Non-Cantorian Set Theory

[n 1963 it (;vas proved hyp othesis

that a celebrated 7nathematical

put forward by Georg Cantor could not be proved. This profound

developl11ent is explained by analogy with non-Euclidean ge07netry

by Paul J. Cohen and Reuben Hersh

T
he abstract theory of sets is cur­ an infinite number of distinct elements; esting only if it could be shown that not
rently in a state of change that in for example, the set of all "natural" num­ all infinite sets have the same cardinal­
several ways is analogous to the bers (1, 2, 3 and so on) is infinite. So too ity. It was this that was Cantor's first
19th-century revolution in geometry. As is the set of all the points on a given line great discovery in set theory. By his fa­
in any revolution, political or scientific, segment. mous diagonal proof he showed that the
it is difficult for those participating in Cantor pointed out that even for in­ set of natural numbers is not equivalent
the revolution or witnessing it to fore­ finite sets it makes sense to talk about to the set of points on a line segment
tell its ultimate consequences, except the number of elements in the set, or at [see illustration on opposite pageJ.
perhaps that they will be profound. One least to state that two different sets have Thus there are at least two different
thing that can be done is to try to use the same number of elements. Just as kinds of infinity. The first, the infinity of
'
the past as a guide to the future. It is with finite sets, we can say that two the natural numbers (and of any equiv­
an unreliable guide, to be sure, but bet­ sets have the same number of elements alent infinite sets), is called aleph nought
ter than none. -the same "cardinality"-if we can (�o). Sets with cardinality �o are called
'Ve propose in this article to use the match up the elements in the two sets countable. The second kind of infinity
oft-told tale of non-Euclidean geometry one for one. If this can be done, we call is the one represented by a line segment.
to illuminate the now unfolding story of the two sets equivalent. Its cardinality is designated by a lower­
nonstandard set theory. The set of all natural numbers can case German c (c), for "continuum." Any
A set, of course, is one of the simplest be matched up with the set of all even line segment, of arbitrary length, has
and most primitive ideas in mathemat­ numbers, and also with the set of all cardinality c [see illustration on page
ics, so simple that today it is part of the fractions [see illustration belowJ. These 106J. SO does any rectangle in the plane,
kindergarten curriculum. No doubt for two examples illustrate a paradoxical any cube in space, or for that matter
this very reason its role as the most fun­ property of infinite sets: an infinite set all of unbounded n-dimensional space,
damental concept of mathematics was can be equivalent to one of its subsets. whether 11 is 1, 2, 3 or 1,000!
not made explicit until the 1880's. Only In fact, it is easily proved that a set is
then did Georg Cantor make the first
nontrivial discovery in the theory of sets.
infinite if, and only if, it is equivalent to
some proper subset of itself. O infinities
nce a single step up the chain of
has been taken, the next
To describe his discovery we must All of this is engaging, but it was not follows naturally. \Ve encounter the no­
first explain what we mean by an infinite new with Cantor. The notion of the car­ tion of the set of all subsets of a given
set. An infinite set is merely a set with dinality of infinite sets would be inter- set [see illustration on page 111J. If the

2., 4, 6, 8., 10, 12, 14, 16, 18, '20, 22, •••

t t t t t t t t t t t
1, 2, 3, 4, 5, 6, 7.., 8, 9, 10, 11, •••

t t t t t t t t t t t
1/1, 1;2, 2/1, 1;3, 2/2, 3/1, I,ll, �3., 3/2, 4/1, 1/5, •••

SET IS TERMED COUNTABLE if it can be matched one for one by the German mathematician Georg Cantor 0845-1918); they are
with the natural numbers (middle row). Thus the set of all even not in their natural order but in order according to the sum of the
numbers (top row) is countable. The set of all fractions (bottom numerator and the denominator. Both examples show that an infi.
row) is also countable. The fractions shown here are the ones used nite set, unlike a finite set, can be e«(uivalent to one of its subsets.

104

© 1967 SCIENTIFIC AMERICAN, INC

 1. .18347984639001 ...
original set is called A, this new set is
called the power set of A and is written
2A. And just as we obtain the power set
2"1 from A, we can next obtain 2(:!A)
from 2A, and so on as long as we please.
Cantor proved that whether A is finite
2. .36948570110924 ...
3. 504722 00173996...
or infinite, 2·{ is never equivalent to A.
Therefore the procedure of forming the .
set of all subsets generates an endless
chain of increasing, nonequivalent in­
finite sets. In particular, if A is the set
of natural numbers, then it is easy to 4. .99801230109487 .. .

prove that 2·� (the set of all sets of nat­

ural numbers) is equivalent to the con­
tinuum (the set of all points on a line 5. .00102305497 610 ...
segment). In brief,

6. .51546798371238...
7. .551198 71350426...
At this point a question may occur
to the reader. Is there an infinite set with
cardinality between X 0 and c? That is,
is there on a line segment an infinite set • •
• •
of points that is not equivalent to the • •
whole segment, and also not equivalent
to the set of natural numbers? SET OF REAL NUMBERS IS UNCOUNTABLE, as Cantor showed in his famous diagonal

