We have at least 2 different kinds of infinities:
1. Integers and all the other countable numbers - That is things you can pair with integers.
For instance pairs: {1, 1}, {1,2} ...
These can be easily paired integers along diagonals:
{1 1} {1 2} {1 3}
{2 1} {2 2}
{3 1}
We can easily pair like so:
1 -> {1 1}
2 -> {1 2}
3 -> {2 1}
4 -> {1 3}
5 -> {2 2}
...
So pairs [fractions being one of them with equivalence class notion on top] are no 'bigger' than integers.
2. On the other hand real numbers are really bigger than integers!
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| Courtesy:Wikipedia |
There is a nice way to prove this due to George Cantor.
In fact we need to only consider real numbers in {0, 1}
if it is countable, We can write this as:
1-> 0.0000...0
2-> 0.0000...1
0.0000..2
..
0.9999..9
Now if we go along the diagonal and change 0 to 1 and everything else [1, 2,...9] to 0 we get a number which differs from all these numbers in atleast one position and it is not mappable to an integer!
So real numbers are indeed bigger, in fact way way bigger! In fact they are not even real despite their name!
Now the continuum hypothesis asks whether there is an infinity strictly between integers and reals or reals are the next big infinity. Of course hypothesis claims it is true.
No one has been able to prove or disprove this in more than a century!
