THE INFINITE
Thomas Jech
The Pennsylvania State University
Until the 19th century the study of the in nite was an exclusive domain of
philosophers and theologians. For mathematicians, while the concept of in nity
was crucial in applications of calculus and in nite series, the in nite itself was to
paraphrase Gauss just \the manner of sp eaking." Indeed, 18th century mathe-
matics, even with its frequent use of in nite sequences and in nitesimals, had no
real need to study the concept of in nity itself.
The increasing use of abstract functions of real and complex variable and the
emerging theory of real numb ers in the work of Cauchy,Weierstrass and Dedekind
brought ab out the necessity to use in nity not only as \a manner of sp eaking" but
rather intro duce into mathematical usage the concept of in nite set.
The earliest mathematician to study sets as ob jects of a mathematical theory
was Bernhard Bolzano in the 1840's. He gave a convincing philosophical agrument
for the existence of in nite sets and was also rst to explicitly formulate the concept
of one-to-one corresp ondence b etween sets. It seems that Bolzano was not aware
of the p ossibility that there may exist in nite sets of di erent size, and that as
some historians b elieve he envisioned a mathematical universe where all in nite
sets are countable. While some of his ideas were clearly ahead of his time, he did
Research for this article was supp orted in part byagrant from the International Research&
Exchange Board IREX , with funds provided by the Andrew W. Mellon Foundation, the National
Endowment for the Humanities, and the U.S. Department of State. None of these organizations
is resp onsible for the views expressed. The author is grateful to the Kurt Godel So ciety in Vienna
and to the Center for Theoretical Studies in Prague for the opp ortunity to discuss the topics of this
article. I am particularly thankful to Professor Petr Vopenka for his interpretation of Bolzano's
work .
Typ eset by A S-T X
M E
consider paradoxical the fact that the set of all p ositiveintegers can b e in a one-to-
one corresp ondence with a \smaller" subset, a fact that had b een already p ointed
out by Galileo centuries earlier.
It was Georg Cantor in the 1870's who realized the signi cance of one-to-one
corresp ondence and who embarked on a systematic study of abstract sets and of
cardinal and ordinal numb ers. His was also the rst application of set theoretical
metho ds, namely his \nonconstructive" pro of of the existence of transcendental
numb ers. The signi cance of Cantor's pro of lies in the fact that it employs, in
its pro of of uncountabilityof R, the idea that any, arbitrary, b ounded increasing
sequence of real numb ers has a least upp er b ound. This assumption is related to
the concept of real numb ers as Dedekind cuts in Q, or as limits of Cauchy sequences
of rationals, or as in nite decimal expansions. These three p ossible approaches to
the de nition of real numb ers are all equivalent in a sense and all require the
existence of arbitrary in nite sets of integers.
Very few mathematicians ob ject to the idea of an arbitrary set of integers. We
will argue later that the idea of existence of \large" cardinal numb ers that some
mathematicians nd ob jectionable the \never-never land" in the words of one
detractor is conceptionally no di erent from Cantor's idea that is inherent in his
pro of.
A natural question arising from Cantor's theory of cardinal numb ers is the fol-
lowing: Do there exist sets of real numb ers that are uncountable, yet not equivalent
in cardinality to the set of all real numb ers? The conjecture that no such sets exist
has b ecome known as the Continuum Hypothesis. Hilb ert attached such imp ortance
to this question that it gured on the top of his famous list of problems in 1900.
The continuum problem has eventually b een solved in a nontraditional way.It
was proved that the continuum hyp othesis is neither provable nor refutable, thus
b eing independent.
To show that a mathematical statement is indep endent, one has to sp ecify a
THE INFINITE 3
system of p ostulates, axioms of the theory, which are then shown to b e insucient
to prove or to refute the statement. For instance, the parallels p ostulate is indep en-
dent of the rest of Euclid's p ostulates. The pro of of indep endence is accomplished
by exhibiting models such as the Bolyai-Lobacevskij mo del of non-Euclidean ge-
ometry . So clearly the assertion that the continuum hyp othesis is indep endent
requires that we sp ecify the axiom system.
There is a more fundamental reason for axiomatizing set theory. With very
few insigni cant exceptions, all mathematical arguments and concepts can b e, in
principle, reduced to concepts and statements formalized in the language of set
theory.Thus an axiom system for the theory of sets is in fact an axiom system for
all of mathematics.
