Friday, December 17, 2010

Thy Vast Domain





GEORG CANTOR
1845 - 1918



Infinite, Thy vast domain; Everlasting is Thy reign.
Holy God, We Praise Thy Name


To see a world in a grain of sand, And a heaven in a wild flower,
Hold infinity in the palm of your hand, And eternity in an hour.
William Blake


I see it, but I don't believe it.
Georg Cantor


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Faith is where you find it. Let me show you where I find some of mine. It won't take long. Come with me to the place where mathematics, philosophy, and religion meet. But first, a question or two....

Consider the set of natural numbers: {1,2,3,4,5...}
Now consider the same set, less the number 1: {2,3,4,5...}
Is this new set smaller?

The set of all numbers between 0 and 1 is an infinite set.
The set of all numbers between 0 and 2 is also an infinite set.
Is this set twice as large?

Does infinity exist?

How big is it?

Does infinity come in different "sizes?"

Think about these for a bit before you read on.


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Answers: no, no, yes, BIG, yes.


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Georg Cantor, the father of set theory, was a fascinating man. He made a life's work out of exploring the infinite, with many surprising results. His proofs are remarkably simple to follow, considering. The mark of his genius was in seeing the way to demonstrate his assertions. Rather than reproduce his work here, let me tell you what I learned from it.

Say I had a bunch of balloons and a group of children. How could I know if I had one balloon for each child? Well, I could count the balloons and the children. That would work. Or, I could just give one balloon to each child. If, at the end, I had no balloons or children left over, then I had exactly enough. The two sets were equal. That is a one-to-one correspondence.

To compare infinite sets, we attempt to set up a one-to-one correspondence between them. If this can be shown to exist, we say the sets have the same cardinality. If it does not exist, they do not. With me? Here are some interesting conclusions:

Any line segment contains the same number of points. In fact, any line segment, of any size, contains the same number of points as the entire number line!

The infinite plane has the same cardinality as the number line. Adding a second dimension does not make the set larger.

In fact, adding any number of dimensions does not make the set larger!

The natural numbers {1,2,3,4,5...} are a countable set. There are an infinite number of them, of course, yet they can be, in Cantor's term, denumerated. Rational numbers: quotients of two integers of the form p/q, where p and q share no common factors, are also a countable set.

Cantor showed that there are as many rational numbers as natural numbers. The sets have the same cardinality. Doesn't this seem strange?

In fact, there are many correspondences which seem counterintuitive, for example:

There are as many even numbers as natural numbers.

There are as many perfect squares as natural numbers.

Isn't this odd? We can exclude half the set of natural numbers and still have a subset the same size? We can exclude almost all of the set and still have a subset the same size? Interesting, no? It gets better, though.

Bear with me for a minute and trust me on this: For a finite set n, we will define its power set, P(n), as the set of all its possible subsets. A set containing n elements will have 2n elements in its power set. For example: let n = {a,b,c}; then P(n) = { {}{a} {b} {c} {a,b} {a,c} {b,c} {a,b,c,}}. 23 = 8 elements. See?

Cantor proved that this holds true in all cases: the power set P(n) is strictly greater than the original set, even if the original set is infinite! What can this mean?

What it means is there are an infinite number of infinities, each infinitely greater than the last. Let's call the set of natural numbers N. Then P(N) is infinitely greater, P(P(N) greater still, P(P(P(N) even greater than that. And so on into the night.

Wow. Whoa. And ouch. Does your brain hurt yet?

One last thing: The set of natural numbers is a countable set. Cantor showed, in his diagonalization proof, that the set of real numbers is an uncountable set, infinitely greater than the set of natural numbers. The cardinality of the continuum is the same over any interval, however small. If you choose any number, there is no next number. No matter what second number I choose, there is always another one closer: infinitely, and always. And this is just the beginning.

Isn't that beautiful?


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I think all of this is incredibly beautiful and comforting. The infinite most certainly exists, yet it exists in very special ways. Cantor himself was a devout Lutheran, who saw his work as a message to the world from God. I don't think he was far off. If an apostle is a messenger, then he was an apostle of sorts. He held infinity in the palm of his hand and could not believe what he saw there. Was it God? No, I don't believe so; but I believe he saw God's fingerprints.

Faith is where you find it and Scriptures abound.


Respectfully Yours,


Cricket




6 comments:

Buck said...

Does your brain hurt yet?

Short answer: Yes!

Any number of algebra, geometry, and trig teachers tried unsuccessfully to convey the beauty of math to me. I must be missing a critical gene or something coz even something as elementary as balancing my checkbook causes my eyes to glaze over.

Nice try, tho, Cricket!

lime said...

oh sugar, my brain hurt as soon as i saw an equation. my brain just doesn't work that way. that said, i delight in the infinite number of ways god can reveal himself. you see him in math. i see him in the magnitude of the night sky.

Cricket said...

Hi Buck -

Thanks for stopping by, as always.

I'll make one more try, just for fun. For myself, I find in mathematics a certain harmony, or symmetry or things like that which, to me, are quite beautiful. I think Cantor's proofs, for example, are quite elegant in their simplicity, which belies the time and effort it took him to think of them, of course. It's the kind of thing that seems so obvious, but only after someone else has pointed it out.

You might also look here:

http://users.erols.com/ziring/mandel.html

And just enjoy these pictures of Mandelbrot and Julia sets. Don't worry about the descriptions of what the sets are, this would probably just irritate you. Just look at the pictures and see what these numbers do. It's truly amazing.

What I find, for myself, in these excursions into set theory, is something fascinating, inspiring, even reassuring. It is that element of recursion that appeals to me so much - the fact that we can't escape the infinite. It is there in the largest things and the smallest, and it exists in infinite variety. To me, it is the fingerprint of God.

My thoughts are not as your thoughts... Right? Think about this:

115665448797814148703201028178798741354146879813546876143546876843576874357687867811429288841417136841387861748718958614

What number is this? Well, I don't know. It's too big to even have a name, I think. It would be difficult for you to even tell me every digit in it without making a mistake. It's pretty much too big to even hold in our minds. Yet it most certainly exists.

Now think of this: if we were to travel to infinity, this number is nothing. We would pass it in an eye-blink and leave it far behind.

Wow. Whoa. And yes, ouch.


Hi Lime -

Yep. As I said, faith is where you find it. :-)

Suldog said...

Although I have a hard time grasping that a set is the same size if you subtract from it, I believe it.

I can grasp infinity going forward - after all, I know nothing else, as I exist now and have no problem imagining always doing so from this point onward - but grasping infinity going backwards is beyond me. By the same token, it is where my faith resides. Even though I can't truly imagine it, it is the only logical conclusion. If there were a beginning to everything, including God, then what came before? What was the cause of it all happening? Until I can grasp a first cause, infinity in reverse is the only answer. And there is nothing that fits that, logically, other than an infinite creator for the other stuff that follows.

I am always humbled in that regard.

Barbara Shallue said...

God's fingerprint - beautiful!

ds said...

WHOA!! I had no idea you'd written this. And yes, my head hurts.
Will have to come back to digest (if possible; even my digestive tract is mathophobic) this better.
Math & poetry (metrics), physics and art (music). Yes, the highest forms of both sides of the brain are those that combine both sides of the brain. Now my head hurts even more--I hope somehow you understood that mess.
Fascinating post; I am sorry that I missed it.
In the realm of bizarre coincidence, "Infinite thy vast domain" was the hymn we sang at the close of Mass every Sunday. Every. Sunday. For years. One more huge blast to the head...
thank you.