If the continuum hypothesis were false

[graciousgroove]

If we accept that there does indeed exist a set whose cardinality is between \aleph_0 and \aleph_1, what would such a set look like?

I know that in ZM-C we can choose to either add the continuum hypotheses or not, but if we chose to negate it, that means that there definitely is a set greater than the natural numbers but less than the real numbers... what would such a set look like? How could we construct it?

 

[micromass]

If we accept that there does indeed exist a set whose cardinality is between \aleph_0 and \aleph_1, what would such a set look like?

There will never be a set whose cardinality is between $\aleph_0$ and $\aleph_1$. The continuum hypothesis does not state this. The continuum hypothesis states that there is no set whose cardinality is between $|\mathbb{N}|=\aleph_0$ and $|\mathbb{R}| = 2^{\aleph_0}$. In other words, it says that $\aleph_1 = 2^{\aleph_0}$.

I know that in ZM-C we can choose to either add the continuum hypotheses or not, but if we chose to negate it, that means that there definitely is a set greater than the natural numbers but less than the real numbers... what would such a set look like? How could we construct it?

That should probably be ZFC. It is very difficult to say what such a set should look like. The reason for that is that the proof that the continuum hypothesis is independent of ZFC is a very difficult proof. It was done first by Cohen who won the fields medal for this. If you read the proof, then you will see how they construct the set in question (or rather, how they construct the entire model for the set theoretic universe!)

 

[rubi]

If you don't want to read Cohen's original proof, you might want to take a look at the last chapter of the book "Introduction to set theory" by Hrbacek and Jech (not to be confused with "Set theory" by Jech!). It explains Cohen's method on an intuitive level without requiring knowledge about model theory. (Of course you can't expect it to be rigorous.)

 

[gopher_p]

To me, the argument of Cohen was less about the construction of a set that was "between cardinalities" than it was about showing that the axioms of ZFC were permissive of a universe where it "looked like" the "set" was between cardinalities. Everyone knows that the "between cardinalities" set is actually countable, it's just that the universe in which this set lies lacks a bijection that would establish countability. It's like a murder trial in which the judge, prosecution, defense, general public, etc. all know the defendant is guilty, but the jurors all vote "not guilty" because they were denied a key piece of evidence due to a legal technicality; to the jurors, it looks like the defendant didn't do it even though it's clear to anyone that has all the facts that he totally did.

That's just my take on it, though.