Generally, we all know what the word infinity means;
something that is endless. But of course, what exactly is
endless? And can anything really be it? Even as child, we faintly understand the concept
of infinity. When we competed over who had the
most of something, we might have said that we have infinity of
it, thinking that we trumped all possible higher numbers. That
is until someone else might have said that they have infinity plus
one. Then, we might have said quick-wittedly that we have infinity
times infinity. As children, this is a playful, meaningless
game, and we mostly know it at the time. We know that you can’t
have anything bigger than infinity. But the irony is, you technically
can. You can’t actually add one to infinity or
time’s infinity by infinity and end up with a bigger infinity.
Infinity isn’t even a number. It’s a concept. It’s a
quality of a quantity in a set. A quantity so large that it can’t be
conceived or measured because it lacks an end point that would allow
for a conclusive value. However, despite this, we do have some
understanding of how infinity works and how some infinities
are different than others.
Despite how confusing it might sound, there are in fact as
many even numbers as there are all the whole numbers. There is
also the same amount of negative numbers as there are of all the
integers; integers including both all the positives and all the
negatives. There are also as many prime numbers, 2, 3, 5, 7, and
so on, as all the evens, odds, negatives, and positives put
together. This phenomenon continues to work in a variety of
applications as a result of the concept of infinity. In order to
understand how, it’s important to understand at least a couple
other general terms and concepts. Number sets are considered to be the same
size when a one to one correlation can be made between all
the elements in both sets. In other words, if you had a room with
exactly 8 chairs and in each chair, there was one person, both
the set of chairs and the set of people would be the same size.
You would know this to be true without even needing to count
them. For the number of chairs and the number people to
be equal, you would just have to know that everyone is in a chair,
and there are no more chairs. Order doesn’t necessarily matter.
Additionally, it is also important to note that a number
set is considered countable only so long as every number within
it can be listed if the lister had an infinite amount of time to
do so. Of course, no one has an infinite amount of time, but
theoretically and mathematically it just has to be possible if
someone did. And so, since there is an infinite amount of
countable natural numbers; 1,2,3 and so on. And there is also an
infinite amount of countable integers; -3,-2,-1,0,1,2,3 and so
on. By lining them all up, side-by-side, in what is called a
bijection, you create two number sets that, although the numbers
themselves don’t match, their cardinality, or quantity of
elements in their sets do line up one-to-one, forever.
This same relative function applies to the infinite amount
of even numbers, prime numbers, and so on throughout all the
rational numbers. The intuitive assumption that there must be
twice as many natural numbers, which include both evens and
odds, as opposed to just the evens or just the odds, is
completely fair, but incorrect. All countable infinite sets are
exactly the same size no matter what they are made of.
To help illustrate this and the increasingly nonsensical yet mathematically possible nature
of infinity, a German mathematician of the 19th and 20th century by the name of David
Hilbert, created a thought experiment known as Hilbert’s Infinite Hotel Paradox. In
which, Hilbert describes a hotel that fits an infinite amount of people in an infinite
number of rooms, each room being numbered in standard order with the natural numbers.
Starting at room 1, then 2, 3 and so on, into forever.
One night, when the hotel is completely full, a man walks
in looking for a room. Even though there is already a person in
every one of the infinite rooms, the hotel manager accommodates
the new guest by simply moving every current guest to the next
room over. Their room number, n, to n+1. The guest in room 1
goes to room 2. The guest in room 2 goes to 3, and so on. Since
there is an infinite number of rooms, there is no issue. And so,
infinity plus 1, or any finite number, equals the same infinity.
That one might seem obvious. However, if that same night, a bus
came that had a countably infinite amount of more people looking
to stay at the hotel, Hilbert also showed that even here, there
would be still be room to fit everyone. By having all current
guests move to the room 2 times their current room, from n to
2n, all the infinitely many odd numbered rooms would become
empty and thus, the infinite amount of new guests could be
assigned to the infinite amount of available odd numbered rooms,
and everyone, new and current, would be allocated to a unique
finite room number. A countable infinity plus another countable
infinity equals the same infinity. Lastly, Hilbert showed that even if a countably
infinite amount of more buses, each with a countably
infinite amount more people, pulled up, the hotel could still accommodate
all current and new guests. The manager could do this
a number of different ways. One being the utilization of the infinite
amount of prime numbers. Since there are an infinite amount
of countable prime numbers, the night manager could assign all
current guests to a new room that is the first prime number, 2,
raised to power of their current room number. And so, the guest
in room 1 would be
moved to 2^1, or room 2. The guest in room 2 would be moved 2^2, or room 4, and so on.
Since each unique prime number raised to a different power produces a different whole
number, the manager could then assign all the prime numbers following the number 2 to
each bus, and each person within each bus would go to the room equal to their bus number
raised to the power of their seat number. In other words, bus 1 would be assigned the
next prime number, 3. The second bus would be assigned the 3rd prime number, 5, and so
on. And so, the person in seat 1 in the first bus would go to room 3^1, or room number 3.
In the second bus, the person in seat 3, for example, would go to room 5^3 or room 125.
This would continue on forever, fanning out every infinity into the infinite hotel evenly
and with no overlap in any room. Every passenger on every bus will be assigned to a unique
room with a finite number. Thus, even here, the same size infinity remains.
