countability, intermediate values, and Cantor's sizes of infinity I understand that there is a difference in the "size" of infinities describing integers and real numbers - formalised by Cantor.
Countability
One way this is explained in tutorials is that the integers are countable, whereas the real numbers are not.
By countability, I understand that there is a 1-to-1 correspondence with the integers. This is why the even numbers have the same size of infinity as the integers themselves.
Intermediate Values
A related concept is the ability to find a value between any two others. For real numbers, I can always find a value between two close values, say 1.001 and 1.002, and this means I can't establish a 1-to-1 correspondence with the integers.
For integers, I can't always find an intermediate value. For example, there is no integer between 2 and 3.
(I don't know the technical term for the existence of in-between-values - continuity? compactness?)
Question
This question is asking for conceptual clarity.
Does countability - eg of integers - imply the first Cantor infinity $\aleph_0$?
Does the ability to find intermediate values - eg for real numbers - imply the second Cantor infinity $\aleph_1$?
I read that the rational $p/q$ are countable. However I can always find an intermediate value between any two different rationals. Does that mean the size of infinity is different to the above two?
I am not a trained mathematician so would appreciate responses that minimise use of terminology.

Unlike countability, "intermediate values" is not a concept that applies to a set directly, but only when an order structure is applied on top of a set.  It is possible to order the natural numbers in a way where intermediate values can be found, and it is possible to order the rational numbers in a way where intermediate values cannot be found.
