Cardinality and Infinities
Question from Andy Hunter:
Does it make sense to talk about equalities among infinities? What I mean is does the word equal only apply to finite sets?
That's a good question. And it has a tough answer, if I can think of one.
I sort of skipped through all the trouble of infinity (mostly because it's confusing), and assumed most of the ideas of 'sizes' and 'equality' to be true. But maybe we should talk more about it.
Infinity is a very tricky subject. First of all, what is infinity? Most people will say "never ending" or "the largest number", but for me that really means nothing. What is "never ending"? Or more concerning, what is "the largest number"? There's no answer I can give for what infinity is. But as far as natural numbers go (and for even numbers too), we can get around saying that the set is 'infinite' (whatever that means) by instead simply saying that there is no largest natural number. That is, we can get as large as we want, simply by keep adding 1.
Now you're right, how is it possible for infinite things to be equal? I mean, they're infinite. Well, the idea is that many things are infinite, but some things can be more infinite (if that makes sense). And in the same way, some things may be equally infinite. Now this isn't the same as when we say two finite things are equal. I mean: 12 = 7+5 = 4x3 = 12 and that's it. But it's not so easy to think of equality with infinite sets the same way. Simply put, we can't assign the 'size' of these infinite sets an actual number. Now keep in mind that we never really assigned the natural numbers N = {1, 2, 3, . . .} or the even numbers L = {2, 4, 6, 8, . . .} a size. We simply said that their sizes, whatever they are, are equal. And that's really the best we can do. So in short: we can't assign the sizes of infinite sets a number, the best we can do is say that their sizes are equal (or that one is bigger than the other).
In fact, if we were to be rigorous about it, we shouldn't even talk about the 'size' of these sets at all. 'Size' is just a loose term we use. We really should say cardinality. And how we define cardinality is exactly the technique we used: if we can create a one-to-one correspondence between two sets (like we did for the natural numbers and even numbers) then those two sets have equal cardinality. And this is nothing more than a definition.
But if you think about it, you can see and convince yourself why we associate cardinality with 'size'.
Does it make sense to talk about equalities among infinities? What I mean is does the word equal only apply to finite sets?
That's a good question. And it has a tough answer, if I can think of one.
I sort of skipped through all the trouble of infinity (mostly because it's confusing), and assumed most of the ideas of 'sizes' and 'equality' to be true. But maybe we should talk more about it.
Infinity is a very tricky subject. First of all, what is infinity? Most people will say "never ending" or "the largest number", but for me that really means nothing. What is "never ending"? Or more concerning, what is "the largest number"? There's no answer I can give for what infinity is. But as far as natural numbers go (and for even numbers too), we can get around saying that the set is 'infinite' (whatever that means) by instead simply saying that there is no largest natural number. That is, we can get as large as we want, simply by keep adding 1.
Now you're right, how is it possible for infinite things to be equal? I mean, they're infinite. Well, the idea is that many things are infinite, but some things can be more infinite (if that makes sense). And in the same way, some things may be equally infinite. Now this isn't the same as when we say two finite things are equal. I mean: 12 = 7+5 = 4x3 = 12 and that's it. But it's not so easy to think of equality with infinite sets the same way. Simply put, we can't assign the 'size' of these infinite sets an actual number. Now keep in mind that we never really assigned the natural numbers N = {1, 2, 3, . . .} or the even numbers L = {2, 4, 6, 8, . . .} a size. We simply said that their sizes, whatever they are, are equal. And that's really the best we can do. So in short: we can't assign the sizes of infinite sets a number, the best we can do is say that their sizes are equal (or that one is bigger than the other).
In fact, if we were to be rigorous about it, we shouldn't even talk about the 'size' of these sets at all. 'Size' is just a loose term we use. We really should say cardinality. And how we define cardinality is exactly the technique we used: if we can create a one-to-one correspondence between two sets (like we did for the natural numbers and even numbers) then those two sets have equal cardinality. And this is nothing more than a definition.
But if you think about it, you can see and convince yourself why we associate cardinality with 'size'.
I know that's all confusing. But I hope it helped.
posted by Sina at
12:51 AM
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