Set Theory: Should You Believe?
N J Wildberger
School of Maths UNSW
Sydney NSW 2052 Australia
webpages: http://web.maths.unsw.edu.au/~norman
"I protest against the use of infinite magnitude as something completed, which is never permissible in mathematics. Infinity is merely
a way of speaking, the true meaning being a limit which certain
ratios approach indefinitely close, while others are permitted to increase without restriction." (Gauss)
"I don’t know what predominates in Cantor’s theory - philosophy
or theology, but I am sure that there is no mathematics there."
(Kronecker)
"...classical logic was abstracted from the mathematics of finite sets
and their subsets...Forgetful of this limited origin, one afterwards
mistook that logic for something above and prior to all mathematics,
and finally applied it, without justification, to the mathematics of
infinite sets. This is the Fall and original sin of [Cantor’s] set theory
..." (Weyl)

Modern mathematics as religion
Modern mathematics doesn’t make complete sense. The unfortunate consequences include difficulty in deciding what to teach and how to teach it, many
papers that are logically flawed, the challenge of recruiting young people to the
subject, and an unfortunate teetering on the brink of irrelevance.
If mathematics made complete sense it would be a lot easier to teach, and
a lot easier to learn. Using flawed and ambiguous concepts, hiding confusions
and circular reasoning, pulling theorems out of thin air to be justified ‘later’
(i.e. never) and relying on appeals to authority don’t help young people, they
make things more difficult for them.
If mathematics made complete sense there would be higher standards of
rigour, with fewer but better books and papers published. That might make
it easier for ordinary researchers to be confident of a small but meaningful
contribution. If mathematics made complete sense then the physicists wouldn’t
have to thrash around quite so wildly for the right mathematical theories for
quantum field theory and string theory. Mathematics that makes complete sense
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tends to parallel the real world and be highly relevant to it, while mathematics
that doesn’t make complete sense rarely ever hits the nail right on the head,
although it can still be very useful.
So where exactly are the logical problems? The troubles stem from the
consistent refusal by the Academy to get serious about the foundational aspects
of the subject, and are augmented by the twentieth centuries’ whole hearted and
largely uncritical embrace of Set Theory.
Most of the problems with the foundational aspects arise from mathematicians’ erroneous belief that they properly understand the content of public school
and high school mathematics, and that further clarification and codification is
largely unnecessary. Most (but not all) of the difficulties of Set Theory arise
from the insistence that there exist ‘infinite sets’, and that it is the job of mathematics to study them and use them.
In perpetuating these notions, modern mathematics takes on many of the
aspects of a religion. It has its essential creed–namely Set Theory, and its
unquestioned assumptions, namely that mathematics is based on ‘Axioms’, in
particular the Zermelo-Fraenkel ‘Axioms of Set Theory’. It has its anointed
priesthood, the logicians, who specialize in studying the foundations of mathematics, a supposedly deep and difficult subject that requires years of devotion
to master. Other mathematicians learn to invoke the official mantras when
questioned by outsiders, but have only a hazy view about how the elementary
aspects of the subject hang together logically.
Training of the young is like that in secret societies–immersion in the cult
involves intensive undergraduate memorization of the standard thoughts before
they are properly understood, so that comprehension often follows belief instead
of the other (more healthy) way around. A long and often painful graduate
school apprenticeship keeps the cadet busy jumping through the many required
hoops, discourages critical thought about the foundations of the subject, but
then gradually yields to the gentle acceptance and support of the brotherhood.
The ever-present demons of inadequacy, failure and banishment are however
never far from view, ensuring that most stay on the well-trodden path.
The large international conferences let the fellowship gather together and
congratulate themselves on the uniformity and sanity of their world view, though
to the rare outsider that sneaks into such events the proceedings no doubt
seem characterized by jargon, mutual incomprehensibility and irrelevance to
the outside world. The official doctrine is that all views and opinions are valued
if they contain truth, and that ultimately only elegance and utility decide what
gets studied. The reality is less ennobling–the usual hierarchical structures
reward allegiance, conformity and technical mastery of the doctrines, elevate
the interests of the powerful, and discourage dissent.
There is no evil intent or ugly conspiracy here–the practice is held in place
by a mixture of well-meaning effort, inertia and self-interest. We humans have
a fondness for believing what those around us do, and a willingness to mold our
intellectual constructs to support those hypotheses which justify our habits and
make us feel good.

