In reading another question (Explaining Infinite Sets and The Fault in Our Stars) it got me thinking about the way that you can prove that the number of numbers between 0 and 1 and between 0 and 2 are the same. (apologies if my terminology is a bit woolly and imprecise, hopefully you catch my drift though).

The way it is proved is that you can show that there is a projection of all the numbers on [0,1] to [0,2] and vice versa. I'm good with this.

However I then got to thinking that you can also create a projection that takes all the numbers from [0,1] and maps them to two numbers from [0,2] by saying for a number x it can go to x or x+1. This is reversible to so you can say that you can find a pair of numbers in [0,2] such that they differ by one and the lowest is a member of [0,1].

Why is it that this doesn't prove that there are twice as many numbers in [0,2] than in [0,1]. It seems to me that this is the crux of why it runs counter to intuition but I can't work out the flaw.

Or is it just in the nature of infinity that infinity*2 is still the same infinity and thus its just that infinite is "weird"?

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    $\begingroup$ There are twice as many numbers in $[0, 2]$ as in $[0, 1]$. It's just that, as you say, twice infinity is still infinity. $\endgroup$ – Qiaochu Yuan Jul 15 '14 at 8:45
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    $\begingroup$ It was "discovered" by Galileo Galilei - see Galileo's Paradox - that : "the ideas of less, equal, and greater apply to finite sets, but not to infinite sets. In the nineteenth century, using the same methods, Cantor showed that this restriction is not necessary. It is possible to define comparisons amongst infinite sets in a meaningful way (by which definition the two sets he considers, integers and squares, have "the same size"), and that by this definition some infinite sets are strictly larger than others." $\endgroup$ – Mauro ALLEGRANZA Jul 15 '14 at 8:48
  • $\begingroup$ There's a question somewhere that talks about different ways of measuring the 'size' of sets. One of them, which you allude to, is that which coincides with the length of the intervals (a particularization of the measure theoretic version of size). The other one is the set theoretic version. I will tag the question (elementary-set-theory) because I'm sure Asaf knowns this question. Edit: Nevermind, he's here. $\endgroup$ – Git Gud Jul 15 '14 at 8:49
  • $\begingroup$ To go further you can read about the cardinality of sets: en.wikipedia.org/wiki/Cardinal_number $\endgroup$ – Yves Daoust Jul 15 '14 at 8:51
  • $\begingroup$ @GitGud: I lol'd. :-) $\endgroup$ – Asaf Karagila Jul 15 '14 at 8:54

The crux is the fact that you don't specify how you measure "size" of an infinite set.

In the case of the real numbers, and even more so when we consider intervals, we can measure their length. In which case $[0,2]$ is twice as long as $[0,1]$ and therefore twice as large.

If you want a "raw" measurement of how large a set is, then you reduce to the notion of cardinality, in which case we only care about bijections and therefore $[0,1]$ and $[0,2]$ and in fact $\Bbb R$ itself all have the same size.

There is still a problem with your argument. The fact that you can map each number to two different numbers (or rather, map exactly two numbers to the same number) is not a good argument for "there are twice as many elements" (which implies a strict inequality, to my ears anyway). For example, consider $\Bbb N$ and map every even element $2k$ to $k$, and every odd element, $2k+1$ to $k$ as well, and of course $\Bbb N$ does not have strictly more elements than $\Bbb N$.

You also have that each natural number has exactly two numbers which map to it, but it still doesn't mean that there are twice as many natural numbers as there are natural numbers. That's just not good mathematics.

