I heard an expressions, some infinities are larger than others recently, and they stated that it was proved to be so. I haven't been able to find this proof, and I'm not as much of a math enthusiast as I was earlier in my life, but I don't believe this statement to be true. I also do not know how to format the math symbols here, so forgive me if I'm a bit vague.

Given a set of numbers S1 = [0, 1], we can say that there are an infinite amount of numbers in that set. If the previous statement, some infinities are larger than others is true then we can also say some infinities are smaller than others and also some infinities are equal in size to others. With this in mind, we could say that if S2 = [1, 2], then there are just as many numbers as there are in S1, as there are in S2, because the range is the same. If both of the previous statements are true, than we could define S3 = S1 + S2, which would yield all numbers in the set [0, 2], and we could say that S3 is larger than both S1 and S2, since it contains all numbers in S1 and S2. With this in mind, we could also say that the set [0, 0.1] has 1/10 as many numbers (but still infinite) as S1.

This is where I lose understanding. What this is saying is that we can have infinity == infinity + infinity == infinity / 10. But because the law of identity states that $A = A$, then how can some infinities are larger than others be true? My understanding is that infinity will always equal infinity, and there are just as many numbers between $[0, 0.1]$ and $[0, 1000]$. If infinity does not equal infinity, then it violates the law of identity. By saying some infinities are larger than others it seems to me like we're saying infinity is the set of all numbers, and there exists a set whose size is larger than infinity.

Am I wrong in this understanding?

  • $\begingroup$ Have you read anything about it online? Have you read one of the threads here, which usually end up having some good answers? $\endgroup$ – Asaf Karagila Dec 15 '14 at 19:39
  • $\begingroup$ Just from the "Related" list on the right side of the screen, just by their titles. One, and two, and three. And I am more than certain that there are others, perhaps many others. $\endgroup$ – Asaf Karagila Dec 15 '14 at 19:40
  • $\begingroup$ Actually $S_3$ is not larger, at least not in the sense of "cardinality." Comparing the sizes (cardinalities) of infinite sets requires you to suspend some of your intuitions about numbers/sizes. I prefer the term cardinality because it is a technical term, and "size" is an intuitive term. $\endgroup$ – Thomas Andrews Dec 15 '14 at 19:40
  • $\begingroup$ Yes, but those questions usually end up with an example of how you can use a bijection to compare two different things (ie. a bushel of corn for a sheep), which results in a statement that there are more real numbers than there are integers. but how can this be possible if there are an infinite number of each? $\endgroup$ – user185205 Dec 15 '14 at 19:42
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    $\begingroup$ No, Cardinality is not at all an integer. Infinity is not a number. $\endgroup$ – Thomas Andrews Dec 15 '14 at 19:55

I think that the issue here is mostly linguistic.

Saying that something is infinite simply says that it is not finite. Saying that I have more than two students in my class tells you nothing about whether there are three, or six, or 42 students in my class.

It is true that both $\Bbb N$ and $\Bbb R$ are infinite. But it tells you nothing about comparing their sizes in a meaningful way.

There are several ways to measure the size of a set, depending on the context. In the case of subsets of $\Bbb R$ we can ask what is their length, or how can we approximate their size using things which have length (namely, intervals). This gives us measure theory, and it turns out that under some reasonable assumptions we cannot assign "length" to every set of real numbers.

The same can be done in $\Bbb R^2$, where now length is replaced by area, or in $\Bbb R^3$ with volume, and so on.

If you want only to consider sets of natural numbers there are ways to measure the size of those sets, in a fashion that assigns bigger and smaller sets some notion of largeness that fits some sort of intuition.

But those are just sets of real numbers, or subsets of the space, or something like that. What about much larger sets? Like the set of all sets of reals? Or all sets of reals which have "length" assigned to them, or so on? What about sets of those sets, or sets of sets of sets of sets of those... etc.

At some point, all the nice structure that the real numbers and related objects carry with them goes away and disappears. Bijections, they stay with you forever. So we measure the size of sets using functions which have certain properties, namely they are injective and surjective.

We say that two sets have the same size, or same cardinality if there is a bijection between them. This bijection does not need to preserve any given structure. The natural numbers look nothing like the rational numbers, but both are countably infinite, for example.

And as it turns out, there is a bijection between $[0,1]$ and $[0,2]$ as intervals of the real line; but there is no bijection between $\Bbb N$ and $\Bbb R$. Since there is an injection from $\Bbb N$ to $\Bbb R$, this means that there cardinality of $\Bbb N$ is strictly smaller than that of $\Bbb R$. Cantor's theorem also tells us that if $X$ is any given set, then $\mathcal P(X)$ which is the set of all subsets of $X$, has a larger cardinality than $X$. So any set has more subsets than it has elements. Even infinite ones.

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  • $\begingroup$ Awesome, i believe it was purely linguistics that was confusing me. Thanks for the explanation. $\endgroup$ – user185205 Dec 15 '14 at 19:57
  • $\begingroup$ My pleasure. Tomorrow I have to go and remind this to my students anyway. :-) $\endgroup$ – Asaf Karagila Dec 15 '14 at 19:57

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