countability, intermediate values, and Cantor's sizes of infinity

Question

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.

Answer 1

$\Bbb Q$ is a dense order (as the in-between property is named) but still countable (it has a bijection with $\Bbb Z$ or $\Bbb N$) so being a dense order or not is independent of the cardinality $\aleph_0$ or continuum. A bijection need not preserve order and cardinality is just determined by bijections. The fact that $\Bbb R$ has a dense order is not the reason no bijection with $\Bbb Z$ exists. It's more subtle.

Answer 2

Yes, $|\Bbb N|=\aleph_0$. There is a bijection (one-to-one correspondence) between $\Bbb N$ and $\Bbb Q$, so $|\Bbb Q|=\aleph_0$ as well; the usual order on $\Bbb Q$ is simply not relevant to the question of its cardinality. Indeed, if $h:\Bbb N\to\Bbb Q$ is a bijection, we can define a new order on $\Bbb N$ that makes it look just like $\Bbb Q$: for $m,n\in\Bbb N$ let $m\preceq n$ if and only if $h(m)\le h(n)$. Now we have an ordering of $\Bbb N$ with respect to which we always can find a natural number between any two natural numbers. Given $m,n\in\Bbb N$ with $m\prec n$, we know that $h(m)<h(n)$. Let $q=\frac12\big(h(m)+h(n)\big)$; then $q\in\Bbb Q$, and $h(m)<q<h(n)$. The map $h$ is a bijection, so there is a natural number $k$ such that $h(k)=q$, so $m\prec k\prec n$.

It is independent of the usual axioms of set theory whether $|\Bbb R|=\aleph_1$: it is consistent with them that $|\Bbb R|=\aleph_1$, but it is also consistent with them that $|\Bbb R|=\aleph_2$, $|\Bbb R|=\aleph_3$, or indeed that $|\Bbb R|=\aleph_n$ for any positive integer $n$, and there are infinitely many other even larger possibilities.

Answer 3

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.