axiom of choice 31

the representation “2 1 0 1,” and it is true that

.

See also

BASE OF A NUMBER SYSTEM

.

axiom (postulate) A statement whose truth is

deemed self-evident or to be accepted without proof is

called an axiom. The name comes from the Greek

word axioma for “worth” or “quality.” The alterna-

tive name “postulate” comes from postulatum, Latin

for “a thing demanded.”

One of the great achievements of the great Greek

geometer E

UCLID

and his contemporaries of around 300

B

.

C

.

E

. was to recognize that not every statement in

mathematics can be proved: certain terms remain unde-

fined, and basic rules (postulates) about their relation-

ships must simply be accepted as true. One must

develop a mathematical theory with a “big bang,” as it

were, by simply listing a starting set of assumptions.

From there, using the basic laws of reasoning, one then

establishes and proves further statements, or

THEO

-

REM

s, about the system.

For example, in a systematic study of E

UCLIDEAN

GEOMETRY

, the terms point, line, and plane are unde-

fined, and one begins a systematic study of the subject

by studying a list of basic axioms that tells us how

these quantities are meant to interrelate. (One axiom

of Euclidean geometry, for instance, asserts that

between any two points one can draw a line.) All the

results presented in a typical high-school text on geom-

etry, for example, are logical consequences of just five

principal assumptions.

In

SET THEORY

, the terms set and element of a set

are undefined. However, rules are given that define the

equality of two sets, that guarantee the existence of cer-

tain sets, and establish the means of constructing new

sets from old ones. In

NUMBER THEORY

, P

EANO

’

S POS

-

TULATES

provide a logical foundation to the theory of

numbers and arithmetic.

A statement in a mathematical system that appears

true, but has not yet been proved, is called a conjecture.

See also

DEDUCTIVE

/

INDUCTIVE REASONING

; E

RNST

F

RIEDRICH

F

ERDINAND

Z

ERMELO

.

axiom of choice First formulated by German mathe-

matician E

RNST

F

RIEDRICH

F

ERDINAND

Z

ERMELO

(1871–1953), the axiom of choice is a basic principle of

SET THEORY

that states that from any given collection C

of nonempty sets, it is possible to construct a set Sthat

contains one element from each of the sets in C. The set S

is called a “choice set” for C. For example, if Crepre-

sents the three sets {1,2,3}, {2,4,6,8,…}, and {5}, then

S= {1,6,5} is a choice set for C. So too is the set S= {2,5}.

The axiom of choice has been considered counter-

intuitive when interpreted on a practical level: although

it is possible to select one element from each of a finite

collection of sets in a finite amount of time, it is physi-

cally impossible to accomplish the same feat when pre-

sented with an infinite collection of sets. The existence

of a choice set is not “constructive,” as it were, and use

of the axiom is viewed by mathematicians, even today,

with suspicion. In 1938 Austrian mathematician K

URT

G

ÖDEL

proved, however, that no contradiction would

ever arise when the axiom of choice is used in conjunc-

tion with other standard axioms of set theory.

Zermelo formulated the axiom to prove that every

ordered set can be well-ordered. The axiom of choice

also proves (and in fact is equivalent to) the trichotomy

law, which states that for any pair of

REAL NUMBERS

a

and b, precisely one of the following holds:

i. a> b

ii. a< b

iii. a= b

Although this statement, on one level, appears obvious,

its validity is fundamental to the workings of the real

numbers and so needs to be properly understood.

See also

WELL

-

ORDERED SET

.