This question occurred to Cantor, but proof. Here a sample of the set is listed in decimal form and at random. If one takes the
first digit of the first number, the second digit of tbe second and so on (color), one obtains
he was unable to find any such set. He
a real number whose infinite decimal expansion is .1640277 . . . . If one randomly cbanged
concluded-or rather conjectured-that
every digit in tbe expansion, one might get .2751388 . . . . A moment's thought will show that
no such thing exists. This guess of Can­ tbe new number is different in at least one place from every number on the list. Hence the
tor's acquired the name "the continuum number was not present on the list, and it has been proved that the list was incomplete.
hypothesis." Its proof or disproof was
first on the celebrated list of unsolved
mathematical problems drawn up by exactly 11 English words," having the are the paradoxes of Zeno, which re­
David Hilbert in 1900. Only in 1963 was peculiar property that they themselves vealed to the Greeks unsuspected com­
it finally settled. It was settled, however, satisfy their defining property; in other plexities in intuitive concepts of lines and
in a sense utterly different from what words, sets that contain themselves as points. vVe can draw an analogy: As
Hilbert had in mind. elements. We call them R sets, the R Russell found a contradiction in the un­
To tackle this problem one could no standing for Russell. Then there are all restricted use of the intuitive concept of
longer rely on Cantor's definition of a other sets-sets that do not belong to set, so Zeno had found contradictions in
set as "any collection into a whole of themselves. Call them the non-R sets. the unrestricted use of the intuitive con­
definite and separate objects of our in­ Now, said Russell, consider the collection cepts of "line" and "point."
tuition or our thought." In fact, this defi­ of all non-R sets. (The word "collection" In its beginning with Thales in the
nition, seemingly so transparent, turned is introduced here simply as a convenient sixth century B.C. Greek geometry had
out to conceal some treacherous pitfalls. synonym for "set.") Call this set M. Then relied on an unspecified intuitive con­
An instance is the sad experience suf­ M is either an R set or a non-R set. But if cept of "line" and "point." Some 300
fered by Gottlob Frege in 1902. Frege M is a non-R set, then it belongs to M, years later, however, Euclid had given.
was about to publish a monumental work by definition of M, so that it is an R set, these concepts an axiomatic treatment.
in which arithmetic was reconstructed by definition of R sets. This is a contra­ For Euclid geometric objects were still
on the foundation of set theory, that diction. On the other hand, if M is an R intuitively known real entities, but inso­
is, on the foundation of "intuitive" set set, then by definition of M it does not far as they were the subject of geometri­
theory as it was then known on the basis belong to M. It does not belong to itself, cal reasoning they were specified by
of Cantor's work. At this point Frege re­ that is, it is not an R set, which is again certain unproved assertions ("axioms"
ceived a letter from the young Bertrand a contradiction. and "postulates"), on the basis of which
Russell, which he acknowledged by add­ The moral is this: The free use of all their other properties were supposed
ing this postscript to his treatise: "A sci­ Cantor's intuitive notion of a set can lead to be proved as "theorems." \Ve do not
entist can hardly meet with anything to contradictions. Set theory can serve know if, and to what extent, this devel­
more undesirable than to have the foun­ as a secure foundation for mathematics opment was a response to paradoxes such
dation give way just as the work is fin­ only if a more sophisticated approach is as Zeno's. There is no doubt, however,
ished. In this position I was put by a employed to steer clear of antinomies, as that to the Greeks geometry was made
letter from Mr. Bertrand Russell as the contradictions of the type proposed by much more secure by virtue of depend­
work was nearly through the press." Russell later came to be known. ing (at least so they believed and intend­
Russell's blow consisted in pointing ed) only on logical inference from a small

I
out a simple conundrum. There are two t has happened before that unwelcome number of clearly stated assumptions.
kinds of sets. First there are those, such paradoxes have intruded into a seem­ The analogous development for set
as "the set of all objects describable in ingly clear mathematical theory. There theory took not 300 years but only 35. If

105

© 1967 SCIENTIFIC AMERICAN, INC

Cantor played the role of Thales-the that through a given point there can be acceptance of the parallel postulate is an
founder of the subject, who was able tu drawn precisely one line parallel to a act of faith about how lines would act if
rely on intuitive reasoning alone-then given line. The difficulty with this state­ they were extended to infinity. It turns
the role of Euclid was played by Ernst ment as an axiom is that it does not have out that from the innocent-seeming ax­
Zermelo, who in 1908 founded axio­ the self-evident character one prefers in iom of choice some unexpected and ex­
matic set theory. Of course, Euclid was the foundation stones of a mathematical tremely powerful conclusions follow. For
really only one of a long succession of theory. In fact, parallel lines are defined example, we are able to use inductive
Creek geometers who created "Euclide­ as lines that never meet, even if they reasoning to prove statements about the
an geometry"; so also Zermelo was only are extended indefinitely ("to infinity"). elements in any set, in much the same
the first of half a dozen great names in Since any lines we draw on paper or on way that mathematical induction can be
the creation of axiomatic set theory. a blackboard have finite length, this is used to prove theorems about the nat­
Just as Euclid had listed certain prop­ an axiom that by its nature cannot be ural numbers 1, 2, 3 and so on.
erties of points and lines and had regard­ verified by direct observation of the The axiom of choice played a special
ed as proved only those theorems in senses. Nonetheless, it plays an indispens­ role in set theory. Many mathematicians
geometry that could be obtained from able role in Euclidean geometry. For thought its use should be avoided when­
these axioms (and not from any possibly many centuries a leading problem in ge­ ever possible. Such a form of axiomatic
intuitive arguments), so in axiomatic set ometry was to p1'ove the parallel postu­ set theory, in which the axiom of choice
theory a set is regarded simply as an un­ late, to show that it could be obtained is not assumed to be either true or false,
defined object satisfying a given list of as a theorem from the more self-evident would be one on which almost all mathe­
axioms. Of course, we still want to study Euclidean axioms. maticians would be prepared to rely. In
sets (or lines, as the case may be), and In abstract set theory, it so happens, what follows we use the term "restricted
so the axioms are chosen not arbitrarily there also was a particular axiom that set theory" for such an axiom system. vVe
but in accord with our intuitive notion of some mathematicians found hard to use the term "standard set theory" for
a set or a line. Intuition is nonetheless swallow. This was the axiom of choice, the theory based on the full set of axioms
barred from any further formal role; only which says the following: If a is any col­ put forward by Zermelo and Abraham
those propositions are accepted that fol­ lection of sets lA, B, . . . }, and none of Fraenkel: restricted set theory plus the
low from the axioms. The fact that ob­ the sets in a is empty, then there exists a axiom of choice.
jects described by these axioms actually set Z consisting of precisely one element

