Wednesday, August 7, 2013

Infinity


"Oh, what does it matter? It is like infinity + 1" , I said
involuntarily.
"What do you mean- infinity + 1?" 
"Well, infinity +1 would still be infinity. It's an incremental change. Doesn't change a thing in the world", I went on.
And then my friend stumped me by quoting one of my own favorite lines- "But it made a difference to that one!"
"Inspiration story wise, I love that line. But mathematically speaking infinity+1 cannot be bigger than infinity. It just doesn't work that way."

Sparing you the rest of the conversation, let me get to my point. Infinity is really a fascinating idea. When somebody tells you- we are in an infinite cycle of birth and death (theologically speaking, or speaking of physics), I immediately think "But when was the first time you were born?" It's a chicken and egg problem- if I want to tell myself the universe is infinite I wonder how it all started. But if I try to reconcile that it is in fact, finite- then my mind goes "But what was there before it all started?" Well, it looks like I'd never get it. Maybe it is not just unknown but unknowable (ya, now I am an infinity agnostic :P)

Anyway, when I spoke of infinity+1, I spoke like it's any other number. Which, of course it isn't. When someone says God has infinite powers, they don't mean he has 9^(9^9) powers or something. They usually mean endless. There is just one infinity. And it's endless. All-encompassing. At least that's what I thought of infinity until yesterday... Yesterday, I happened to read a little about Modern Set Theory and cardinal numbers, and then I realized, infinity is not just a concept of endlessness- there are infinite infinities in this world. 

Wait, whaaa? 
(Click on Read more for more)

Think of a simple question- Is the set of natural numbers bigger than the set of real numbers? If you didn't think too much about it you would have immediately responded- "No". The set of natural numbers is definitely a proper subset of the set of real numbers, and thus definitely smaller. But think again- both sets are infinite!
From a very commonly known fact from basic math, we have gone to "There is not just one infinity, but two. And one of them smaller than the other." In fact, one could extend this and create more infinite sets that are larger than the set of real numbers (Edit 1: I was wrong about the example of complex numbers. Thanks, Mohan for pointing out! say all complex numbers).

I found this mind-blowing because it's something I have known all along (ever since I learnt about real numbers I guess) and yet I never thought about it. I very carelessly tossed the word 'infinite' around, like it was just any other number. 

Well, at this point, you might think "What the heck! So what if math defines infinite infinities. Math doesn't need to really correlate to real life. Isn't it all in our heads?" The simple answer is "Of course it is all in our heads. But why should that mean it is not real?"

The complex answer is this. If there are infinite infinities, when I say God has infinite powers, which infinity do I mean? The one that's equal in size to the set of all real numbers? The one that is the largest among all infinities (that's an absurd concept, of course. But for argument's sake)? The infinity that is equal to the number of infinities there are?! So you see how this simple mathematical idea can change what we think about many things...

 

Well, it's just something to think about. And coming back to that conversation-my friend was partially right. The whole "It made a difference to that one" might actually be true. Rather, it would be "That one made all the difference!" :)
 
Note- 1)
For the mathematically inclined: please read about Cantor's work. His proofs are awesome, especially because some are really simple. Of course, if you are really into math, you probably know this stuff.

2) My last line is not strictly correct mathematically. Infinities do not usually vary by 1 number. They often vary by infinite numbers. In fact, there are people who tried to study whether one infinity is the smallest infinity after another.
That's right. I didn't make up that sentence!
:D
3) About Edit 1: I found it a little weird that the cardinality of all real numbers is the same as that of all complex numbers. Intuition leads you to think that real numbers should be a subset (with all imaginary parts 0). The proof makes sense, but I am yet to reconcile with it philosophically.

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