534 zero

transfinite cardinals), and 2 years later succeeded in

proving the well-ordering conjecture in his 1904 article

“Beweis, dass jede Menge wohlgeordnet werden kann”

(Proof that every set can be well ordered). His success

garnered him international fame and recognition as a

brilliant mathematician. The University of Göttingen

immediately awarded him a full professorship.

In 1905, leaving further work on the continuum

hypothesis aside, Zermelo began the task of axiomatiz-

ing set theory. By identifying a small set of basic

assumptions about the topic, he hoped to resolve some

of the paradoxes about infinite sets that were arising in

the subject. Unfortunately, he found it necessary to

include an “axiom of infinity” asserting, essentially, that

infinite sets exist, along with the extraordinarily contro-

versial

AXIOM OF CHOICE

, which asserts that it is always

possible to select, in one fell swoop, one object from

each set in any infinite collection of sets. Mathemati-

cians were uncomfortable with each of these assump-

tions at the time. Moreover, despite his efforts, Zermelo

was unable to prove that the axioms he put forward

were free from contradictions. Other mathematicians

then began to work on this task, and some 15 years

later, in 1922, mathematicians Adolf Fraenkel and Tho-

ralf Skolem independently developed a refined system of

10 axioms that were deemed of sufficiently good stead.

Zermelo left Göttingen in 1910 to take an

appointment as the chair of mathematics at Zurich

University, where he stayed for 6 years. Zermelo spent

the remainder of his life in Freiburg. He died there on

May 21, 1953.

See also R

USSELL

’

S PARADOX

.

zero (naught) The counting number, denoted 0, used

to signify that no objects are present is called zero. This

number has the arithmetical property that its addition

to any other number does not change that number:

a+0 = a= 0 + afor all values a. Thus zero serves as the

additive

IDENTITY ELEMENT

in arithmetic. Zero is the

only real number that is neither

POSITIVE

nor negative.

It was only recently in the history of mathematics

that zero was recognized as a valid and important math-

ematical entity. In times of antiquity numbers were

thought only to represent counts or magnitudes of quan-

tities. Thus if there were no objects to be counted, there

was no number to consider (and so speaking of zero as a

number consequently had no meaning). The ancient

Egyptian and Greek scholars never used zero in their

computations. The ancient Babylonians were the first to

utilize a notion of zero by using it, in some sense, as a

placeholder when expressing large numbers (much as we

use zeros today to distinguish between 102 and 12.) The

Hindu mathematician B

RAHMAGUPTA

(ca. 598–665) is

credited today as being the first scholar to acknowledge

zero as a valid number and use it in arithmetical compu-

tations. His idea was later expanded upon by M

UHAM

-

MAD IBN

M–

US

–

AAL

-K

HW

–

ARIZM

–

ı (ca. 780–850) in his

famous development of a theory of algebra. Italian

mathematician F

IBONACCI

(ca. 1170–1250) popularized

much of al-Khw–

arizm

–

ı’s work in Western Europe, but it

was not until the Renaissance that the notion of zero

was finally deemed a valid concept.

Arithmetic involving zero can be troublesome.

Although any number can be multiplied by zero (to

produce the answer zero), no number can be divided by

zero. For instance, there can be no answer to the divi-

sion problem 1 ÷ 0. (If 1/0 = x, then multiplying

through by the denominator gives the absurd result: 1

= x×0 = 0.) The quantity 0/0 is also deemed unde-

fined. (One could argue that the answer to this division

problem is 17, since 0 = 17 ×0, or, by the same token,

that the answer is 2 since 0 = 2 ×0. There is no single

well-defined solution to 0 ÷ 0.)

A counting number is considered even if it can be

written as a sum of two equal whole numbers. Since

zero equals 0 + 0, it follows that zero is an even number.

The study of E

XPONENT

s shows that it is appropri-

ate to define x0to be 1 for any real number xother

than zero. (For example, 20= 1 and 1730= 1.)

Although it is true that limx→0+xx= 1, it is not valid to

assume that 00also equals 1. (This

LIMIT

does not exist

if one permits

COMPLEX NUMBERS

in our considera-

tions, and so the notion of raising zero to the zeroth

power is problematic in this setting.) A study of the

FACTORIAL

shows that it is appropriate to define 0! = 1.

A

ROOT

of a function is sometimes called a zero of

the function. A zero function is a function whose out-

put is always zero. For example, the function f(x) =

defined on positive values of xis always

zero. In

GAME THEORY

, a zero-sum game is a game in

which a win for some player or players is always bal-

anced by losses for the remaining players. A zero matrix

is a

MATRIX

all of whose entries are zero.