Comparing numbers between (0 and 2) and (0 and 1).

Question

I read that between any two number , there are infinite numbers. And that infinities are equal (number of numbers between 0 and 1 is equal to number of numbers between 0 and 2). Let the numbers be 0 and 1, 0 and 2 . Let the numbers between 0 and 1 be "x". Let's take any number between 0 and 1, "0.pqrst", there cannot be "1.pqrst" between them. But between 0 and 2, it exists. So it implies for every number between 0 and 1 ie '0.something' there will be '0.something' and '1.something' between 0 and 2. Doesn't it imply that the number of numbers between 0 and 2 is double the number of numbers between 0 and 1?

Please explain if I have any conceptual blunder or anything like that.....

Answer 1

The numbers between 0 and 1 can be placed into one-to-one correspondence with the numbers between 0 and 2. In particular, if $x$ is between 0 and 1, then $2x$ is between 0 and 2, and this is a one-to-one mapping between the two sets. Furthermore, for any number $y$ between 0 and 2, we have $y/2$ between 0 and 1, so it's a bijection - a one-to-one correspondence.

You're right that you can also map the numbers from $(0,1)$ into a subset of the interval $(0,2)$, leaving room to map them also into the remaining subset, as you describe. (Send $x$ to $x$, and also to $x+1$). However, you could also do this in reverse: For any number $y$ on the interval $(0,2)$, send $y$ to $y/4$, and you'll only use up half of the interval $(0,1)$.

This goes to show that infinite sets are tricky to count. There are some useful theorems, such as this: If there is a one-to-one map (not necessarily a bijection) from set A to set B, and also a one-to-one map from set B to set A, then the two sets are the same cardinality (size).

Answer 2

What is implied by "those infinities are equal" is that the sets can be put in bijection.

In your case, just consider function $f:x\mapsto 2x$. It is clearly a bijection from $[0,1]$ to $[0,2]$.

This is not evident if you try to "count" those sets, because they are infinite. So "double infinite" is the same as "infinite" in this case, but "infinite integers" is much less than "infinite reals".

To better understand what's at stake here, google "cardinals", "ordinals", and maybe "Hilbert's hotels and buses".

Answer 3

It took quite a while for mathematicians to figure out how to deal consistently with infinitely large sets.

The fundamental idea, due to Cantor, is that two sets are "the same size" if there's a way to pair their elements up with no omissions and no duplicates.

The classic example is that there are the same number or positive even integers as positive integers: pair $2$ with $1$, $4$ with $2$, $6$ with $3$, and so on.

Your conceptual blunder is in thinking that a set can't be the same size as a proper subset of itself. In fact, that is essentially the definition of a set of infinite size.

In your example, when you pair $x$ with $2x$ you match every number between $0$ and $1$ with a number between $0$ and $2$, so there are the same number of each kind of number.

You can read much more about this idea.

https://en.wikipedia.org/wiki/Bijection

https://en.wikipedia.org/wiki/Georg_Cantor#Set_theory