2

The problem with foundations
The reason that mathematics doesn’t make complete sense is quite easy to
explain when we look at it from the educational side. Mathematicians, like
everyone else, begin learning mathematics before kindergarten, with counting
and basic shapes. Throughout the public and high school years (K-12) they are
exposed to a mishmash of subjects and approaches, which in the better schools
or with the better teachers involves learning about numbers, fractions, arithmetic, points, lines, triangles, circles, decimals, percentages, congruences, sets,
functions, algebra, polynomials, parabolas, ellipses, hyperbolas, trigonometry,
rates of change, probabilities, logarithms, exponentials, quadrilaterals, areas,
volumes, vectors and perhaps some calculus. The treatment is non-rigorous,
inconsistent and even sloppy. The aim is to get the average student through the
material with a few procedures under their belts, not to provide a proper logical
framework for those who might have an interest in a scientific or mathematical
career.
In the first year of university the student encounters calculus more seriously and some linear algebra, perhaps with some discrete mathematics thrown
in. Sometime in their second or third year, a dramatic change happens in the
training of aspiring pure mathematicians. They start being introduced to the
idea of rigorous thinking and proofs, and gradually become aware that they
are not at the peak of intellectual achievement, but just at the foothills of a
very onerous climb. Group theory, differential equations, fields, rings, topological spaces, measure theory, operators, complex analysis, special functions,
manifolds, Hilbert spaces, posets and lattices–it all piles up quickly. They
learn to think about mathematics less as a jumble of facts to be memorized
and algorithms to be mastered, but as a coherent logical structure. Assignment
problems increasingly require serious thinking, and soon all but the very best
are brain-tired and confused.
Do you suppose the curriculum at this point has time or inclination to return
to the material they learnt in public school and high school, and finally organize
it properly? When we start to get really picky about logical correctness, doesn’t
it make sense to go back and ensure that all those subjects that up to now have
only been taught in a loose and cavalier fashion get a proper rigorous treatment?
Isn’t this the appropriate time to finally learn what a number in fact is, why
exactly the laws of arithmetic hold, what the correct definitions of a line and a
circle are, what we mean by a vector, a function, an area and all the rest? You
might think so, but there are two very good reasons why this is nowhere done.
The first reason is that even the professors mostly don’t know! They too
have gone through a similar indoctrination, and never had to prove that multiplication is associative, for example, or learnt what is the right order of topics
in trigonometry. Of course they know how to solve all the problems in elementary school texts, but this is quite different from being able to correct all the
logical defects contained there, and give a complete and proper exposition of
the material.
The modern mathematician walks around with her head full of the tight
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logical relationships of the specialized theories she researches, with only a rudimentary understanding of the logical foundations underpinning the entire subject. But the worst part is, she is largely unaware of this inadequacy in her
training. She and her colleagues really do believe they profoundly understand
elementary mathematics. But a few well-chosen questions reveal that this is
not so. Ask them just what a fraction is, or how to properly define an angle,
or whether a polynomial is really a function or not, and see what kind of nonuniform rambling emerges! The more elementary the question, the more likely
the answer involves a lot of philosophizing and bluster. The issue of the correct
approach to the definition of a fraction is a particularly crucial one to public
school education.
Mathematicians like to reassure themselves that foundational questions are
resolved by some mumbo-jumbo about ‘Axioms’ (more on that later) but in
reality successful mathematics requires familiarity with a large collection of ‘elementary’ concepts and underlying linguistic and notational conventions. These
are often unwritten, but are part of the training of young people in the subject. For example, an entire essay could be written on the use, implicit and
explicit, of ordering and brackets in mathematical statements and equations.
Codifying this kind of implicit syntax is a job professional mathematicians are
not particularly interested in.
The second reason is that any attempt to lay out elementary mathematics
properly would be resisted by both students and educators as not going forward,
but backwards. Who wants to spend time worrying about the correct approach
to polynomials when Measure theory and the Residue calculus beckon instead?
The consequence is that a large amount of elementary mathematics is never
properly taught anywhere.
But there are two foundational topics that are introduced in the early undergraduate years: infinite set theory and real numbers. Historically these are very
controversial topics, fraught with logical difficulties which embroiled mathematicians for decades. The presentation these days is matter of fact–‘an infinite set
is a collection of mathematical objects which isn’t finite’ and ‘a real number is
an equivalence class of Cauchy sequences of rational numbers’.
Or some such nonsense. Set theory as presented to young people simply
doesn’t make sense, and the resultant approach to real numbers is in fact a joke!
You heard it correctly–and I will try to explain shortly. The point here is that
these logically dubious topics are slipped into the curriculum in an off-hand
way when students are already overworked and awed by all the other material
before them. There is not the time to ruminate and discuss the uncertainties
of generations gone by. With a slick enough presentation, the whole thing goes
down just like any other of the subjects they are struggling to learn. From then
on till their retirement years, mathematicians have a busy schedule ahead of
them, ensuring that few get around to critically examining the subject matter
of their student days.