I
may exist in the real world is irrelevant each from A, from B and so on through n 1938 this subject was profoundly il-
to the process of formal deduction (al­ all the sets in a. For instance, if a consists luminated by Kurt Code!. Codel is
though it is essential to discovery). of two sets, the set of all triangles and best known for his great "incomplete­
vVe agree to act as if the symbols for the set of all squares, then a clearly sat­ ness" theorems of 1930-1931 [see "Co­
"line," "point" and "angle" in geometry, isfies the axiom of choice. We merely del's Proof," by Ernest Nagel and James
or the symbols for "set," "is a subset of" choose some particular triangle and some R. Newman; SCIENTIFIC AMERICAN,
and so on in set theory, are mere marks particular square and then let these two June, 1956]. Here we refer to later work
on paper, which may be rearranged only elements constitute Z. by Codel that is not well known to non­
according to a given list of rules (axioms Most people find the axiom of choice, mathematicians. In 1938 Codel proved
and rules of inference). Accepted as like the parallel postulate, intuitively the following fundamental result: If re­
theorems are only those statements that very plausible. The difficulty with it is stricted set theory is consistent, then so
are obtained according to such manipu­ in the latitude we allow a: "any" collec­ is standard set theory. In other words,
lations of symbols. (In actual practice tion of sets. As we have seen, there are the axiom of choice is no more dangerous
only those statements are accepted that endless chains of ever bigger infinite than the other axioms; if a contradiction
clearly could be obtained in this manner sets. For such an inconceivably huge col­ can be found in standard set theory, then
if one tuok enough time and trouble.) lection of sets there is no way of actually there must already be a contradiction
choosing one by one from all its member hidden within restricted set theory.

N
ow, in the history of geometry one sets. If we accept the axiom of choice, But that was not all Codel proved.
postulate played a special role. This our acceptance is simply an act of faith vVe remind the reader of Cantor's "con­
was the parallel postulate, which says that such a choice is possible, just as our tinuum hypothesis," namely that no in-

p s

--��---- ��--------�----------�=-----�----��---------- -w�----------�__-L

INFINITE LINE AND FINITE LINE SEGMENT can also be match between points on S and points on L. As the ray changes di­
shown to have a one-to-one correspondence. Here P is the center of rection from left to right no point is omitted from either S or L.
a semicircle S that is tangent to an infinite line L. A ray from P cuts Thus a one-to-one correspondence exists between the points on an
S at only one point. In this way rays from P give a one-to-one infinite line and the points on a finite segment of arbitrary length.

106

© 1967 SCIENTIFIC AMERICAN, INC

 Just getting there will be like shooting a bee in flight
with an air rifle from a whirling merry-go-round at a range
of 100 yards. Getting back will be even tougher.
But any analogy is far too simple. To return, the lunar module must port services in such vital areas as
For NASA's lunar-landing Projec t take off from the moon and rendez­ communications, timing, instrumenta­
Apollo beggars the imagination. vous with the command module, tion, computer programming, reliabil­
One day in the early 1970's a Saturn which then must make the voyage ity, technical information, and logis­
V booster, developing 7.5 million back to earth. tics for NASA at its Kennedy, Hunts­
pounds of thrust, will launch three Meanwhile, a corps of engineers, ville, and Houston space installations.
astronauts in combined command and scientists and technicians back on earth The success of NASA's Project
lunar modules toward the moon, al­ will be praying, sweating and doing Apollo will be another tremendous
most 240,000 miles away. the jobs they've trained for-and doing breakthrough in mankind's knowl­
Once there, the command module them right. edge, and every American will have
continues its moon orbit while the More than two thousand of them good reason to stand a bit taller.
lunar module ferries two of the astro­ will be from Federal Electric Corpora­ International Telephone and Tele­
nauts to the moon's surface. tion, an ITT subSidiary, providing sup- graph Corporation, N.Y., N.Y. 10022.

ITT
© 1967 SCIENTIFIC AMERICAN, INC
© 1967 SCIENTIFIC AMERICAN, INC
This philosopher wants tomorrow's
students to get the best teaching
possible-with or without computers.

What's he doing at IBM?

