A thought:
There are an infinite number
of infinities, and some infinities are bigger than others. A simple statement.
A mind-blowing statement. A true statement [for the proper definitions of each
of those words, at least].
The procedure that
mathematicians rely upon to compare the size of two collections of stuff (this
striving to be both incredibly general and exactly specific can be a source of
confusion to the uninitiated - but therein dwells the true power of
mathematics) is as follows:
We match things, one by one.
Yes, just like in Pre-School. But it works! Let's go on:
Think about counting. Suppose
you need to get some apples from the store. If you are buying a single apple,
say for a quick bite, you don't even need to think about counting – we are able
to innately see the single object and see it as one thing.
Maybe you're buying a few
apples, for an apple pie, let's say. Almost all people can do the same for 4
apples as you did for one. Beyond that, you need to count.
What are you doing as you
count these apples? One Apple. Two Apples. Three Apples. What a mathematician
would tell you if asked (and do we love to be asked!) is that you are matching,
one by one, the counting numbers to the apples. You are making pairs: 'Green
Apple and One' 'Red Apple and Two' 'Red-with-funny-shape Apple and Three'.
Generally speaking, two
collections of things are the same size if I can pair all of them up with no
bachelors left from either set.
In everyday life, infinity is
generally used as more-or-less equivalent to “without end”. This is a very good
definition, and it is a decent way to describe what is meant by the 'infinitely
large' and 'infinitely small' one might first (and for all too many, also last)
encounter in undergraduate Calculus.
For this, however, a
different way of thinking about infinity will develop. It is important to note
that neither view is true or false. In mathematics, the cake of the same
general idea may be baked and sliced in many different ways, each appropriate
to a different meal, each perfectly delicious in its own domain. Onward to the
Cake!
Consider this: are there more
fractions (or equivalently, all decimals that have after some point repeat the
same exact sequence without end (this conveniently includes both 1/3 =.33333...
and ½ = .500000....)) than there are counting numbers (1, 2, 3, 4,
...hahaha...)? The shocking answer is NO! There are just as many! Basically,
you can construct a grid, kind of like a multiplication table, and at each box,
instead of multiplying, you write the corresponding fraction, so that row 5,
column 7 would contain the fraction 7/5 (or 3.5, if you prefer). Of course,
there is no end to this grid. And here is the magic: the matching. Even though
each row and each column go on forever, each diagonal (going from the bottom
left to upper right) is of a finite length. These I can can match too! I can
match each fraction to exactly one counting number by zig-zagging back and
forth. And so, each counting number is matched up with its destined fraction,
and they live happily ever after.
There are two hurdles to be
wary of: One is that care must be taken too not double count – ½ is equivalent
to 2/4, and both appear on this list. Second is that it is the process of
matching itself that must be defined and shown here, as an actual
implementation would be terminally lengthy.
Thus, you have counted to
infinity, and shown that there are just as many fractions as there are counting
number themselves. How astounding! What madness! – the first column (1/1, 2/1,
3/1, and so on) are the counting numbers themselves! The counting
numbers are just as numerous as their extended family, when including
themselves in the census. Mere chemical hallucinogens have nothing
on mathematics.
But now, here is the final
magic, the wonder! There is an infinity that we can't count by this
matching method! The irrationals – the numbers that cannot be written as
fractions (hence ir – ratio – nal). Part and parcel of this fact is that their
decimal expansions (in other words, when I state as many subsequent digits as I
can before I drive my audience to insanity) are rather wild – they never
settle down into a pattern. This is not to say that they are random in the
manner of thrown dice – the calculation of any digit you please is quite
well-defined and orderly [although one can certainly ask for one that is...
annoying to calculate – simple and well defined does not imply easy]. Rather,
it is the case that instructions to build a real number are not going to be
like 1/6, whereupon I might instruct you to write '.' and then '16' eternally,
or dare I say 1/7 – a Sisyphean sentence most cruel of '.' then '142857' until
Hades itself is overturned!
How? Here is How! Here is
Cantor's magic! Imagine, dear reader, that some shady salesman offers you a
once-in-a-lifetime deal: the magical list of all Irrational Numbers [in no
particular order] – and they are bulleted, on by one:
1.
√48 = 6.92820323028.....
2.
√2 = 1.4142135623.....
3.
π = 3.14159265.....
4.
√2 = 2.6457513....
5.
and so on...
Now, say you suspect that he might
be selling you adder oil – so you consider for a couple millenia and say
(probably to the FTC):
“Your claim is false! There
is an irrational not on your list!”
First, for brevity, let's say
that this novel number is between 0 and 1. This way we only have to deal with
one side of the decimal point. Then:
Irrational # 1's 1rst digit
after the decimal is 9: my number is now 0.0
Irrational # 2's 2nd digit
after the decimal is 1: my number is now 0.02
Irrational # 3s 3rd digit
after the decimal is 1: my number is now 0.022
Irrational # 4's 4th digit
after the decimal is 7: my number is now 0.0228
.
.
.
and so on.
Now! Your number doesn't
match any number on this cheat's list! It can't be the first, the first
digit mismatches, and on and on and on. These – they are more numerous than
those! And so, we named the small infinity ℵ0 [Aleph Null] and this larger
infinity ℵC. There are infinitely many
infinities, but that is a story for another day.
And so: We, these transient
specks on a speck huddling by a spark in the vast darkness - Yet we have seen
beyond even ℵ0 - and so we journey onward
forevermore, light into the darkness.
If you're interested in
learning more on this topic, I highly recommend Everything and More, by
David Foster Wallace.
