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Chapter 1: On Infinity
MATTHEW: Concepts of the infinite variety can be found throughout
mathematics. First of all, there is the endless supply of natural numbers.
There is the real number line that extends to infinity in both the positive
and negative directions. The Euclidean and hyperbolic planes are
infinite.1 We also have infinite series, infinite integrals, and countless
other examples. Now, I know that philosophy is often concerned with the
definitions of important terms. I would like to ask you to consider
“infinity.”
PHILIP: I have often heard it defined as “that which is without bound.” In
mathematics I would assume it is usually referring to boundless
quantity, but in philosophy this could mean boundless duration,
boundless existence, or boundless power.2 A problem that I have with
this definition, however, is that it is not falsifiable. Since it is being
defined in the negative sense – based on what it doesn’t have – it is
impossible to prove that infinity as a concept does not exist. You would
have to find the beginning and end, the bounds, of everything in existence.
MATTHEW: That would be what James Thomson calls a “supertask.”3
PHILIP: So that may be a problem with the definition I mentioned, because
then the concept of infinity is invincible from attack. 4
MATTHEW: True, but in mathematics the definition can be salvaged and
the problem resolved from the other direction – we can directly prove the
existence of an infinite quantity, thus affirming the existence of the
concept itself. Suppose there was a finite set 1,2,3,..., N containing all of
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the natural numbers in ascending order.5 If we consider N  1, we see
that it is a natural number through closure under addition, but it is not
included in our original set because it is larger than the set’s greatest
element. Thus, the complete finite set cannot exist and there must be
infinitely many natural numbers. That is the simplest and most intuitive
example I can think of, though many other things can be proven to be
infinite in the same fashion.6 And in modern set theory, this notion of the
infinite successiveness of the integers is a foundational axiom. In fact, it
is so fundamental that Peano’s first axiom declares “each integer has a
unique successor.”
PHILIP: Okay, the proof makes sense to me, but I’m still not sure if I’m
completely comfortable with the definition we started with. Maybe we’ll
hit upon a better one as we continue.
MATTHEW: At this point I’m not as concerned with the definition as I am
with some seeming contradictions that exist. For instance, it can be
shown that the set of natural numbers is larger than the set of even
numbers, but it can also be shown that the two sets are the same size.
Galileo Galilei actually stumbled across this logical oddity near the turn
of the 17th Century.7 First, we can consider the vastly familiar even
numbers – 2, 4, 6, etc. It is certainly true that the set of even numbers is
entirely contained within the natural numbers – 1, 2, 3, 4, etc. We would
say the evens are a subset of the natural numbers. But observe that the
set of natural numbers contains elements other than the evens, namely,
the odd numbers, and there are a lot of them. So we can say that the
evens are a proper subset of the natural numbers, and the quantity of
natural numbers must be greater than that of the evens as a result.
PHILIP: I’m with you so far.
MATTHEW: On the other hand, let’s imagine the entire set of natural
numbers being laid out in a single row, and likewise for the entire set of
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even numbers. We can place the elements of these two infinite sets into
one-to-one correspondence with each other. Now we can evaluate their
sizes by determining which row runs out first. But it becomes clear that
for every natural number there is an even number associated with it
through the one-to-one correspondence. And for every even there will
always be a natural number. In this way, it becomes clear that the two
sets are the same infinite “size.”8
PHILIP: Okay, but isn’t that a contradiction. How can something be
strictly larger than something else and simultaneously be the same size?
MATTHEW: Even though it smells like a contradiction, it’s not. The first
argument is based on invalid logic, and it is the second that is currently
accepted. The problem with claiming that an infinite set is larger than
another infinite set because it contains additional elements is that finite
comparisons such as “greater than” and “less than” are exactly that –
finite. It is fallible to assume that finite conventions extend directly into
the infinite realm without the need for modifications or reinterpretations.
As Galileo realized, “the values of ‘equal, ‘greater,’ and ‘less’ are not
applicable to infinite, but only finite, quantities.” Indeed, during Galileo’s
time, those concepts did not carry into infinity. It took an entire