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Infinite sets
I think we can agree that (finite) set theory is understandable. There are many
examples of (finite) sets, we know how to manipulate them effectively, and the
theory is useful and powerful (although not as useful and powerful as it should
be, but that’s a different story).
So what about an ‘infinite set’ ? Well, to begin with, you should say precisely
what the term means. Okay, if you don’t, at least someone should. Putting
an adjective in front of a noun does not in itself make a mathematical concept.
Cantor declared that an ‘infinite set’ is a set which is not finite. Surely that
is unsatisfactory, as Cantor no doubt suspected himself. It’s like declaring that
an ‘all-seeing Leprechaun’ is a Leprechaun which can see everything. Or an
‘unstoppable mouse’ is a mouse which cannot be stopped. These grammatical
constructions do not create concepts, except perhaps in a literary or poetic
sense. It is not clear that there are any sets that are not finite, just as it is
not clear that there are any Leprechauns which can see everything, or that
there are mice that cannot be stopped. Certainly in science there is no reason
to suppose that ‘infinite sets’ exist. Are there an infinite number of quarks or
electrons in the universe? If physicists had to hazard a guess, I am confident the
majority would say: No. But even if there were an infinite number of electrons,
it is unreasonable to suppose that you can get an infinite number of them all
together as a single ‘data object’.
The dubious nature of Cantor’s definition was spectacularly demonstrated
by the contradictions in ‘infinite set theory’ discovered by Russell and others
around the turn of the twentieth century. Allowing any old ‘infinite set’ à la
Cantor allows you to consider the ‘infinite set’ of ‘all infinite sets’, and this
leads to a self-referential contradiction. How about the ‘infinite sets’ of ‘all
finite sets’, or ‘all finite groups’, or perhaps ‘all topological spaces which are
homeomorphic to the sphere’ ? The paradoxes showed that unless you are very
particular about the exact meaning of the concept of ‘infinite set’, the theory
collapses. Russell and Whitehead spent decades trying to formulate a clear and
sufficiently comprehensive framework for the subject.
Let me remind you that mathematical theories are not in the habit of collapsing. We do not routinely say, “Did you hear that Pseudo-convex cohomology
theory collapsed last week? What a shame! Such nice people too.”
So did analysts retreat from Cantor’s theory in embarrassment? Only for a
few years, till Hilbert rallied the troops with his battle-cry "No one shall expel us
from the paradise Cantor has created for us!" To which Wittgenstein responded
"If one person can see it as a paradise for mathematicians, why should not
another see it as a joke?"
Do modern texts on set theory bend over backwards to say precisely what
is and what is not an infinite set? Check it out for yourself–I cannot say
that I have found much evidence of such an attitude, and I have looked. Do
those students learning ‘infinite set theory’ for the first time wade through The
Principia? Of course not, that would be too much work for them and their
teachers, and would dull that pleasant sense of superiority they feel from having
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finally ‘understood the infinite’.
The bulwark against such criticisms, we are told, is having the appropriate
collection of ‘Axioms’ ! It turns out, completely against the insights and deepest
intuitions of the greatest mathematicians over thousands of years, that it all
comes down to what you believe. Fortunately what we as good modern mathematicians believe has now been encoded and deeply entrenched in the ‘Axioms
of Zermelo—Fraenkel ’. Although there was quite a bit of squabbling about this
in the early decades of the last century, nowadays there are only a few skeptics.
We mostly attend the same church, dutifully repeat the same incantations, and
insure our students do the same.
Let us have a look at these ‘Axioms’, these bastions of modern mathematics.
In what follows, X and Y are ‘sets’.
1. Axiom of Extensionality: If X and Y have the same elements, then
X =Y.
2. Axiom of the Unordered Pair: For any a and b there exists a set {a, b}
that contains exactly a and b.
3. Axiom of Subsets: If φ is a property (with parameter p), then for any X
and p there exists a set Y = {u ∈ X : φ(u, p)} that contains all those u in X
that have the property φ.
4. Axiom of Union: For any X there exists a set Y = ∪X, the union of all
elements of X.
5. Axiom of the Power Set: For any X there exists a set Y = P (X), the set
of all subsets of X.
6. Axiom of Infinity: There exists an infinite set.
7. Axiom of Replacement: If F is a function, then for any X there exists a
set Y = F [X] = {F (x) : x ∈ X}.
8. Axiom of Foundation: Every nonempty set has a minimal element, that
is one which does not contain another in the set.
9. Axiom of Choice: Every family of nonempty sets has a choice function,
namely a function which assigns to each of the sets one of its elements.
All completely clear? This sorry list of assertions is, according to the majority of mathematicians, the proper foundation for set theory and modern mathematics! Incredible!
The ‘Axioms’ are first of all unintelligible unless you are already a trained
mathematician. Perhaps you disagree? Then I suggest an experiment–inflict
this list on a random sample of educated non-mathematicians and see if they
buy–or even understand–any of it. However even to a mathematician it should
be obvious that these statements are awash with difficulties. What is a property? What is a parameter ? What is a function? What is a family of sets?
Where is the explanation of what all the symbols mean, if indeed they have any
meaning? How many further assumptions are hidden behind the syntax and
logical conventions assumed by these postulates?
And Axiom 6: There is an infinite set!? How in heavens did this one sneak in
here? One of the whole points of Russell’s critique is that one must be extremely
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careful about what the words ‘infinite set’ denote. One might as well declare
that: There is an all-seeing Leprechaun! or There is an unstoppable mouse!
Just to get you thinking about whether in fact you understand the ‘ Axioms’,
consider the set
A = {a} .
As we do. Please stop reading for a moment, and just consider this set.
Thanks for considering it. Ah, but someone has a question! Yes? You would
like to know what a is? Very well, I will tell you. I am not sure if I am legally
obligated to (am I?), but I will tell you anyway–a is itself a set, also a very
simple one, with just two elements, called a1 and a2 . Thus
a = {a1 , a2 } .
Can we move on now? Wait, someone insists on knowing: what are a1 and a2 ?
They are also sets, also with two elements each, so that
a1
a2