The axiomatization of set theory based on Cantor's \de nition" of sets turns
out to b e inconsistent, for rather trivial reasons. The idea is to consider as sets all
collections of ob jects that satisfy an arbitrary \prop erty". This approach is clearly
untenable, as exempli ed by the \set"
fa : a=2 ag
whose existence leads to a logical contradiction.
Since the 1920's, the widely accepted system of axioms for set theory is the
Zermelo-Fraenkel axiomatic set theory ZF . It consists of several axioms and ax-
iom schemata that encompass all uses of sets in mathematics. This axiomatic
theory is almost universally accepted as the logical foundation of current mathe-
matics.
Most axioms of Zermelo-Fraenkel are construction principles that merely formal-
ize certain manipulations with sets. The two notable exceptions are the axiom of
in nity and the axiom of choice.
The axiom of in nity p ostulates the existence of an in nite set or, equivalently,
the existence of the set N of all natural numbers. This axiom is of course the
essence of set theory.Ifwe replace the axiom of in nityby its negation, we get a
4 THOMAS JECH
theory equivalenttoPeano arithmetic, the axiomatic theory of elementary number
theory.
Historically, the most interesting axiom of ZF is the axiom of choice. Unlike the
other axioms, it is highly nonconstructive, as it p ostulates the existence of choice
functions without giving a sp eci c description of such functions. It was intro duced
by Zermelo in his pro of of well-ordering, and is in fact equivalent to the principle
that states that every set can b e well ordered. As it seems imp ossible to de ne a
well ordering of the set R, among others, this principle was violently opp osed by
some mathematicians of the rst half of this century.
Some of the ob jections are quite understandable in lightofsuch consequences of
the axiom of choice as the Hausdor -Banach-Tarski paradoxical decomp osition of
the unit ball. It should b e noted that the axiom of choice is indep endent of the rest
of ZF by the work of Godel and Cohen , which puts set theory in situation roughly
similar to that of Euclidean v. non-Euclidean geometry.From the formalist's p oint
of view it do es not makeany di erence whether the axiom of choice is accepted as an
axiom or not. In view of its usefulness in algebra and analysis, the prevailing opinion
favors the axiom. I would like to p oint out that even such an inno cuous statement
as \the union of countably many countable sets of real numb ers is countable"
requires the axiom of choice .
It is of these Zermelo-Fraenkel axioms that the continuum hyp othesis was shown
to b e indep endent. First, in 1938, Kurt Godel constructed a mo del of set theory
\the constructible universe" in which the continuum hyp othesis holds. Godel's
mo del is a canonical mo del; it is the least p ossible mo del, whichany other mo del
that contains all ordinal numb ers has to include. In 1963, Paul Cohen discovered
a metho d of constructing \generic extensions" of mo dels of set theory, including
mo dels in which the continuum hyp othesis fails. Cohen's metho d turned out to b e
quite general and resulted in a numb er of indep endence pro ofs. It so happ ens that
anumb er of conjectures in di erent areas of mathematics cannot b e decided on the
THE INFINITE 5
basis of Zermelo-Fraenkel axioms, and have b een proved indep endentby Cohen's
metho d. In addition to the continuum hyp othesis wemention Suslin's Problem,
Borel's Conjecture, Whitehead's Problem and Kaplansky's Conjecture.
In view of the apparent incompleteness of Zermelo-Fraenkel axioms a reasonable
suggestion o ers itself, namely to nd a complete system of axioms, in whichevery
p ossible statement of mathematics would b e formally decidable. This was the so
called Hilbert's Program. A mortal blow to this program were the two Incomplete-
ness Theorems of Godel.
The First Incompleteness Theorem shows in e ect that there is no hop e for a
complete axiomatization of mathematics. It states that every axiomatic theory that
includes elementary arithmetic is incomplete if it is consistent . In particular, b oth
Peano arithmetic and Zermelo-Fraenkel set theory are incomplete, and will remain
incomplete even if we add further axioms.
Godel's pro of of the Incompleteness Theorem yields an undecidable sentence, but
unlike the Continuum Hyp othesis in set theory,Godel's sentence is not particularly
meaningful. It do es provide a starting p oint in the search for easily understandable
statements expressible in the language of arithmetic that are indep endentofPeano
arithmetic. Such statements have indeed b een found, by Harrington and Paris.