rethinking in the form of the one-to-one correspondence test, known as
Hume’s Principle, which is utilized now for comparison purposes now.
PHILIP: So under Hume’s Principle, an infinite set can have a proper
subset that is also infinite.
MATTHEW: Exactly. And that idea can even be used to form a new
definition, as suggested by Dedekind – a set has infinitely many elements
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when it contains a proper subset that can be placed in a one-to-one
correspondence with itself.
PHILIP: With this definition, we can again say that the natural numbers
are infinite because we found a one-to-one subset with the evens. You
also mentioned the real number line. Can we now say that the real
numbers are infinite because they contain the natural numbers, which
are obviously infinite?
MATTHEW: We can say the real numbers are infinite, but we have to be
careful. If we use the Dedekind-type definition, we have to find a subset
within the real numbers that is the same “size” as the real numbers –
meaning a one-to-one correspondence exists between the two. In more
specific terms, the subset would have to have the same cardinality as the
real numbers. Cantor proved that no such correspondence exists
between the naturals and the reals, implying they have different
cardinalities.9
PHILIP: So we have to find some other subset that has the correct
correspondence.
MATTHEW: Yeah, and the simplest example is just the interval of real
numbers from 0 to 1. It’s plain to see that it’s a subset, and even though
it’s not plain to see, there actually is a one-to-one correspondence
between the two sets.10
PHILIP: You said Hume’s Principle is currently the standard practice, but
it seems to lead to some fairly counter-intuitive results. I’m not saying
that it invalidates the principle, though, because human intuition can
often lead us astray.
MATTHEW: I believe Galileo was realizing the same thing when he
compared the naturals and the evens. He said that difficulties arise when
we try to comprehend infinity “with our finite minds.”11
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PHILIP: I’ve been reading about a similar thing in Plato’s Parmenides. The
problem there is that people try to conceptualize the Forms by comparing
them to material objects, but as soon as you try to materialize the Forms
you’ve already missed the point. Even Socrates was guilty of this. At first
he said that the Forms were as the day, because day can be in many
places at once without losing its day-ness. Then he went a step further
and said that it must also be like a sail, laid out over everything as the
day lays out over the Earth. But this was his big error because different
parts of the sail would be over different parts of the Earth, and if you
removed a particular section of the sail you would have altered the
original. This materialization, which the human mind is constantly
tempted to perform, is not an accurate way to think about the Forms. A
Form can give out shares of itself without altering itself.
MATTHEW: Infinity is difficult to grasp for the same reason – it can lose a
part of itself but remain infinite. However, that is not true for things we
come in contact with in everyday life.
PHILIP: Which is why philosophers are leery of sensory data. We can
constantly see parts being removed from the whole, and consequently,
the whole is altered. But this does not extend to the Forms.
MATTHEW: And we can constantly see numbers being subtracted from or
divided in half, and consequently, the original number is reduced. But
this does not extend to infinity, because infinity minus any finite value is
still infinity, and infinity divided in half is still infinity.
PHILIP: Have we stumbled upon another definition of infinity?
MATTHEW: Let’s hear it.
PHILIP: Maybe we can say that if a quantity has a part of itself removed
and remains the same quantity, then it is infinite.
MATTHEW: Interesting. We should think about that one a bit more.12 But
before we do, I wanted to bring up the fact that infinity is not just this
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side-concept in mathematics where weird wild stuff happens. It was
actually the careful consideration of the infinite and its reciprocal, the
infinitesimal,13 that allowed mathematics to rigorize itself to the point
they have today. In particular, it was the limit concept that increased the
logical merit of our continuous functions and our calculus, among other
things.
PHILIP: What are some examples of this limit notion?
MATTHEW: Well, we’ve seen how dangerous it can be to try and deal
directly with infinity. So mathematicians will instead introduce a limit.
They will consider what happens as a variable approaches infinity. This
keeps them on firmer logical ground and it allows them to do some pretty
powerful mathematics. For instance, we can determine some identities
with infinity, and we can nearly divide by zero. We can also work with
functions going to infinity at different rates using l’Hopital’s rule.14
 1  
lim x  1  lim x
x 
x 