= {a11, a12 }
= {a21, a22 } .

And, before you ask, each of the elements a11 , a12 , a21 and a22 is itself a set,
with also exactly two elements. Does the pattern continue? Suppose it does,
would that make A legitimate? But suppose it doesn’t, and that I refuse to
reveal a pattern, perhaps because non exists. In modern mathematics we are
allowed to consider patterns that do not have any pattern to them. In such a
case does A still exist? Does it exist if I invoke some appropriate new ‘Axiom’ ?
The Zermelo-Fraenkel ‘Axioms’ are but the merry beginning of a zoo of
possible starting points for mathematics, according to modern practitioners.
The ‘Axiom of Choice’ has numerous variants. There is the ‘Axiom of Countable
Choice’. ‘The Axiom of Dependent Choice’. There is the ‘Axiom that all subsets
of R are Lebesgue measurable’ (which contradicts the ‘Axiom of Choice’). Not
to mention all the higher possible axioms concerned with large cardinals. You
can mix and match as you please.
I have been a working mathematician for more than twenty years, and none
of this resembles in any way, shape, or form the subject as I have come to experience it. In my studies of Lie theory, hypergroups and geometry, there has never
been a point at which I have pondered–should I assume this postulate about
the mathematical world, or that postulate? Of course one makes decisions all the
time about which definitions to focus on, but the nature of the mathematical
world that I investigate appears to me to be absolutely fixed. Either G2 has an
eleven dimensional irreducible representation or it doesn’t (in fact it doesn’t).
My religious/ philosophical/Axiomatic position has nothing to do with it. So I
am confident that a view of mathematics as swimming ambiguously on a sea of
potential Axiomatic systems strongly misrepresents the practical reality of the
subject.

7

Does mathematics require axioms?
Occasionally logicians inquire as to whether the current ‘Axioms’ need to be
changed further, or augmented. The more fundamental question–whether mathematics requires any Axioms–is not up for discussion. That would be like trying
to get the high priests on the island of Okineyab to consider not whether the
Divine Ompah’s Holy Phoenix has twelve or thirteen colours in her tail (a fascinating question on which entire tomes have been written), but rather whether
the Divine Ompah exists at all. Ask that question, and icy stares are what you
have to expect, then it’s off to the dungeons, mate, for a bit of retraining.
Mathematics does not require ‘Axioms’. The job of a pure mathematician is
not to build some elaborate castle in the sky, and to proclaim that it stands up
on the strength of some arbitrarily chosen assumptions. The job is to investigate
the mathematical reality of the world in which we live. For this, no assumptions
are necessary. Careful observation is necessary, clear definitions are necessary,
and correct use of language and logic are necessary. But at no point does one
need to start invoking the existence of objects or procedures that we cannot see,
specify, or implement.
The difficulty with the current reliance on ‘Axioms’ arises from a grammatical confusion, along with the perceived need to have some (any) way to continue
certain ambiguous practices that analysts historically have liked to make. People
use the term ‘Axiom’ when often they really mean definition. Thus the ‘axioms’
of group theory are in fact just definitions. We say exactly what we mean by
a group, that’s all. There are no assumptions anywhere. At no point do we or
should we say, ‘Now that we have defined an abstract group, let’s assume they
exist’. Either we can demonstrate they exist by constructing some, or the theory
becomes vacuous. Similarly there is no need for ‘Axioms of Field Theory’, or
‘Axioms of Set theory’, or ‘Axioms’ for any other branch of mathematics–or
for mathematics itself!
Euclid may have called certain of his initial statements Axioms, but he had
something else in mind. Euclid had a lot of geometrical facts which he wanted
to organize as best as he could into a logical framework. Many decisions had
to be made as to a convenient order of presentation. He rightfully decided that
simpler and more basic facts should appear before complicated and difficult
ones. So he contrived to organize things in a linear way, with most Propositions
following from previous ones by logical reasoning alone, with the exception of
certain initial statements that were taken to be self-evident. To Euclid, an
Axiom was a fact that was sufficiently obvious to not require a proof. This is a
quite different meaning to the use of the term today. Those formalists who claim
that they are following in Euclid’s illustrious footsteps by casting mathematics
as a game played with symbols which are not given meaning are misrepresenting
the situation.
We have politely swallowed the standard gobble dee gook of modern set
theory from our student days–around the same time that we agreed that there
most certainly are a whole host of ‘uncomputable real numbers’, even if you
or I will never get to meet one, and yes, there no doubt is a non-measurable
8