Their examples are combinatorial principles, expressible in the language of arith-
metic, and true i.e. provable in set theory but unprovable in Peano arithmetic. It
remains to b e seen whether some of the unproved conjectures in elementary number
theory may p erhaps b e examples of undecidable statements.
The Second Incompleteness Theorems dashes all hop es that the axiom of in n-
ity can ever b e justi ed by elementary means. The theorem states that in every
axiomatic theory that includes Peano arithmetic, the statement that the theory
1
itself is consistent which is expressible in Peano arithmetic is unprovable if the
theory is consistent . Thus Peano arithmetic cannot prove its own consistency, and
1
The statement can b e formulated roughly as follows: There exists no nite sequence of sen-
tences that constitutes a formal pro of of a contradiction.
6 THOMAS JECH
neither can Zermelo-Fraenkel. On the other hand, the axiom of in nityprovides,
in set theory,away to prove the consistency of arithmetic or of set theory without
the axiom of in nity .
Using the in nite set N, one can de ne the usual op erations + and and thus
obtain a mo del of Peano arithmetic, thereby establishing its consistency.AsPeano
arithmetic or the theory of nite sets which is equivalent to it cannot prove the
consistency of Peano arithmetic, and set theory do es, it is a consequence of Godel's
Second Theorem that the axiom of in nity cannot b e proved consistent relativeto
the other axioms of Zermelo-Fraenkel. Therefore, our acceptance of the existence of
in nite sets, even of in nite sets of integers, is an act of faith, faith that set theory
is not contradictory.
The theory of cardinal numb ers initiated byCantor some 100 years ago proved
to b e unusually rich and profound. While the basic arithmetic of cardinals, addition
and multiplication, follows very simple rules, exp onentiation of cardinals makes an
interesting mathematical theory. Already the Continuum Hyp othesis is a problem
@
0
where @ is the cardi- in cardinal exp onentiation. Since the cardinalityof R is 2
0
@
0
= @ , where @ is the nalityof N, the continuum hyp othesis can b e stated as 2
1 1
least uncountable cardinal.
A central problem of mo dern set theory is to describ e the b ehavior of the cardinal
@
function 2 . One outcome of the work following Cohen's discovery was a solution
of this problem for regular cardinals @ Easton's Theorem : the result is that
@
sub ject to some trivial constraints, the value of 2 when @ is regular is quite
@
when @ arbitrary.Incontrast to Easton's theorem, the problem of calculating 2
is singular the Singular Cardinals Problem is far from b eing solved, despite some
sp ectacular results Silver, Magidor, Shelah .
The cardinal numb er of an in nite set A is called singular if A is the union
S
of a smaller numb er of smaller sets, i.e. A = A where b oth I and each A
i i
i2I
have a smaller cardinality than A. Cardinals that are not singular are regular.For
THE INFINITE 7
instance, @ is regular: the set N is not the union of nitely many nite sets. On
0
the other hand, the cardinal @ = supf@ : n =0; 1; 2; ...g is singular, as it is the
! n
limit of a countable sequence of smaller cardinals.
Which brings us to the most imp ortant issue in presentday set theory: large
cardinals. When we consider a typical limit cardinal, i.e. a cardinal @ whose index
is a limit ordinal, suchas @ ; @ ; @ and so on, we notice that such cardinal
! ! ! +!
1 2
is singular. However, as Hausdor already observed some 70 years ago, it is not
inconceivable that a limit cardinal could b e regular. Hausdor called such cardinals
inaccessible. Inaccessible cardinals, if they exist, have to b e extremely large, much
greater than cardinals that come up in ordinary mathematical practice. Moreover,
their existence is unprovable in Zermelo-Fraenkel.
The fact that inaccessible cardinals cannot b e proved to exist is a manifestation
of Godel's Second Incompleteness Theorem. If we assume that an inaccessible
cardinal exists then we can prove that set theory is consistent. Thus not only one
cannot prove in ZF that an inaccessible exists, but one cannot even prove that its
existence is consistent. The situation is analogous to the pro of of consistency of
nite set theory from the axiom of in nity.Thus adopting inaccessible cardinals
constitutes a similar act of faith as adopting in nite sets.