2
x
lim  lim x
x  2
x 
1

x 0 x
lim
2x 2  7
4x
2
 lim

2
x  3 x  12 x
x  6 x  12
3
lim
PHILIP: This reminds me of a famous paradox in the history of philosophy.
MATTHEW: Let’s hear it.
PHILIP: They are known as Zeno’s paradoxes and there are three of them.
But I know that calculus supposedly answers one, so I’ll see if I can
remember it.
Consider an arrow that has just been released from a bow. We can
imagine viewing the arrow’s movement within a certain portion of time,
say the arrow’s first second of flight. We can further divide this time and
look at only the first half second, or the first quarter second. We can even
imagine viewing this arrow in the window of a single moment, where it is
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absolutely motionless because it has no time in which to move. Now,
since time is a collection of moments, the arrow’s flight comprises many
moments just like the one we isolated. But the arrow’s movement in each
moment is zero, and the sum of many zeros is zero. Therefore, motion of
the arrow, or anything else for that matter, should be impossible.15
MATTHEW: I think I see how calculus addresses this problem, because the
notion of isolating a single moment of time is analogous to “zooming in”
on a distance function. This can be done with the limit of secant lines,
which gives us the tangent line. And the reason the tangent line idea
works is because smooth, continuous functions are what we call locally
linear – when you zoom in on them enough, they lose any curvature that
they may have and appear as a straight line. From there, it is easy to find
the slope of a straight line and subsequently, the slope of the function.
What Zeno was doing with the arrow was similar, because he was
zooming in to a single moment. By imagining the change in time being
zero – what he called a moment – he saw in his mind’s eye a motionless
arrow, and then he concluded that its velocity at this moment was zero.
But this was his error, because the arrow can have an instantaneous
velocity even when the change in time is zero. Going back to our tangent
line analogy, this process of zooming does result in a line, but it does not
have to be a horizontal line. Sending the change in x to zero does not
necessarily send the slope to zero as well. Thus, Zeno’s arrow can have a
velocity at a particular moment, you just are not able to see it within that
moment.
PHILIP: So this is yet another example of problems arising through a lack
of care when dealing with infinite values, in this case, infinite divisions of
time.
MATTHEW: I would say so, yes.
PHILIP: Well, I think we’ve learned a lesson of infinite value in that regard.
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1
The spherical plane is also infinitely large, though in a different manner than the Euclidean and hyperbolic
planes. The Euclidean and hyperbolic planes are both infinite and unbounded, while the spherical plane is
bounded due to its positive curvature. If, on an infinite time scale, you were to travel in a straight line in
elliptic geometry, you would eventually arrive back at your starting position. This is not true of Euclidean
or hyperbolic geometry. For more, see Appendix D.
The theistic god of Judaism, Christianity, and Islam is often ascribed all of these characteristics –
boundless duration, boundless existence, and boundless power. In this way, the concept of god is an
embodiment of infinity. Ironically, philosophers have utilized a rejection of infinity in arguments for god’s
existence. For example, Thomas Aquinas claimed that the chain of causes could not go infinitely backward
so there must be a “first cause,” namely god. His argument is essentially as follows: infinity cannot exist,
therefore infinity exists.
2
3
A supertask involves infinitely many steps and therefore can never be completed. One such task arises in
Russell’s “Grand Hotel” paradox which is discussed in this chapter. Supertasks are also found in Zeno’s
paradoxes, discussed in note 14.
4
The existence of infinity is rejected by Leopold Kronecker in his philosophy of finitism. In this strict
branch of constructivism, it must be possible to construct all mathematical concepts in a finite number of
steps. Kronecker and his critical view of mathematical conventions are best known through his words,
“God created the natural numbers, all else is the work of man.”
5
This is possible due to the well-ordering principle of the natural numbers.
6
Euclid of Alexandria used a similar argument over 2,000 years ago to prove that the set of prime numbers
is infinite. His argument can be paraphrased thusly: Suppose the finite set A, B, C,..., D contains every
prime number. By considering the number N  A  B  C  ...  D  1 it can be shown that either N or its
divisor G is a prime that was excluded from the set, which implies that a finite set containing all primes
does not exist and there is an infinitude of primes.
7
For more from Galileo, see Dialogues Concerning Two New Sciences, translated by Crew and del Salvio.
8
For another seeming contradiction, see Appendix A.
9
The natural numbers are countably infinite. Cantor proved that the real numbers are uncountably infinite
and therefore strictly larger than the natural numbers. For a sketch of this proof, see Appendix B.
The function y  (2 x  1) ( x  x 2 ) can serve as a correspondence between the real numbers and the
open interval from 0 to 1. Utilizing the one-to-one conceptualization of infinite size, it has also been shown
that the interval of real numbers from 0 to 1 is the same “size” as a square built upon that interval. This
counter-intuition is similar to the fact that the natural numbers are the same “size” as the rational numbers.
10
A perfect example of our minds failing to comprehend infinity resides in Bertrand Russell’s Grand
Hotel. Russell begins by asking us to think about a hotel with a finite number of rooms. When these rooms
are full it is obvious that no other guests can be accommodated. Russell then beckons us to consider a hotel
with an infinite number of rooms, and suppose again that the rooms are full. The unexpected result is that
this Grand Hotel can still accept another guest. You simply ask the person in Room 1 to move to Room 2,
the person in Room 2 to move to Room 3, and so forth. This opens up Room 1 for the new occupant.
Furthermore, you could accommodate infinitely many more guests by asking the person in Room 1 to move
to Room 2, the person in Room 2 to move to Room 4, the person in Room 3 to move to Room 6, and so on,
opening up an infinite number of rooms – the odds. Though this may seem like a paradox, it is actually just
another example of infinity behaving differently than we would expect from our familiar knowledge of the
finite.
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12
Johann Bernoulli relied on a version of this definition when considering the sum of the harmonic series.
He found that 1  1/ 2  1/ 3  1/ 4  ...  A while simultaneously 1  1 / 2  1 / 3  1 / 4  ...  A  1. He
then concluded that the sum must be infinite (the series diverges) because it remains the same whether 1 is
added or removed.
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13
An infinitesimal is a conceptual value that is greater than zero but less than any positive real number.
Initially, the calculus developed by Leibniz used dx to represent this concept as an infinitely small change
in x. This can lead to some pleasing results. For example, consider f x   x 2 . Then, using this
infinitesimal, we have
f ( x) 
f  x  dx   f  x 
dx
2

