How do the terms “countable” and “uncountable” not assume the continuum hypothesis?

Question

1. Every countable set has cardinality $\aleph_0$.
2. The next larger cardinality is $\aleph_1$.
3. Every uncountable set has cardinality $\geq 2^{\aleph_0}$

Now, an infinite set can only be countable or uncountable, so how does this concept not negate the possible existence of a set $S$ such that $\aleph_0<|S|<2^{\aleph_0}?$

I am guessing the issue is that claim $(3)$ is in fact not necessarily true. If so, I would be glad to hear more on why this is not the case. I've only started looking into this area of mathematics recently, so pardon me if my question is naive.

Answer 1

The word "uncountable" should be seen as equivalent to "not countable." So any infinite set that can't be put into bijection with $\Bbb{N}$ is uncountable, regardless of anything involving CH.

Answer 2

When we say that a set is finite if there is a bijection between the set and a proper initial segment of the natural numbers. We say that a set is infinite if it is not finite.

Similarly, we say that a set is countable if there is an injection from that set into the set of natural numbers. We say that it is uncountable if it is not countable. That is all.

The continuum hypothesis is a statement about a particular set and its cardinality. It has nothing to do with the definition of countable, uncountable or cardinals in general.