what does it mean ' infinity comes in different sizes'?

Question

What does mean when mathematicians say infinity comes in different sizes ?

Answer 1

Here we mean that two sets are the same size of infinite if there exists a one-to-one and onto transformation from one set to the other.

For example: There are an equal number of integers even integers by the transformation

$$ T : \mathbb{Z} \longrightarrow 2\mathbb{Z} $$

Defined by $$ T(a) = 2a \ \ \ \text{ for all}\ \ \ a \in \mathbb{Z}. $$

This is equivalent to saying we can list the items one by one, try this with two finite sets to see how it works. We can also list all rational numbers one by one, give this a try. So there are the same number of rationals as integers.

Now we can't list all reals, meaning there are more reals than integers and rationals.