function, despite the fact that no one can tell us what it is, and yes, there surely
are non-separable Hilbert spaces, only we can’t specify them all that well, and
it surely is possible to dissect a solid unit ball into five pieces, and rearrange
them to form a solid ball of radius two.
And yes, all right, the Continuum hypothesis doesn’t really need to be true
or false, but is allowed to hover in some no-man’s land, falling one way or the
other depending on what you believe. Cohen’s proof of the independence of the
Continuum hypothesis from the ‘Axioms’ should have been the long overdue
wake-up call. In ordinary mathematics, statements are either true, false, or
they don’t make sense. If you have an elaborate theory of ‘hierarchies upon
hierarchies of infinite sets’, in which you cannot even in principle decide whether
there is anything between the first and second ‘infinity’ on your list, then it’s
time to admit that you are no longer doing mathematics.
Whenever discussions about the foundations of mathematics arise, we pay
lip service to the ‘Axioms’ of Zermelo-Fraenkel, but do we every use them?
Hardly ever. With the notable exception of the ‘Axiom of Choice’, I bet that
fewer than 5% of mathematicians have ever employed even one of these ‘Axioms’
explicitly in their published work. The average mathematician probably can’t
even remember the ‘Axioms’. I think I am typical–in two weeks time I’ll have
retired them to their usual spot in some distant ballpark of my memory, mostly
beyond recall.
In practise, working mathematicians are quite aware of the lurking contradictions with ‘infinite set theory’. We have learnt to keep the demons at bay,
not by relying on ‘Axioms’ but rather by developing conventions and intuition
that allow us to seemingly avoid the most obvious traps. Whenever it smells
like there may be an ‘infinite set’ around that is problematic, we quickly use
the term ‘class’. For example: A topology is an ‘equivalence class of atlases’.
Of course most of us could not spell out exactly what does and what does not
constitute a ‘class’, and we learn to not bring up such questions in company.
There is also the useful term ‘category’. Consider the ‘category of all finite
groups’. Given any set a whatsoever, I can create a one element set A = {a}
whose single element is a. Then I can define A to be a group, by defining a·a = a.
Thus for every set a, there is a group with one element which determines a. So
if you believe that the ‘set of all sets’ doesn’t make good sense, then how can the
‘category of all finite groups’ be any better? Do category theorists begin their
lectures to the rest of us with a quick primer as to what the term ‘category’
might precisely mean? Does the audience get nervous not knowing? Back in the
good old nineteenth century they probably did, but nowadays those who attend
research seminars regularly are quite used to taking for granted abstractions
that they feel incapable of understanding.
Another good example arises from the usual definition of a function. Although the official doctrine is that a function is prescribed by a domain (a set)
and a codomain (a set) as well as a rule that tells us what do with an element of
the former to get an element of the latter, we know that in practice the domain
and codomain can be dispensed with in shady circumstances, or the term can be
replaced by the somewhat more flexible ‘functor’, particularly in category the9

ory. To illustrate–when we define the fundamental group π (X) of a topological
space X, we instinctively know that it is better not to write
π : Top → Group
because chances are Top and Group are not ‘properly defined infinite sets’. We
just employ the everyday understanding of a function, namely that it suffices
to say what kind of an object it inputs, what kind of an object it outputs, and
what it does precisely to an input to get an output. No need to have all the
possible inputs and outputs arranged in front of us neatly as two sets. This kind
of understanding can be usefully extended to many more mundane situations.
Do you really think you need to have all the natural numbers together in a set
to define the function f (n) = n2 + 1 on natural numbers? Of course not–the
rule itself, together with the specification of the kinds of objects it inputs and
outputs is enough. As computer scientists already know.