x  dx   x 2

dx

x 2  2 xdx  dx 2  x 2
dx

2 xdx  dx 2
dx
 2 x  dx
 2 x.
The work is easy to follow and the result is the expected function; however, Bishop Berkeley noted the
logical inconsistency in this approach because dx is first treated as nonzero (when it occupies the
denominator of the fraction) and then later treated as zero (when it is discarded in the last step). Therefore,
under the guidance of Weierstrass and others, the definition of the derivative was reinterpreted as a limiting
case of secant lines approaching the tangent line. Likewise, the integral is viewed as the limiting case of
Riemann sums. There is a modern branch of mathematics that has formalized Leibniz’s original concept of
the infinitesimal into what are known as hyperreal numbers, and this theory, though unpopular, has been
used to reinterpret much of analysis.
L’Hopital’s rule states that if you are evaluating the limit of the quotient of two differentiable functions,
and both functions have a value of zero or infinity at the limit boundary, then the limit of the functions is
equal to the limit of their derivatives.
14
Zeno’s other two most famous paradoxes have to do with Achilles never being able to catch a tortoise in
a foot race, and the impossibility of traveling any distance. If Achilles is trying to catch up from behind a
tortoise, he must first arrive at the spot where the tortoise is currently. But in the time it takes Achilles to
reach this spot, the tortoise has moved forward. Now, before he can pass, the Greek hero must arrive at the
new location of the tortoise. But in the time it takes for this to happen, the tortoise will have moved ahead
again, and so forth. Zeno’s argument concerning travel – known as the dichotomy paradox - is that before a
mover can get to his destination he must get halfway there, and before he can get halfway there he must get
a quarter of the way, ad infinitum. Thus, the mover has to cover infinitely many distances which is a
supertask (see note 3) and is impossible. The solutions to these “paradoxes” will not be presented here.
They are left as exercises for the reader.
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