0

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.....

Andrés E. Caicedo
  • 79,201
Meet Vyas
  • 9
  • the notion to "count" the numbers in a set is defined through bijection, that is, we says that a set $A$ and $B$ have the same cardinality if exists a bijection between $A$ and $B$. This is the way mathematicians approach to this counts. – Masacroso May 08 '17 at 16:34
  • What you need to do is carefully define what it means for two things to be the same size. So: here, I have a bag of rocks and a field of sheep. Can you describe a method without counting rocks or sheep that lets me know if I have more, fewer, or the same number of rocks as sheep? Shepherds who could not count could have used this method to determine if they had lost a sheep. – Eric Lippert May 08 '17 at 16:35
  • Two sets have the same cardinality if there exists a one-to-one, onto function from one set to the other. The function $f : (0,1) \to (0,2)$ given by the formula $f(x)=2x$ satisfies the requirements, and so the sets $(0,1)$ and $(0,2)$ have the same cardinality. In general, every set of infinite cardinality has the same cardinality as some proper subset; your description of $(0,2)$ and its proper subset $(0,1)$ is a good example of this phenomenon. – Lee Mosher May 08 '17 at 16:36

3 Answers3

0

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).

G Tony Jacobs
  • 31,218
0

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".

Nicolas FRANCOIS
  • 4,846
0

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

Ethan Bolker
  • 95,224
  • 7
  • 108
  • 199
  • Mr. Ethan Bolker, between 0 and 1 there exist "0.pqrst..." And between (0 and 2) "0.pqrst.." and "1.pqrst...." exist. So for every number (ie 0.something) between 0 and 1 , there exist "0.Something" and "1.something" between 0 and 2. Doesn't it imply that number of numbers between 0 and 2 is double the number of numbers between 0and 1. What is the problem with this logic ? – Meet Vyas May 09 '17 at 06:22
  • 1
    No. The whole point of Cantor's way of "counting" is that "double the number" s just one of those paradoxes about infinite sets that you have to learn make no sense. There are not twice as many numbers as even numbers, there are the same amount of each. Read the wikipedia page, and other discussions of infinity. – Ethan Bolker May 09 '17 at 10:34