Why real numbers are a joke
According to the status quo, the continuum is properly modelled by the ‘real
numbers’. What is a real number? Let’s start with an easier question: What is
a rational number? Here comes set theory to our aid. It is, according to some
accounts, nothing but an equivalence class of ordered pairs of integers. Thus
when my six year old daughter uses the fraction 2/3, what she is really doing is
using the ‘equivalence class’
2/3 = {[2, 3] , [4, 6] , [−22, −33] , [14, 21] , [86, 129] , · · · } .
Good grief. But let’s carry on. A Cauchy sequence of rational numbers is a
sequence
λ = [r1 , r2 , r3 , · · · ]
where each of the ri is a rational number (of the kind just mentioned) with the
property that for all ε > 0 there exists a natural number N such that if n and
m are bigger than N, then
|rn − rm | ≤ ε.
But here is a very important point: we are not obliged, in modern mathematics,
to actually have a rule or algorithm that specifies the sequence r1 , r2 , r3 , · · · . In
other words, ‘arbitrary’ sequences are allowed, as long as they have the Cauchy
convergence property. This removes the obligation to specify concretely the
objects which you are talking about. Sequences generated by algorithms can
be specified by those algorithms, but what possibly could it mean to discuss a
‘sequence’ which is not generated by such a finite rule? Such an object would
contain an ‘infinite amount’ of information, and there are no concrete examples
of such things in the known universe. This is metaphysics masquerading as
mathematics.

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To get you used to the modern magic of Cauchy sequences, here is one I just
made up:
µ = [2/3, 2/3, 2/3, 2/3, 2/3, 2/3, · · · ] .
Anyone want to guess what the limit is? Oh, you want some more information
first? The initial billion terms are all 2/3. Now would you like to guess? No, you
want more information. All right, the billion and first term is 475. Now would
you like to guess? No, you want more information. Fine, the next trillion terms
are all 2/3. You are getting tired of asking for more information, so you want me
to tell you the pattern once and for all? Ha Ha! Modern mathematics doesn’t
require it! There doesn’t need to be a pattern, and in this case, there isn’t,
because I say so. You are getting tired of this game, so you guess 2/3? Good
effort, but sadly you are wrong. The actual answer is −17. That’s right, after
the first trillion and billion and one terms, the entries start doing really wild
and crazy things, which I don’t need to describe to you, and then ‘eventually’
they start heading towards −17, but how they do so and at what rate is not
known by anyone. Isn’t modern religion fun?
So now what is a real number? It is an equivalence class of Cauchy sequences!
That’s right, not just one, not just two, but an entire equivalence class of them.
We can’t even list the elements of such a ‘class’, since each and every one of them
contains an ‘uncountable’ number of Cauchy sequences. So of course we have
already absorbed the ‘infinite set theory’ to make sense of these statements, and
we still ought to ‘explain’ the equivalence relation. Let’s forego that, and just
present a representative example. Here is a real number, where I have saved
considerable space by not presenting rational numbers in their full glory:
{[2/3, −14, 1/3, 2/3, · · · ] , [4/9, 4, −4/17, 2458, · · · ] , [78, 2/29, 3, 4, 5/3, · · · ] , · · · } .
Like to guess what real number this is? You’re right! It is 5π + e. However did
you know?
Now that you are comfortable with the definition of real numbers, perhaps
you would like to know how to do arithmetic with them? How to add them,
and multiply them? And perhaps you might want to check that once you have
defined these operations, they obey the properties you would like, such as associativity etc. Well, all I can say is–good luck. If you write this all down
coherently, you will certainly be the first to have done so. On top of the manifold ugliness and complexity of the situation, you will be continually dogged by
the difficulty that in all these sequences there does not have to be a pattern–
they are allowed to be completely ‘arbitrary’. That means you are unable to
say when two given real numbers are the same, or when a particular arithmetical statement involving real numbers is correct. Even a simple statement like
1 + 1 = 2 will cause you consternation, since you have to phrase everything in
terms of unending Cauchy sequences, and in the absence of solid conventions for
specifying infinite sequences, you will wrestle with the question of whether the
Cauchy sequence [1, 1, 1, · · · ] really does represent 1, or perhaps just appears to
from this end of things.

