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.
It’s larger than the rational numbers in the sense that it is a strict superset. Cardinality is what a lot of people reach for when they are talking about “larger” or “smaller”, but there are lots of other useful concepts which we can translate to “larger” and “smaller”.
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.
> “closeness” does not necessarily exist in a vector space.
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.
There is a 1-to-1 mapping between the sets, so by definition that means they have the same cardinality.
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:
..etc. The formula for this mapping would be floor(n/2)*(-1)^(n%2). Clearly everything in the left set has exactly one corresponding item in the right set and vice versa, so they must be the same "size", even though the right set contains every item in the left set and then some.
> that was the question I had - clearly it's smaller than reals, but how and why is this field larger than rational numbers?
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.)
Not really. The words are used in closely analogous ways. But the "hypothesis" in "continuum hypothesis" is part of the name of the continuum hypothesis, carried over from a time when we didn't know the answer.
Names are just names. Euclid's Algorithm is an algorithm. The Division Algorithm is a theorem.
"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."
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.
Suggestion of overturning it? It's a proof. That article would be headlined "disproving the independence of the continuum hypothesis" or some such; it would be huge news, not somebody's fun blog post.
Given that continuum hypothesis is independent of ZF, it would be very interesting to find a set of interest to a general audience whose cardinality lies between the rationals and the reals.
A "theorem" is a proved proposition, so there is no such thing as a theorem that 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:
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.