Says Bruse Moncreiff, a philosopher by training: "You'll find computers on many

campuses today, doing many things. And one current experiment which may prove
increasingly important is computer-assisted instruction (CAl). But we must learn
from the teachers-let them decide if computers can be useful as a teaching aid."
That's why IBM's Moncreiff spends most of his time working with teachers. He
finds divergent points of view about CAl, but some things are clear.
"Today, both children and adults must be better educated, to survive in a world
where change has become the norm," says Moncreiff. "And because each person
is different from the next, one ideal method of teaching is one-to-one- individual
instruction."
It's impossible to have a teacher for every student. But it might be possible for
teachers to at least approach this ideal with the help of computers.
In one experimental method, a student sits at a typewriter that
is linked to a computer. The computer types out ques­
tions. The student types back answers. If he's
right, he gets a more difficult question. If
he's wrong, he gets a hint; and if he really
bogs down, the suggestion, "better
see your teacher." The teacher can
help the student where he needs it,
while the other members of the
class continue uninterrupted.
"The computer's role as a
teaching aid demands hard
t h o u g h t ," s a y s M o n c r e i f f.
"Promising as it may seem to
us, we must take our lead
from those who know the
most-the teachers them­
selves. It is they who will
have the final say."

IBM ®

© 1967 SCIENTIFIC AMERICAN, INC

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© 1967 SCIENTIFIC AMERICAN, INC

 {.,�,e}
finite cardinal exists that is greater than
Xo and smaller than c. Code! also
showed that we can safely take the con­
A:
tinuum hypothesis as an additional ax­
iom in set theory; that is, if the continu­
um hypothesis plus restricted set theory
implies a contradiction, then again there

{ J
must already be a contradiction hidden
within restricted set theory. This was a
half-solution of Cantor's problem; it was
not a proof of the continuum hypothesis

{.} {.,�}
but only a proof that it cannot be dis­
proved.
To understand how Codel achieved
his results we need to understand what
2A:
{�J (.,e)
is meant by a model for an axiom sys­
tem. Let us return for a moment to the
axioms of plane geometry. If we take
these axioms, including the parallel pos­
tulate, we have the axioms of Euclidean

{e} (�,e) {.,�,e}

geometry; if instead we keep all the
other axioms as before but replace the
parallel postulate by its negation, we
have the axioms of a non-Euclidean ge­
ometry. For both axiom systems-Eu­
clidean and non-Euclidean-we ask: Can SET OF ALL SUBSETS OF A GIVEN SET is illustrated. The square, triaugle and circle at
top form the three-element set A. This set has 2'\ or 8, subsets (provided that the whole set
these axioms lead to a contradiction?
and the empty set are somewhat improperly included). This new set consisting of eight ele­
To ask the question of the Euclidean
ments is called the power set of A, and it is denoted 2". If A has n elements, the power
system may seem unreasonable. How
set of A has 2n elements. If A is infinite, 2·<[ is also infinite, and it is not equivalent to A.
could there be anything wrong with our
familiar, 2,OOO-year-old high school ge­
ometry? On the other hand, to the non­ surface in the sense we have just de­ a coordinate system. As many college
mathematician there certainly is some­ scribed. In other words, we now see that freshmen learn, to each point in the
thing suspicious about the second axiom if the axioms of non-Euclidean geometry plane we can associate a pair of numbers:
system, with its denial of the intuitively led to a contradiction, then so would the its x and y coordinates. Then with each
plausible parallel postulate. Nonetheless, ordinary Euclidean geometry of spheres line or circle we can associate an equa­
from the viewpoint of 20th-century lead to a contradiction. Thus we have a tion: a relation between the x and y
mathematics the two kinds of geometry relative proof of consistency; if Euclid­ coordinates that is true only for the
stand more or less on an equal footing. ean three-dimensional geometry is con­ points on that line or circle. In this way
Both are sometimes applicable to the sistent, then so is non-Euclidean two­ we set up a correspondence between ge­
physical world and both are consistent, dimensional geometry. vVe say that the ometry and elementary algebra. For
in a relative sense we shall now explain. surface of the Euclidean sphere is a every statement in one subject there is a
First we show that non-Euclidean ge­ model for the axioms of non-Euclidean corresponding statement in the other. It
ometry is consistent. In order to do this geometry. (In the particular model we follows that the axioms of Euclidean ge­
we merely replace the word "line" every­ have used the parallel postulate fails ometry can lead to a contradiction only
where by the phrase "great circle," a because there are no parallel lines. It is if the rules of elementary algebra-the
line formed on the surface of a sphere by also possible to construct a surface, the properties of the ordinary real numbers­
a plane passing through the center of the "pseudosphere," for which the parallel can lead to a contradiction. Here again
sphere. We now regard the axioms as postulate is false because there is more we have a relative proof of consistency.
statements about points and great circles than one line through a point parallel to Non-Euclidean geometry was consistent
on a given sphere. Moreover, we agree a given line.) if Euclidean geometry was consistent;
to identify each pair of diametrically op­ The invention of non-Euclidean geom­ now Euclidean geometry is consistent if
posite points on the sphere as a single etry, and the recognition that its COll­ elementary algebra is consistent. The
point. If the reader prefers, he can im­ sistency is implied by the consistency Euclidean sphere was a model for the
agine the axioms of non-Euclidean ge­ of Euclidean geometry, was the work non-Euclidean plane; the set of pairs of
ometry rewritten, with the word "line" of many great 19th-century mathemati­ coordinates is in turn a model for the
everywhere replaced by "great circle," cians; we mention the name of Bernhard Euclidean plane.
the word "point" everywhere replaced Riemann in particular. Only in the 20th vVith these examples before us we can
by "point pair." Then it is evident that century was the question raised of say that Coders proof of the relative
all the axioms are true, at least insofar whether or not Euclidean geometry itself consistency of the axicm of choice and
as our ordinary notions about the sur­ is consistent. of the continuum hypothesis is analogous
face of a sphere are true. In fact, from to Hilbert's proof of the relative consist­