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  • $\begingroup$ Your objection to the argument relies on the fact that it is nonsense to say there are twice as many natural numbers as there are natural numbers, which in turn presumes some intuition about infinity. But this sort of objection is backwards. You must first pick a formal notion of "there are twice as many as" and then stick with it. There's no real basis in your answer for saying it's just not "good mathematics": we run into counterintuitive consequences of formal notions all the time. You would need to explain why the other formal notions are more useful than this formal notion. $\endgroup$ – symplectomorphic Jul 15 '14 at 11:46
  • $\begingroup$ @symplectomorphic: No, my objection is based on the fact that "twice as many" is translated to a strict inequality; rather to a weak inequality. The natural numbers cannot have strictly more elements than the natural numbers. I'm sorry that you disagree with that notion. I also edited my answer to clarify this point. $\endgroup$ – Asaf Karagila Jul 15 '14 at 11:53
  • $\begingroup$ I fail how to see how that objection differs from my description of it. Maybe you just mean that instead of it being nonsense, it is just false; fine. The issue I take is with your naive explanation of "translation" here, and what the criterion for what distinguishes "good" math from bad. Yes, if you want "twice as many as" to preserve a strict inequality, then the OP's formal regimentation won't work. But we give up aspects of concepts all the time when we formalize. The OP's proposal is one way of regimenting the notion. $\endgroup$ – symplectomorphic Jul 15 '14 at 12:02
  • $\begingroup$ (Cont) Your objection is just that it doesn't capture some other sense of the natural language talk, which it happens to preserve for finite sets. That is not the right criterion. Your conceitedness in "I'm sorry that you disagree with that notion" is both unwarranted and misses the point. I don't disagree with that notion. $\endgroup$ – symplectomorphic Jul 15 '14 at 12:02
  • $\begingroup$ I'm sorry if my words come out as conceit, it wasn't my intention. I still don't quite understand what would ease your mind, would it be if I remove "nonsense" and write "plain false"? I don't want to do that, and I prefer to use a stronger language because it will help whoever reads this remember that translating from natural language to a formal language is a process which requires precision to be added to the statement, since the statement can be interpreted in two ways (strict or weak inequality) it has no well-defined meaning. It has no sense. It is... nonsense. $\endgroup$ – Asaf Karagila Jul 15 '14 at 12:06

Your last sentence is the best answer to your question possible. Yes, indeed, infinities are "weird" and are not intuitive when you first encounter them.

Any infinite set is, in fact, in effect "twice as big" as itself. For example, even $[0,1]$ contains twice as many elements as $[0,1]$, since you can map $x$ to the pair $(\frac x2, \frac x2 +\frac12)$.

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For finite sets the simplest way is to count the elements to have a proper concept of size.

For infinite sets it gets more tricky. In naive set theory one compares sets by trying to establish one to one mappings, like you wrote, in which case the sets are considered to be of the same size.

Looking at $d = b - a$ to compare intervals $[a,b]$ or $[0,1] \subset [0,2]$ does not help in the context of comparing the number of their elements relative to each other. They all end up as large as $[0,1]$ (for $a\ne b$).

Regarding the last line of the question:

While $2 \cdot \infty$ might yield just $\infty$ in your case and is counter-intuitive, or $\mbox{card}(\mathbb{N}^n) = \mbox{card}(\mathbb{N})$ which I found remarkable (link), you will find funny results for $\infty - \infty$ (see certain quantum field theoretic calculations) and enlarge already infinite sets $A$ with the power set construction $2^A$, getting into different orders of infinity (which remarkably is the reason why there are uncomputable programs).

Why is it "weird" or counter-intuitive? Personally, I tend to the biological explanation. It is us not the subject. Our brain is working fine for our environment and us living in it, which is

  • finite,
  • mostly flat,
  • rather slow (compared to the speed of light),
  • has not that much gravity (compared to the conditions on the surface of a neutron star),
  • is not too small and not too large

So we seem to have more difficulties grasping everything which is not like that, like infinities, theory of relativity and quantumn mechanics.

If we had to grapple the last billion years with infinite objects in our physical world, I believe we wouldn't be that surprised often.

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  • $\begingroup$ You should differ between "field theoretic" and "quantum field theoretic". $\endgroup$ – Asaf Karagila Jul 15 '14 at 11:22
  • $\begingroup$ @AsafKaragila Thanks. $\endgroup$ – mvw Jul 15 '14 at 11:28
  • $\begingroup$ You shouldn't edit things from your iPad! ;-) $\endgroup$ – Asaf Karagila Jul 15 '14 at 11:30
  • $\begingroup$ I am still not sure if I should drop autocorrection or not. It is frustrating. $\endgroup$ – mvw Jul 15 '14 at 11:31

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