Infinite sets
2010-12-20 03:07 am![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
archershipsWithin the logic of infinite sets, apparently larger infinite sets are actually the same size as seemingly smaller infinite sets. For example, the set of all natural numbers (1, 2, 3...) is the same size as the set of all even numbers (2, 4, 6...), though you might think it'd be twice the size, because the first natural number can be mapped onto the first even number, the second onto the second, and so on, to infinity, without any natural numbers left over.
via markvernon.com
I find this very hard to wrap my mind around, even though the logic seems sound.
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Date: 2010-12-20 12:32 pm (UTC)no subject
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Date: 2010-12-20 11:45 pm (UTC)no subject
Date: 2010-12-21 12:00 am (UTC)There are still some people that argue that infinity is not really well-defined, but most mathematicians and philosophers think that since we haven't run into any actual contradictions since the development of a few conceptual distinctions in the early 20th century, the remaining "paradoxes" are just weird things that we can investigate to try to understand better. (Like Zeno's paradoxes, which have been known since ancient times, but don't really prove that motion is impossible.) But no one wants anyone to just take any of this on faith!
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Date: 2010-12-21 04:11 am (UTC)So, two ways of thinking about infinity (the simple mapping you mention on the one hand, and extrapolating to progressively higher numbers on the other hand) give different answers.
There's a good reason why the mapping way of thinking about it is the "right" way of thinking about it though, and extrapolating to higher numbers is just unreliable.
Trying to extrapolate by comparing the density of numbers is unreliable because density is different than total quantity. The set of even numbers 2,4,6,8, and 10 has the same size as the set of numbers 1,2,3,4, and 5. It's not that they are different sizes, it's just that one of them has the numbers spaced closer together.
Imagine you started with a barbed wire fence that had 100 wooden posts on it, and then you decided to have a contractor come in and for some reason change the spacing, by making the same 100 posts twice as close together. No change in the number of posts, they're just closer together. But what if the number of posts wasn't 100, it was infinite and they stretched all the way infinitely in both directions as far as you could see (and all the way further)? Then the contractor could also (hypothetically) come in and change the spacing to be twice as close, and you still wouldn't have any change in the total number, you'd just have a change in the density... how far apart they are. Of course, in that case it would take an infinite amount of time for him to dig up the old posts and move each one, one at a time =) But hopefully, that makes it seem a bit more natural, and highlights why you can't rely on looking at the spacing to tell you how many there are total.
you could say that it's weird that x/2 = x for sufficiently high values of x, and move along.
x/2 = x is uncontroversial for infinity values of x, but what is even more interesting is that there are some mathematicians, physicists, and philosophers who argue that x/2 = x even for some very large but *finite* numbers. This school of thought is called "ultra finitism".
For example, an ultra-finitist would argue that 10^(10^100)/2 = 10^(10^100). Actually, everyone agrees that 10^(10^100)/2 = 10^(10^100) for all practical purposes, it's just that most people would say it equals that *approximately* whereas ultra-finitists argue that it equals that exactly.
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Date: 2010-12-21 05:09 am (UTC)"But what if the number of posts wasn't 100, it was infinite [...]";
"But hopefully, that makes it seem a bit more natural [...]"
Not really. It seems pretty much the same, intuitionally. But once we get outside the range of direct human experience, I'm not sure there's any reason to expect intuition or what seems natural to matter. The fact that infinity doesn't follow the same rules as regular numbers bothers me far more than arguments about density.
"there are some mathematicians, physicists, and philosophers who argue that x/2 = x even for some very large but *finite* numbers. This school of thought is called "ultra finitism"."
Wikipedia suggests that ultrafinitists would argue that such very large numbers aren't defined or don't exist in the same sense that numbers with relevance to the physical world do. So "10^(10^100)/2 = 10^(10^100)" would be the case only in the sense that both are undefined, which seem to me like a very weak sort of equality, and it's not clear to me that self-described ultrafinitists would subscribe to your statement, for that reason.
As may be clear from my objections above, I'm not sure I think the concept of infinity makes sense, but I don't know enough math to have any but the most simplistic arguments, so (knowing this) I very, very rarely argue about it. :)
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Date: 2010-12-21 05:29 am (UTC)Wikipedia suggests that ultrafinitists would argue that such very large numbers aren't defined or don't exist in the same sense that numbers with relevance to the physical world do. So "10^(10^100)/2 = 10^(10^100)" would be the case only in the sense that both are undefined, which seem to me like a very weak sort of equality, and it's not clear to me that self-described ultrafinitists would subscribe to your statement, for that reason.
Hmmm... well, I'm not sure precisely what the term ultra-finitism entails so I could be talking about something else.
But I do know that I've run into at least one physics professor (whom I published a paper with one summer) who simply stated this as fact... that 10^(10^100) = 10^(10^100), in his words "even in principle, not just in practice". He explained further that large numbers have very different mathematical properties than small numbers, and for numbers of the form 10^(10^x), where x is at least 2 order of magnitude greater than 1, numbers can only be defined up to an overall constant factor, and discussing differences between a number like 10^(10^100) and the same number multiplied by a constant is "entirely meaningless".
My advisor also routinely said very similar things, although he may not have explained the reasoning quite as explicitly all in one sitting as this other professor did. This particular comment came up during a conversation I was having with the other professor over what to tell the students about numbers this large (for a class he was teaching and I was TA'ing, called Statistical Mechanics)... and that's what he told me to tell them.
Looking at the ultrafinitism wikipedia page I get the impression that it is at least very closely related, although perhaps what the physics professors I'm familiar with believe is not exactly the same as ultrafinitism.
I'm not sure why both of the professors I've co-authored with are so convinced that x = x/2 for very large numbers, as that seems like it would be somewhat controversial with most mathematicians. But given that there is a school called "ultrafinitism" I get the impression it is at least something some mathematicans believe. And possibly the same thing as what ultrafinitists believe.
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Date: 2010-12-21 05:30 am (UTC)10^(10^100) = 10^(10^100)
arg, meant to write 10^(10^100) = 10^(10^100)/2 (one googleplex equals half a googleplex)
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Date: 2010-12-21 05:40 am (UTC)As may be clear from my objections above, I'm not sure I think the concept of infinity makes sense
I think there is a reasonable case to be made that it is simply a "useful fiction" that helps with a lot of calculations. I guess when you work with it long enough though, it starts to seem real in the same way finite numbers are, and you get more of an intuition for what sort of properties of infinite numbers are the same as finite numbers, and what sorts of properties are different.
You mention that it bothers you that "the rules are different". I wouldn't say the rules are different, just that they behave differently because they are a different kind of number... the basic rules are the same, they just have different consequences.
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Date: 2010-12-20 09:28 pm (UTC)no subject
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Date: 2010-12-20 04:19 pm (UTC)The real mindblower is when you learn about Cantor's diagonalization argument to the unindexablity of the real numbers. It's an extremely elegant and insightful proof by contradiction ("you say [foo]? OK, if [foo] is true, then [bar], [bat], and therefore [blah]. But, [blah] is untrue or contradicts [foo], so [foo] is false").
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Date: 2010-12-20 05:23 pm (UTC)no subject
Date: 2010-12-20 06:03 pm (UTC)Now, with the same paper, you could try and write all the real numbers down (which includes the naturals), from 0 to positive infinity... but I could come by and note easily that you missed a number (using Cantor's argument). And when you added that number, you've still missed more. And more. That's uncountable/unindexable infinity!
Yes, it's a little hard, but math can be fun, too.
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Date: 2010-12-20 06:13 pm (UTC)no subject
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