T by David Hilbert. Hilbert's solution

the axioms of Euclidean solid geometry his question was asked and answered ency of Euclidean geometry. In both in­
one can easily prove as theorems that the stances the standard theory was justified
surface of a sphere is a non-Euclidean was a simple application of the idea of in terms of a more elementary one. Of

III

© 1967 SCIENTIFIC AMERICAN, INC

course, no one ever seriously doubted the non-Euclidean geometry use the same not disprovable is added the fact that it
reliability of Euclidean geomeh-y, where­ axioms, with the one exception of the is also not provable.
as such outstanding mathematicians as parallel postulate, so standard ("Cantori­ Both Code!'s result and the new dis­
L. E. J. Brouwer, Hermann Weyl and an") and nonstandard ("non-Cantorian") coveries require the construction of a
Henri Poincare had grave doubts about set theory differ only in one axiom. Non­ model, just as the consistency proofs for
the axiom of choice. In this sense Code!'s Cantorian set theory takes the axioms of geometry that we have described re­
result had a much greater impact and restricted set theory and adds not the quired a model. In both cases we want
significance. axiom of choice but rather one or another to prove that if restricted set theory is
The analogous development with re­ form of the negation of the axiom of cOllsistent, then so is standard set theory
spect to non-Euclidean geometry-what choice. In particular we can take as an (or nonstandard theory).
we might call non-Cantorian set theory­ axiom the negation of the continuum Code!'s idea was to construct a model
has taken place only since 1963, in the hypothesis. Thus, as we shall explain, for restricted set theory, and to prove
work of one of the authors of this article there now exists a complete solution of that in this model the axiom of choice
(Cohen). What is meant by "non-Can­ the continuum problem. To Code!'s dis­ and the continuum hypothesis were the­
torian set theory"? Just as Euclidean and covery that the continuum hypothesis is orems. He proceeded in the following
way. Using only the axioms of restricted
set theory [see illustration on page 114],
we are guaranteed first the existence of
at least one set (the empty set) by Axiom
2; then by Axiom 3 and Axiom 4 we are
guaranteed the existence of an infinite
sequence of ever larger finite sets; then
by Axiom 5, the existence of an infinite
set; then by Axiom 7, of an endless se­
U)
w quence of ever larger (nonequivalent)
cr:
<t infinite sets, and so on. In essentially this
cr:
co way Codel specified a class of sets by the
-'
manner in which they could actually be
<t
U
constructed in successive steps from
o
-'
Simpler sets. These sets he called the
"constructible sets"; their existence was
guaranteed by the axioms of restricted
set theory. Then he showed that within
the realm of the constructible sets the
axiom of choice and the continuum hy­
pothesis can both be proved. That is to
say, first, from any constructible collec­
tion ll' of constructible sets (A, B, ...) one
can choose a constructible set Z consist­
ing of at least one element each from A,
B and so on. This is the axiom of choice,
which here might more properly be

I
A B C D E F G called the theorem of choice. Second,
if A is any infinite constructible set, then
there is no constructible set "between"
C>-
cr:
A and 2A (bigger than A, smaller than
<t
cr:
the power set of A and equivalent to nei­
Q;l ther). If A is taken as the first infinite
-'

<t
'?
•
cardinal, this last statement is the con­
cr: tinuum hypothesis.
I-

I-I ence a
Z
w
u "generalized continuum hy-
)L\STER pothesis" was proved in the case of
A B C D E F G CATALOGl'E constrllctible set theory. Code!'s work
would therefore dispose of these two
questions completely if we were pre­
pared to adopt the axiom that only con­
structible sets exist. Why not do so?
Because one feels it is unreasonable to
insist that a set must be constructed
RUSSELL'S PARADOX is illustrated by supposing that in a certain country it is tbe cus·
according to an y prescribed formula in
tom of librarians to list their books not in a card catalogue but in a looseleaf catalogue;
order to be recognized as a genuine set.
that is, the catalogue itself is a book. Some librarians list the catalogue itself in the cata·
logue (top); some do not (second from top). The first kind of catalogue is called an R.set,
Thus in ordinary (not necessarily con­
after Bertrand Russell; R·sets are sets that include themselves. What happens, however, if structible) set theory neither the axiom
the head librarian of the country decides to make a master catalogue of all the catalogues of choice nor the continuum hypothesis
that do not list themselves? Does his own catalogue belong in the master catalogue or not? had been proved. At least this much was

1 12

© 1967 SCIENTIFIC AMERICAN, INC

 "COMMON NOTIONS"

1. Things that are equal to the same

thing are also equal to one another.

2. If equals are added to equals, the

wholes are equal.