11

Perhaps you would like to consult the usual ‘Constructing the Real Numbers’
section in your favourite calculus text instead. Have a look, and see what passes
for logical thinking in modern mathematics education. Then to really sink your
spirits, open up a ‘rigorous’ analysis text, and thumb through to the critical
section where they explain the continuum–exactly what a real number is and
how one operates with them. This is the heart of the matter–the bedrock on
which modern analysis is built. And in all such books, waffling and ambiguity
is what you find, unless the subject is passed over altogether. Some of them are
honest about it. Others cleverly confuse the issue by allowing talk about ‘sets’ of
rational numbers without any mention of how you actually specify them. It is in
this gap that the logical difficulties lurk. A set of rational numbers is essentially
a sequence of zeros and ones, and such a sequence is specified properly when
you have a finite function or computer program which generates it. Otherwise
‘it’ is not accessible in a finite universe.
This critical issue of describing the points on the continuum should have a
strong connection with notions of computability, but it turns out, according to
the standard dogma, that computable real numbers are just a ‘measure zero
slice’ of ‘all real numbers’. Despite the fact that neither you nor anyone else
has been able to write down a single ‘non-computable real number’ and the
undeniable fact that they never play the slightest role in any actual scientific,
engineering or applied mathematical calculation.
Even the ‘computable real numbers’ are quite misunderstood. Most mathematicians reading this paper suffer from the impression that the ‘computable
real numbers’ are countable, and that they are not complete. As I mention in my
recent book, this is quite wrong. Think clearly about the subject for a few days,
and you will see that the computable real numbers are not countable, and are
complete. Think for a few more days, and you will be able to see how to make
these statements without any reference to ‘infinite sets’, and that this suffices
for Cantor’s proof that not all irrational numbers are algebraic.
When it comes to foundational issues, modern analysis is off in la-la land.

But what about the natural numbers?
Okay, you say, perhaps you have a bit of a point here, but surely you are going
too far in denouncing infinite sets altogether. After all, there is one infinite set
that we can be absolutely sure of, one that is so familiar, so cut and dried, it
is beyond reproach. What about–the set N of all natural numbers you ask??
Have a look, here it is in its glorious entirety:
N = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, · · · } .
Well, perhaps not in its entirety, but we all know what those three little dots
represent, don’t we? All the rest of those numbers, squeezed in between the 18
and the right bracket!
The ancient Greeks believed that the natural numbers are not finite, but that
didn’t mean they agreed that you could put them all together to form a well12

defined mathematical object. A finite set we can describe explicitly and specify
completely–we can list all its elements so there is no possible ambiguity. But
the question is–are we allowed to state that all of the natural numbers are
collectible into one big set?
Some will argue that a mathematician can do whatever she likes, as long
as a logical contradiction doesn’t result. But things are not so simple. Are we
allowed to introduce all-seeing Leprechauns into mathematics as long as they
seem to behave themselves and not cause contradictions? A far better approach
to create beautiful and useful mathematics is to ensure that all basic concepts
are entirely clear and straightforward right from the start. The onus is on us to
demonstrate that our notions make sense, instead of challenging someone else
to find a contradiction.
We’ll now see that the concept of the ‘set of natural numbers’ is neither clear
nor straightforward, but immersed in complexity. The difficulties start when we
leave the familiar and comfortable domain of microscopic natural numbers, and
start pushing on through the sequence in an effort to write down bigger and
bigger numbers. Pretty soon expressing numbers in decimal form like
a = 23, 518, 800, 234, 444, 511, 009
gets uneconomical, and it is easier to use exponential notation. Iteration allows
us to write a tower of three tens:
10

b = 1010 .
Let’s keep on going, and write down the number
10
..
10.

c = 10

10

}1010

where the tower of exponents on the left has altogether b tens. My guess is that
c is already bigger than any number ever used (meaningfully) in mathematics or
science, but I could be wrong. In any case, it’s still early days in our exploration
of N, as we’ve only been at it for five minutes. How about
10
..
10.

d = 10

10
..
10.

}10

10
..
10.

