Incorrect proof of the infinities between 0 and 1 and 0 and 2

Question

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"?

Answer 1

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 $(\frac x2, \frac x2 +\frac12)$.

Answer 2

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.

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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.

Answer 3

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.