3. If equals are subtracted from equals,

the remal nders are equal.

4. Things that coincide with one an­

other are equal to one another.

5. The whole IS greater than the part.

"POSTULATES"

1. (It IS possible) to dl-aw (exactly one)

stl-alght line fmm any pOint to any pOint.

2. (It IS POSSIble) to extend a finite straight

line continuously In a straight line.

3. (It IS pOSSible) to descflbe a cll'cle With

any center and distance.

4. All fight angles are equal to one an­

other.

5. If a straight I tile failing on two straight

lines makes the Interior angles on the
same side less than two fight angles, the
two straight lines. If produced Indefl­
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Here the analogy with the parallel
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axioms are consistent was taken for BOX 20 NEW HOPE, PENNSYLVANIA 18938
granted until quite recently. The ques-

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© 1967 SCIENTIFIC AMERICAN, INC

 V FOR A L L I F A N D O N LY I F E I S A M EM B E R ( E L E M E N T ) OF

3 TH ERE EXISTS V OR EQUALS

31 T H E R E E X I S T S U N I Q U E LY & AND * DOES NOT E Q U A L

u U N ION NOT ¢ THE EMPTY SET

I M P L I ES <::: I S A SUBSET OF

1 . A X I O M O F EXT E N S I O N A L I T Y
"Ix . y ( V Z (Z E X � Z E y) � x - V).
Two sets are equ a l I f a n d o n ly If they have the same members.

2. A X I O M O F T H E N U L L S ET
3x Vy (- y E x).
There eXists a set with no members (the em pty set ).

3 . A X I O M OF U N O R D E R E D P A I R S
"Ix . y 3 z VW (W E Z ...... W = X V w = y) .
If x a n d y are sets, then the ( u n orde red) pair x. y is a set. ( )

4. A X I O M O F T H E S U M S ET O R U N I O N
"I x 3 y V z (z E y ...... 3 t (Z E t & t E X ) )

If x IS a set of sets, the u n ion of all ItS members IS a set. ( For exam ple. If x = ( { a.b c ) ) then the u n ion of the (two) ele m e n ts of x IS
(
the set a.b. c . d . e ) )
{a c d e } .

5. A X I O M O F I N F I N I T Y
3 x (¢ E X & V y ( y E X � Y U ( y } E X).
There exists a set x that con tai n s the em pty set, a n d that i s s u ch that if y belongs t o x. then the u n ion of y a n d y i s also i n x . The ( )
( )
d i stin ction between the element y and the s i ngleton set y IS basic. This axiom guarantees the eXisten ce of i n f i n ite sets.

6n. AXIOM OF REPLACEMENT

Vt, . . t , ( "Ix 3 1 y A n (x . y ; t , . . . t , ) � V u 3v B ( u , v ) ) where B ( u . v ) ::: V r ( r E v <-> 3s (S E U & An(s. r ; t , . . . . t, ) ) ) .
This aXiom is diffi c u lt to restate i n E n glish I t I S called 6 n rather than 6 because I t I S rea l ly a whole fam i ly of aXiom s. We s u ppose that all the
for m u las exp ressib l e In ou r system have been e n u merated ; the n t h IS called A n . Then the axiom of replacement says that If for fixed
t" . . . , t , . An (x, y ; t , ) defines y u niquely as a f u n c t i on of x, say y = q>(x), then for each u the range of q> on u IS a set. This means, roughly , that
a n y ( " reasonab l e " ) p roperty that can be stated in the formal language of the theory can be used to define a set (the set of things
having the stated p roperty).

7 . A X I O M OF T H E P O W E R S ET
"Ix 3 y V z (Z E y ...... Z <::: x).
T h i s aXiom says that there eXists for each x the set y of all s ubsets of x . Although Y I S th u s defi ned by a p roperty, It IS not covered by
the replacement axiom because I t is n ot given as the range of a n y f u n c tion. I n deed, the cardi n ality of y will be greater than that of x,
so that this a X i o m allows u s to const r u ct higher cardi nals.

8. A X I O M O F C H O I C E
I f 0 � A G * ¢ i s a f u n ction defined for all 0 E x , then there exists an other f u n c tion f ( 0 ) for 0 E x , and f ( 0 ) E AG.
This IS the well - k n ow n axiom of choice, which allows u s to do an I n finite amou n t of " choosing" even though we have n o p roperty that
would defin e the choice f u n c tion and th u s e n able us to u se 6n i n stead.

9. A X I O M O F R E G U L A R I T Y
V x 3 y (x = ¢ V (Y E X & V Z (Z E X � - Z E y) ) ) _
Th i s axiom explic i tly p rohibits X E x , for example.

ZERMELO·FRAENKEL AXIOMS FOR SET THEORY are listed. of set theory, a glossary of wbich is given at top. This axiom sys·
In order to state these theorems it is necessary to use the symbols tem was put forward by Ernst Zermelo and Abraham Fraenkel.

