Yep, that's right. Its cardinality is the same as rationals, since it's countable.
Unlike most of the time, I read the article first and now that I'm here, that was the question I had - clearly it's smaller than reals, but how and why is this field larger than rational numbers? Guess it's not.
So when someone says “larger” or “smaller”, your first step might be to try and translate that relationship into a more precise mathematical concept, like cardinality or measure.
Casual terminology also leads to weird discussions. Like when someone asks whether some function is “close” to another, and these functions are defined in terms of vector spaces. Unfortunately, “closeness” does not necessarily exist in a vector space. So the answer may be that the question does not make sense.
The asker will give a definition. For example, two vectors are close if sqrt of dot product of difference of the two vectors is smaller than some number delta.
Ok, did I miss the explanation of that? Or is it something in "part 2" which I didn't see a link to?
Let me give a more common example. Consider these two sets: N, the set of non-negative integers, and Z, the set of all integers. Clearly, everything in N is also in Z, and then some. But N and Z still have the same cardinality, because there are 1-to-1 mappings between the two sets. Here is one example of such a mapping:
N | Z
---------
0 -> 0
1 -> 1
2 -> -1
3 -> 2
4 -> -2
5 -> 3
6 -> -3
Isn't there a theorem that speaks of the existence or non-existence of a set whose cardinality is strictly larger than Q and strictly smaller than R.
And a conjecture that says this theorem might well be unprovable?
The proposition you refer to (which is not a theorem since no proof is known, and in fact it has been shown that this proposition is logically independent of the usual foundations of set theory, so it cannot be proved in that framework) is called the Continuum Hypothesis:
As pdonis points out sidethread, this isn't really a valid question. (Or rather, the question is fine, but the answer to all questions of this form is already well-known, so there's no point in asking this specific question.)
It is not possible to prove that a set is both smaller than the reals and larger than the rationals, because such a set would disprove the continuum hypothesis. (And symmetrically, it isn't possible to prove that no such set exists, because that would be a proof of the continuum hypothesis.)
Sure, even without much of a mathematical background, people generally take it for granted. Which is why it's disappointing that a suggestion of overturning it isn't fulfilled.
Names are just names. Euclid's Algorithm is an algorithm. The Division Algorithm is a theorem.