The most frequent argument against the study of large cardinals raised by non-
sp ecialists is that the sets in question are so large that they havenorelevance
to problems arising in \mainstream" mathematics. Thus the \never-never land"
lab el. This argument is hardly convincing, considering the numerous discoveries
that haveprovided ample evidence of close relationship b etween large cardinals
and diverse mathematical problems. The case in p ointisthemeasureproblem.
Sp eci cally, consider the question whether Leb esgue measure can b e extended to
a -additive measure de ned on all subsets of the real line. By the work of Ulam
and Solovay, this question is equiconsistent with the existence of a certain large
cardinal, called measurable. And a measurable cardinal is inaccessible, and in fact
8 THOMAS JECH
its existence is a considerably stronger p ostulate than the existence of inaccessibles.
The relationship of measurable v. inaccessible cardinals is prototypical in the
b estiary of large cardinals. Firstly,every measurable cardinal is inaccessible. Sec-
ondly, the least inaccessible cardinal is not measurable, and in fact every measurable
cardinal has b elow it a co nal set of inaccessibles. And thirdly app ealing to Godel's
Second Incompleteness Theorem , the consistency of the existence of a measurable
cardinal is unprovable in the theory that p ostulates the existence of inaccessible
cardinals. The theory of large cardinals leads to a plethora of large cardinals that
are more or less arranged in a hierarchy where the relationship b etween two kinds
of large cardinals is like the relationship b etween measurables and inaccessibles
describ ed ab ove.
The large cardinals investigated in set theory arise either by generalizing prop er-
ties of other large cardinals or by analyzing problems in other areas of mathematics,
mostly measure theory, mo del theory or combinatorics. The most convincing evi-
dence for large cardinals, however, comes from descriptive set theory, the study of
\nice" sets of real numb ers.
When investigating various prop erties of sets of reals such as Leb esgue measur-
ability, it is useful to distinguish b etween arbitrary sets and those sets that are
de nable; the de nable sets can b e arranged in a natural hierarchy: Borel, ana-
lytic, pro jective etc. We note in passing that the assumption that all pro jective
sets are Leb esgue measurable is equiconsistent with the existence of inaccessible
cardinals. The main to ol of descriptive set theorists is determinacy of in nite
games. One of the successes of mo dern set theory was the pro of that the hierarchy
of axioms arising from determinacy byvarying the complexity of the payo set
corresp onds very closely to the hierarchy of large cardinal axioms. These results
have inextricably tied large cardinals with prop erties of sets of real numbers.
AsaresultofGodel's Theorem, it will never b e p ossible to prove consistency of
large cardinals. And one cannot of course exclude the p ossibility that some of the
THE INFINITE 9
large cardinals may turn out to b e inconsistent. In the rush to generalize various
large cardinal prop erties in the early 70's, a certain axiom was prop osed that later
was refuted a result of Kunen . It seems unlikely however that a similar fate awaits
other large cardinal axioms. Firstly, these axioms have b een studied for a number
of years, without getting a contradiction. Moreover, the theory leads to an elegant
structure theory of the set theoretic universe, admitting canonical mo dels for large
cardinals, not unlikeGodel's mo del of constructible sets. Secondly, large cardinal
prop erties arise in seemingly unrelated areas. We already mentioned descriptive set
theory; it should also b e noted that the Singular Cardinals Problem is very closely
tied to the large cardinal theory.
To conclude this discourse I would like to p oint out the following. The state-
ment that the existence of some large cardinal say measurable or sup ercompact or
whatever is consistent, is expressible as a sentence in the language of arithmetic.
Thus wehave an arithmetical sentence that is unprovable in set theory even under
the assumption that the large cardinal in question exists, while provable under the
assumption of some bigger large cardinal. Of course, this arithmetical sentence has
no transparent meaning, but it is not inconceivable that a meaningful statement
might b e discovered in the future in analogy with the Harrington-Paris result for
Peano arithmetic . Thus it is conceivable that one might formulate a conjecture of
elementary numb er theory or in nite mathematics, whichwould b e provable under
the assumption of some very large cardinal but not without it.
While this is all very sp eculative, I wish to mention a recent result of Richard
Laver. Using a large cardinal assumption just a tad weaker that the one refuted by
Kunen, Laver gives a decision pro cedure for the word problem for the free algebra
with one generator and one left-distributive binary op eration. So here is a problem
of nite mathematics for which the only known solution employs one of the largest
cardinals that set theorists ever invented.
Department of Mathematics, The Pennsylvania State University, University Park,
PA 16802