>>8238580At first seems counter intuitive, but there is a way to say that one infinite set is bigger then another infinite set.
Consider what we mean when we say that two finite sets have the same size.
You could say they have the same amount of elements because we can assign every element of set A one element of set B.
This idea can be extended to infinite sets and we say to infinite sets are equal if there exists some rule that assigns every element of set A exactly one element of set B.
For example consider the set A={1,2,3,...} and B={2,4,6,...}.
They have the same size because if we take one element from A and multiply it by 2 we get an element from set B.
Two things are important here, firstly this element is unique, so if we take two different elements from set A and multiply them by 2 we have to different elements from set B and secondly if we double every element from set A we will at some point get every element of set B.
These two properties of the rule "multiplying by 2" means that it is called bijective.
And 2 sets are equal if there exists such an bijective rule between them.
For example the set of all natural numbers and the set of Integers are equal, but surprisingly there is also an bijective rule between the set of natural numbers and the ration numbers.
So there exists a way to assign every rational number one natural number but importantly there exists no such rule for the real and natural number which means that the set of the real numbers is bigger then the set of all natural numbers.
The question now is is there a set which is bigger then the natural numbers but smaller then real numbers?
This is exactly what:
means.
This question has been proven to be neither provable nor disprovable and is one example of where the incompleteness theorem manifests. That means that you could not construct such as set but you also cant prove that such a set does not exists.