}· · ·10

10

}1010

where the number of brackets is c? Please think about this number for a few
minutes. This should not be too much of a burden on you, since you routinely
bandy the set N of all natural numbers about.
Next we could introduce e, then f, then some suitable iterate of iterates,
say a1 , then b1 , then eventually a2 and so on, and so on, constrained only by
the limits of our imaginations, and the amount of writing paper at our disposal.
Assuming our imaginations are not a problem, there is the issue of space, for
as we keep going and keep going, we are going to start running out of memory
space to write down our increasingly large numbers. First they will fill a page,
then a book, then our hard drives. Of course we can make our computers bigger
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and our coding more efficient, start dismantling stars and spreading our memory
banks across galaxies. But... the universe is almost certainly finite. Eventually,
you and I may have vaporized and rearranged all the stars, furniture and other
creatures in our quest to write down yet bigger numbers, and now we are starting
to run out of particles with which to extend our galactic hard drive. Suppose
you reduce me to atoms in the interests of science, and perhaps your outer
extremities too. At some point, you are going to write down a number so vast
that it requires all the particles of the universe (except for some minimal amount
of what’s left of you). May I humbly suggest you call this number w, in honour
of the last person you vaporized to create it?
Now here is a dilemma. Once you have written down and marvelled at w in
all its glory, where are you going to find w + 1? From this end of things–the
working end–the endless sequence of natural numbers does not appear either
natural nor endless. And where is the infinite set N?
The answer is–nowhere. It doesn’t exist. It is a convenient metaphysical
fiction that allows mathematicians to be sloppy in formulating various questions
and arguments. It allows us to avoid issues of specification and replace concrete
understandings with woolly abstractions. What seems to be a happy and well
behaved sequence when viewed from the beginning is more like an enormous
fractal when viewed from the other end.
Unlike a, the numbers b, c and d are dramatic anomalies in the zoo of natural numbers, because they can be written down using so little space. Their
complexity, or informational content, is much smaller than they are themselves.
Most numbers are not like this at all. To emphasize this point, let’s make
a crude calculation to bound the number of possible numbers we could write
down by treating the entire universe as an enormous hard-drive, packed row
upon row with elementary particles to encode some gigantic number. Suppose
10
that in one dimension the universe is at most 1010 metres wide, that there are
2
perhaps 1010 dimensions (to make room for future versions of string theory),
3
that the smallest possible particle size is 10−10 metres, and that there are say
10
10 different particles that we could place at any one point in the universe. So
the number of possible configurations of particles filling up all of the universe is
at most
⎞
⎛
2
3
10
⎜
×1010
⎝ 1010

1010

1010 ⎟
⎠

.

Although this is a respectable number, it pales to insignificance when compared
to c. Conclusion: The vast majority of numbers less than c cannot be written
down in our universe. These numbers are completely inaccessible to us, and
always will be. But c can be written down in one line. Numbers ‘close’ to c
in the sense of having expressions that are not all that different from that of c
form little ‘islands of simplicity’ in a sea of overwhelming complexity.
It follows that long before you get to w, you are going to reach numbers whose
prime factorizations are impossible, since some of the factors, if they existed,
would require more room to write down than w. For example c + 23 is almost
surely such a number–I claim it has no prime factorization. Neither you nor I
14

nor anyone ever living in this universe will ever be able to factor this number,
since most of its ‘prime factors’ are almost surely so huge as to be inexpressible,
which means they don’t exist.
Perhaps you believe that even though you cannot write down numbers bigger
than w, you can still abstractly contemplate them! This is a metaphysical claim.
What does a number bigger than w mean, if there is nothing that it counts,
and it can’t even be written down? Believing you can ‘visualize’ an all-seeing
Leprechaun or an unstoppable mouse in your mind, by some melange of images,
descriptive phrases and vague feelings, does not mean they exist. By all means,
write plays and poems about all those numbers beyond w, but don’t imagine you
are doing mathematics. Twentieth century physicists have learnt to disregard
‘concepts’ which are not measurable or observable in some form or another, and
we mathematicians ought to be equally skeptical.
Elementary mathematics needs to be understood in the right way, and the
entire subject needs to be rebuilt so that it makes complete sense right from the
beginning, without any use of dubious philosophical assumptions about infinite
sets or procedures. Show me one fact about the real world (i.e. applied maths,
physics, chemistry, biology, economics etc.) that truly requires mathematics involving ‘infinite sets’ ! Mathematics was always really about, and always will be
about, finite collections, patterns and algorithms. All those theories, arguments
and daydreams involving ‘infinite sets’ need to be recast into a precise finite
framework or relegated to philosophy. Sure it’s more work, just as developing
Schwartz’s theory of distributions is more work than talking about the delta
function as ‘a function with total integral one that is zero everywhere except
at one point where it is infinite’. But Schwartz’s clarification inevitably led to
important new applications and insights.
If such an approach had been taken in the twentieth century, then (at the
very least) two important consequences would have ensued. First of all, mathematicians would by now have arrived at a reasonable consensus of how to
formulate elementary and high school mathematics in the right way. The benefits to mathematics education would have been profound. We would have strong
positions and reasoned arguments from which to encourage educators to adopt
certain approaches and avoid others, and students would have a much more
sensible, uniform and digestible subject.
The second benefit would have been that our ties to computer science would
be much stronger than they currently are. If we are ever going to get serious about understanding the continuum–and I strongly feel we should–then
we must address the critical issue of how to specify and handle the computational procedures that determine points (i.e. decimal expansions). There is
no avoiding the development of an appropriate theory of algorithms. How sad
that mathematics lost the interesting and important subdiscipline of computer
science largely because we preferred convenience to precision!
But let’s not cry overlong about missed opportunities. Instead, let’s get
out of our dreamy feather beds, smell the coffee, and make complete sense of
mathematics.

15