Hilbert’s hotel parable essentially demonstrates that all
positive rational numbers are countable, or listable, and can
always be mapped in a one-to-one correspondence with the
infinite natural numbers. Thus, all countable infinite number
sets are always the same size. Even if you theoretically add or
multiply them all together, you would still end up with the same
infinity. The hotel is always full, and yet, somehow always has
room. However, infinity doesn’t stop there.
Georg Cantor, another German mathematician of the 19th and 20th century, slightly preceding
David Hilbert, is regarded as one the primary, if not the primary leading figure in the realm
of infinity. He helped create the foundation out of which Hilbert’s hotel was built.
But he also found a way to shut the hotel down.
In the process of comparing different infinite sets, Cantor
sought to find out whether or not there could be any infinities
that don’t fit. In other words, are there any number sets that
aren’t countable or listable, and therefore, cannot be mapped in
a one-to-one correspondence with the infinite natural numbers?
All countable infinities, and all conditions of the
infinite hotel deal exclusively with rational numbers, which
include the natural numbers like 1,2,3.., the negative integers,
the fractions, etc.. However, there aren’t just rational
numbers. There are all the real numbers, which are all the
possible numbers that can be plotted on a number line, including
all the numbers between each integer. This includes all rational
numbers but also all the irrational numbers, which are numbers
like pie, the square root of 2, and so on. These numbers go on
forever, infinitely within themselves with no repeating pattern.
And here, Cantor discovered a different kind of infinity. An
infinity that is too big to fit in the infinite hotel.
Cantor discovered this through what is known as Cantor’s
Diagonal Argument, which demonstrates that if you were to create
a list of all the real numbers, you could never complete the
list, even in an infinite amount of time, because the list would
still be missing numbers. He proved this by showing that if you
change the nth digit of each number, as in the digit in the
place that corresponds to the order the number is in the list;
for example, the first digit in the first number on the list,
the second digit in the second number on the list, and so on;
you could do this infinitely over, and every time, a new number
will be created that isn’t already on the list. This is because
the new number will be different than the first number in the
first digit, the second number in the second digit, third in the
third, into infinity. As a result, you could never complete a
list with all the real numbers and thus, never create a one-to-
one to correspondence with the infinite natural numbers. This
means that the infinity of real numbers between just 0 and 1 is
incomprehensibly larger than the infinity of all the numbers on
an infinite list of numbers. In this, Cantor discovered and
proved that there are different types and different sizes of
infinities. Now, some 130 or so years later, we still don’t know
how many and we don’t even know how much bigger the infinity of
the reals is compared to the countable infinities. We just know
that it’s a lot bigger, even though they all go on infinitely.
Some infinities, however, are apparently bigger than others.
This of course warrants the question; are we wrong here?
What does it mean when mathematic proofs don’t fully line up, if
at all, with physical reality, our sensibility, and logic? Is
our sense wrong or is our math wrong? Or both? Math is often considered to be the language
of the universe, and so, if something in math reveals
a quality of reality that seems to exceed the limits of
our comprehension, does that reveal something that nonetheless
exists as a function of objective, physical reality despite our
mind’s inability to comprehend it, or does it reveal that logic
and mathematics are, at least to some degree, contrived or manipulated
by the human mind, and at some point of extreme, fail when
applied to actual reality? At bottom, arguably the concept of
infinity reveals either our genius or our insanity.
On one hand, if the universe has an end, and consequently
every object within it, then no thing and no one can ever truly
exhibit or test or list infinity. And if nothing can ever test
or count or exist infinitely, how can infinity exist at all? And
if it can’t, the mathematic concept of infinity would be a
contradiction with physical reality along with potentially other
aspects of our mathematic understanding. But then, on the
contrary, if the universe itself is infinite, and either loops
back around through itself again and again, over and over; or
perhaps extends outward forever, with no starting or end point;
or perhaps something completely different that we can’t even
begin to fathom. In this case; in the case that infinity does
exist, then so too would infinite possibilities. Our mere existence would thus be both product
and parcel of infinity, like guests existing in one of the
rooms of the infinite hotel of the universe, surrounded
by every other finite possibility and condition of reality, plotted
on a street of infinite other infinite hotels, all at the
same time. And perhaps we get to stay in our room until we
are shifted to the next in order to accommodate another infinity.
Of course, since we can’t know if infinity really exists
and what exactly it might be, we can’t know what infinity truly
implicates in terms of time, existence, and the nature of the
universe. It all remains as unclear as infinity itself. And
perhaps to try to measure the infinite through finite reasoning,
or to try to understand the irrational through a rational system
of mathematic thinking is impossible. It’s not inconceivable, though, that a comprehensive
understanding of infinity exists somewhere at some time in the
future. As of now, however, all we can know is that infinity at
least seems to poke at something. Some potential fringe of our
current perception and understanding. An edge that we stand on
and look out from, like an infant child standing atop a canyon’s
edge, looking out at across a vast landscape that it doesn’t yet
understand or have the words for. We see something, but we don’t
what it is. We are stuck in the dark of space with our numbers
and no one but ourselves to translate them. If infinity does
exist though, there is quite literally an infinite number of
possible scenarios in which we understand
the infinite.