In other words, fractions, integers, positive integers all belong to the set of countable infinities, meaning there is an isomorphic function that bidirectionally maps each positive integer (the count) to every item in the target, countable infinity.
There is no isomorphism between real numbers and any countable set. If you create one before you are 40, you will get a Field Medal.
The isomorphism and the distinction between sets that have them and sets that do not has proven useful.
You could think of infinity as one concept, but it is usefully divided into countable and uncountable versions.
Fractions are countable. Real numbers are not.
In other words, fractions, integers, positive integers all belong to the set of countable infinities, meaning there is an isomorphic function that bidirectionally maps each positive integer (the count) to every item in the target, countable infinity.
There is no isomorphism between real numbers and any countable set. If you create one before you are 40, you will get a Field Medal.
The isomorphism and the distinction between sets that have them and sets that do not has proven useful.
You could think of infinity as one concept, but it is usefully divided into countable and uncountable versions.