The Loewenheim-Skolem Theorem: Any first-order theory which has a
model has a denumerable model.
* The Baire Category Theorem: The reals are not the union of
countably many nowhere dense sets (i.e., the reals are not
meager).
* The Ultrafilter Theorem: Every Boolean algebra has an ultrafilter
on it.
Alternatives to AC
* Accept only a weak form of AC such as the Denumerable Axiom of
Choice (every denumerable set has a choice function) or the Axiom
of Dependent Choice.
* Accept an axiom that implies AC such as the Axiom of
Constructibility ( V = L ) or the Generalized Continuum Hypothesis
(GCH).
* Adopt AC as a logical axiom (Hilbert suggested this with his
epsilon axiom). If set theory is done in such a logical formal
system the Axiom of Choice will be a theorem.
* Accept a contradictory axiom such as the Axiom of Determinacy.
* Use a completely different framework for mathematics such as
Category Theory. Note that within the framework of Category Theory
Tychonoff's Theorem can be proved without AC (Johnstone, 1981).
Test Yourself: When is AC necessary?
If you are working in Zermelo-Fraenkel set theory without the Axiom of
Choice, can you choose an element from...
1. a finite set?
2. an infinite set?
3. each member of an infinite set of singletons (i.e., one-element
sets)?
4. each member of an infinite set of pairs of shoes?
5. each member of inifinite set of pairs of socks?
6. each member of a finite set of sets if each of the members is
infinite?
7. each member of an infinite set of sets if each of the members is
infinite?
8. each member of a denumerable set of sets if each of the members is
infinite?
9. each member of an infinite set of sets of rationals?
10. each member of a denumerable set of sets if each of the members is
denumberable?
11. each member of an infinite set of sets if each of the members is
finite?
12. each member of an infinite set of finite sets of reals?
13. each member of an infinite set of sets of reals?
14. each member of an infinite set of two-element sets whose members
are sets of reals?
The answers to these questions with explanations are accessible
through http://www.jazzie.com/ii/math/index.html
References
Benacerraf, Paul and Putnam, Hilary. "Philosophy of Mathematics:
Selected Readings, 2nd edition." Cambridge University Press, 1983.
Dauben, Joseph Warren. "Georg Cantor: His Mathematics and Philosophy
of the Infinite." Princeton University Press, 1979.
A. Fraenkel, Y. Bar-Hillel, and A. Levy with van Dalen, Dirk.
"Foundations of Set Theory, Second Revised Edition." North-Holland,
1973.
Johnstone, Peter T. "Tychonoff's Theorem without the Axiom of Choice."
Fundamenta Mathematica 113: 21-35, 1981.
Leisenring, Albert C. "Mathematical Logic and Hilbert's
Epsilon-Symbol." Gordon and Breach, 1969.
Maddy, "Believing the Axioms, I", J. Symb. Logic, v. 53, no. 2, June
1988, pp. 490-500, and "Believing the Axioms II" in v.53, no. 3.
Moore, Gregory H. "Zermelo's Axiom of Choice: Its Origins,
Development, and Influence." Springer-Verlag, 1982.
Rubin, Herman and Rubin, Jean E. "Equivalents of the Axiom of Choice
II." North-Holland, 1985.
This section of the FAQ is Copyright (c) 1994 Nancy McGough. Send
comments and or corrections relating to this part to nancym@ii.com.
The most up to date version of this section of the sci.math FAQ is
accesible through http://www.jazzie.com/ii/math/index.html
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