Tuesday, June 22, 2010

Infinity

Infinity is a funny thing. When I was a kid, my mom always used to tell me that she loved me to the moon and back infinity times. Not long ago, it occurred to me that she might as well have said she loved me to the grocery store and back infinity times, or the width of a hair and back infinity times, or to the edge of the galaxy and back infinity times. It's all exactly the same distance. But I guess some infinities sound larger than others.

My kids, especially my 5-year old son, have started using the word infinity to describe everyday things, which almost always produces an amusing mental image in my head. For example, my son will say that he wants a snack and he would like infinity popsicles please, and of course I have to picture the entire known universe filled with popsicles, and then some.

A few days ago, I fixed our Nintendo Wii, which was making a horrible grinding noise, and today it started making the noise again. I told my son that it shouldn't take me very long to fix it again this time, and I should be able to fix it more permanently because I got some practice last time. He said he hoped I wouldn't be fixing the Wii for infinity days... or worse, infinity years.

Of course, infinity years is not at all worse than infinity days. I asked him, "Do you know when you could use the Wii if I had to fix it for infinity days?" He said he didn't know, and I said, "Never!" And I said the same for infinity years. Then I reassured him that it probably wouldn't take anywhere close to that. Then again, whether it takes me one hour or three days, either possibility is equally close to infinity years, which is to say, nowhere close.

My kids also don't tend to use a million or a billion to stand in for a very large number. They usually say a googol. The other day I was remarking to my wife that a googol is probably much larger than the number of atoms in the observable universe, and she wasn't convinced. I did a back of the envelope estimate and came up with about 10^80 atoms in the universe, which by sheer dumb luck turns out to be very close to the actual best estimate we have. And a googol is 100,000,000,000,000,000,000 times larger than that.

So ten times the observable universe is still nowhere close to a googol atoms. One billion observable universes is still about 100,000,000,000 times less. You have to take one hundred billion billion observable universes to get about one googol atoms. And that is equally distant from infinity as the number of atoms in my pinky fingernail. Infinity is a funny thing.

6 comments:

Burk said...

So glad to see you back.

It seems like an infinity of days since you last posted.

Saganist said...

Heheh, or infinity years! I'll confess I've let life interfere, but I do want to get back to posting regularly.

dbd said...

I remember back to my CS days at the Y where an instructor told us about two different infinities: The smaller infinity was the set of integers. The larger infinity was the set of all numbers between 0 and 1. I still haven't figured out (accepted?) that one.

Welcome back.

Christine said...

So glad to see you posting again. Running has simply taken too much of your time. :)

Saganist said...

dbd, that thought was actually in the back of my mind when I wrote "some infinities sound larger than others", but I decided against mentioning it explicitly. I believe your instructor was talking about "countably" infinite sets as opposed to "uncountably" infinite sets. It's been a while, but I think that a countably infinite set is one whose members can in principle be enumerated, such as all the integers. An uncountably infinite set is one whose members cannot be enumerated in principle, such as the set of all real numbers.

... Ah, after a quick trip to Wikipedia, I remember that an infinite set is called uncountable if there is no way to map its members via an injective function onto the set of natural numbers. In other words, there is no way to say "this is the first member of the set, this is the second member of the set, etc." such that you will eventually enumerate all members of the set.

It seems to me that all numbers between 0 and 1 may be countably infinite, as you can enumerate them breadth-first, for example in decimal notation as follows: 0, 1, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.01, 0.02, ... , 0.09, 0.11, 0.12 ... , 0.19, etc. Each level of precision has ten times as many members but in this way they can all be enumerated. At least, I think so - I'm not a mathematician but I play one on the Internet. I'm probably wrong, because I usually am.

But the set of *all* real numbers is uncountably infinite because ... well, now my brain hurts. I need to study up on this stuff again. It's been too long.

Saganist said...

Thanks, Christine. Honestly I think you've hit the nail on the head. Most of my time lately has been either running or recovering from running. Not enough time writing!