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© 1967 SCIENTIFIC AMERICAN, INC

tion that interested geometers was Rather than throw up our hands and
whether or not they are independent, say it is impossible to describe this model
that is, whether the parallel postulate in a nontechnical article, we shall at­
could be proved on the basis of the oth­ tempt at least to give a descriptive ac­
ers. A whole series of geometers tried to count of one or two of the leading ideas
prove the parallel postulate by showing that are involved. Our starting point is
that its negation led to absurdities. It ordinary set theory (without the axiom
B
seems that Carl Friedrich Causs was the of choice). We hope only to prove the
first to see that these "absurdities" were consistency of non-Cantorian set theory
simply the theorems of a new, non-Eu­ in a relative sense. Just as the models of
clidean geometry. But what Causs had non-Euclidean geometry prove that non­
the courage to think he did not have the Euclidean geometry is consistent if Eu­
courage to publish. It was left for Janos clidean geometry is consistent, so we
Bolyai, Nikolai Ivanovich Lobachevsky shall prove that if restricted set theory is
and Riemann to carry out the logical C01 1- consistent, it remains so if we add the
sequences of denying the parallel postu­ statement "The axiom of choice is false"
late. These consequences were the dis­ or the statement "The continuum hy­
covery of "fantastic" geometries that had pothesis is false." We may now assume
as much logical consistency as the Eu­ that we have available as a starting point
clidean geometry of "the real world." a model for restricted set theory. Call
Only after this had happened was it rec ­ this model M; it can be regarded as Co­
ognized that two-dimensional non-Eu­ del's class of constructible sets.
clidean geometry was just the ordinary We know from Coders work that in
Euclidean geometry of certain curved order for the axiom of choice or the con­
surfaces (spheres and pseudospheres). tinuum hypothesis to fail we must add
The analogous step in set theory to M at least one nonconstructible set.
would be to deny the axiom of choice or How to do this? We introduce the letter
the continuum hypothesis. By this we a to stand for an object to be added to M;

mean, of course, that the step would be it remains to determine what kind of
to prove that such a negation is con­ thing a should be. Once we add a we
sistent with restricted set theory, in the must also add everything that can be
same sense in which Codel had proved formed from a by the permitted opera­
that the affirmation was consistent. It is tions of restricted set theory: uniting two
this proof that has been accomplished or more sets to form a new set, forming
in the past few years, giving rise to a the power set and so on. The new col­
surge of activity in mathematical logic lection of sets generated in this way by
whose final outcome cannot be guessed. M + a will be called N. The problem is
Since it is a question of proving the how to choose a in such a way that (1)
relative consistency of an axiom system, N is a model for restricted set theory, as
we naturally think of constructing a M was by assumption , and (2) a is not
model. As we have seen, the relative constructible in N. Only if this is possible
consistency of non-Euclidean geometry is there any hope of denying the axiom
was established when surfaces in Eu­ of choice or the continuum hypothesis. ON SURFACE OF A SPHERE "straight
clidean three-space were shown to be We can get a vague feeling of what line" is interpreted to mean "great circle"
models of two-dimensional non-Euclid­ has to be done by asking how a geometer ( A and B at top). Through any pair of di­
ean geometry. In a comparable way, in of 1850 who was trying to discover the ametrically opposite points ( aa' and bb ' )
order to prove the legitimacy of a non­ pseudosphere might have proceeded. In there pass many great circles. If w e interpret
Cantorian set theory in which the axiom a very rough sense, it is as if he had start­ "point" to mean "point pair," then Euclid's

of choice or the continuum hypothesis is ed with a curve M in the Euclidean first postulate is true. The second postulate

false we must use the axioms of restricted is true if one allows the extended "straight
plane, thought of a point a not in that
line" to have a finite total length, or to re­
set theory to construct a model in which plane, and then connected that point a
trace itself many times as it goes around the
the negation of the axiom of choice or to all the points in M. Since a is chosen
sphere. The third postulate is also true if
the negation of the continuum hypothe­ not to lie in the plane of M, the resulting one understands distance to be measured
sis can be proved as theorems. surface N will surely not be the same as along great circles that can be retraced sev­
the Euclidean plane. Thus it is reason­ eral times; here a "circle" means merely the

I
t must be confessed that construction able to think that with enough ingenuity set of points on the sphere at a given great­
of this model is a complex and delicate and technical skill one could show that it circle distance from a given point_ The
affair. This is perhaps to be expected. In is really a model for a non-Euclidean fourth postnlate is likewise true. Playfair's
Codel's constructible sets, his model of geometry. postulate is false, because any two great cir­

Cantorian set theory, the task was to cles intersect. Thus the sphere is a model of
The analogous thing in non-Cantorian
non-Euclidean geometry. So is the pseudo­
create something essentially the same as set theory is to choose the new set a as a
sphere ( bottom), if straight lines are inter­
our intuitive notion of sets but more nonconstructible set, then to generate a
preted as being the shortest curves connect­
tractable. In our present task we have to new model N consisting of all sets ob­ ing any two points on the surface. On the
create a model of something unintuitive tained by the operations of restricted set surface of the pseudosphere there are many
and strange, using the familiar building theory applied to a and to the sets in M . "straight lines" that pass through a given
stones of restricted set theory. I f this can b e done, i t will have been point and do not cross a given straight line.

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© 1967 SCIENTIFIC AMERICAN, INC

 G EOMETRY STA G E O F D E V ELOPM ENT SET TH EORY

T H A L E S , P Y THAGORAS I N T U I T I V E B A S I S FOR F I R S T T H EO R E M S CANTOR

ZENO P A R A D O X R EV E A L E D RUSSELL

E U DOX U S , E U C L I D A X I O M AT I C B A S I S F O R S TA N D A R D T H E O R Y Z E R M E L O , F RA E N K E L , E T C .

D ESCA R T E S , H I LB E R T S TA N D A R D T H EO R Y S H OW N ( R E LAT I V E LY ) C O N S I S T E N T GODEL

GAUSS, RI EMAN N D I S C O V E R Y O F N O N S TA N D A R D T H EO R I E S C U R R E N T WORK

M I N KOWS K I , E I N S T E I N A P P L I C AT I O N O F N O N S TA N D A R D T H EO R Y ? ?

ANALOGY IN DEVELOPMENT of geometry ( left ) and set theory ometry has been applied in such theories as Einstein's theory 01
( right ) is traced historically. Nonstandard ( non·Euclidean) ge- relativity. Nonstandard s e t theory has y e t t o be applied in physics.

proved that one is safely able to negate We can actually do this in such a way abIes one to construct them all : the no­
the axiom of constructibility. Since Go­ that the elements we add have cardinality tion of "generic" and the rela'Ed noiion
del showed that constructibility implies K of "forcing." Very roughly speaking, ge­
the axiom of choice and the continmun (2 0)
� =2 neric sets have only those properties they
hypothesis, this is the necessary first step 2 are "forced" to have in order to be set­
in negating either of these two state­ from the viewpoint of the model M . like. In order to decide whether a is
ments, Again a rough geometric analogy may be "forced" to have a certain property we
In order to carrv out this first step helpful: To a two-dimensional creature must look at all of N. Yet N is not really
two things must be shown: that a can be living embedded in a non-Euclidean sur­ defined until we have specified at The
chosen so that it remains nonconstruct­ face it would be impossible to recognize recognition of how to make this seem­
ible, not only in M but also in N, and that his world is part of a three-dimen­ ingly circular argument noncircular is
that N, like M, is a model for restricted sional Euclidean space. In the present another key element in the new theory.

\ hat does the history of geometry

set theory. To specify a we take a round­ instance we, standing outside M, can see

V suggest for the future of set theory?

about procedure. ''''e imagine that we that we have thrown in only a countable
are going to make a list of all possible infinity of new elements. They are such,
statements about a, as a set in N. Then however, that the counting cannot be The most remarkable thing about non­
a will be specified if we give a rule by done by any apparatus available in M Euclidean geometry is that it turned out
which we can determine whether or not itself. Thus we obtain a new model N', to be an essential prerequisite for Ein­
any such statement is true. in which the continuum hypothesis is stein's general theory of relativity. Rie­
The crucial idea turns out to be to false. The new elements, which in N' mann created Riemannian geometry for
choose a to be a "generic" element, that play the role of real numbers (that is, the purely abstract purpose of unifying,
is, to choose a so that only those state­ points on a line segment), have cardi­ clarifying and deepening the non-Eu­
ments are true for a that are true for al­ nality greater than 2 K o , and so there is clidean geometry of Lobachevsky, Bolyai
most all sets in M. This is a paradoxical now an infinite cardinal-namely 2Ko ­ and Gauss. This geometry turned out
notion. Every set in M has both pat·ticu­ that is greater than �o and yet smaller to be the indispensable tool for Ein­
lar special properties that identify it, and than c, since in our model N', c is equal to stein's revolutionary reinterpretation of
also general typical properties that it the gravitational force.
tot
shares with almost all the other sets in M. (2 0) Does this example justify an expecta­
2 .
It turns out to be possible in a precise tion that non-Cantorian set theory se me­
way to make this distinction between Since we can construct a model of set day will find a currently unforeseeable
special and generic properties perfect­ theory in which the continuum hypothe­ application in the "real" (that is, non­
ly explicit and formal. Then when we sis is false, it follows that we can add to mathematical) world? No one today
choose a to be a generic set (one with, so our ordinary restricted set theory the would venture an answer. Certainly we
to speak, no special properties that dis­ assumption of the falsity of the contin­ can see (with hindsight) that geometry
tinguish it from any set in M), it follows uum hypotheSiS; no contradiction can has always furnished the essential back­
that N is still a model for restricted set result that was not already present. In ground in which physical events take
theory. The new element a we have in­ the same spirit we can construct models place. In that sense it should perhaps
h'oduced has no troublesome properties for set theory in which the axiom of have been expected tllat fundamental
that can spoil the M we started with. At choice fails. We can even be quite spe­ advances in geometry would find a phys­
the same time a is nonconstructible. cific about which infinite sets it is pos­ ical application. Set theory does not
Any constructible set has a special char­ sible to "choose from" and which are seem today to have any such organic in­
acter-the steps by which it can be con­ "too big to choose from." terrelationship with physics. Still, there
structed-and our a precisely lacks any Whereas Godel produced his results have been some mathematicians (S an­
such individuality. with a single model (the constructible islaw Ulam, for example) who have
To construct a model in which the sets), we have in non-Cantorian set proposed that abstract set theory might
continuum hypothesis is false we must theory not one but many models, each furnish useful models for theoretical
add to M not just one new element a constructed with a particular purpose in physics. At this stage the safest thing is
but a great many new elements, In fact, mind. Perhaps more important than any to refuse to predict anything about the
we must add an infinite number of them. of the models is the technique that en- future-except that it is unpredictable.

1 16

© 1967 SCIENTIFIC AMERICAN, INC

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© 1967 SCIENTIFIC AMERICAN, INC