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.

 

[blue_raver22]

If the continuum hypothesis were false, it would mean that there exists a set whose cardinality is between \aleph_0 and \aleph_1, which are the cardinalities of the set of natural numbers and the set of real numbers, respectively. This would imply that there is an infinite set that is larger than the set of natural numbers but smaller than the set of real numbers.

Constructing such a set would be a challenging task as it would require us to find a way to order the elements in this set in a way that preserves the properties of both the natural numbers and the real numbers. One possible way to construct such a set is by using the concept of a well-ordered set, where every non-empty subset has a smallest element. This would allow us to order the elements in such a way that preserves the properties of the natural numbers, but also includes elements that are not part of the natural numbers.

Another approach could be to use the concept of a countable set, where the elements can be placed in a one-to-one correspondence with the natural numbers. This would allow us to include elements that are not part of the natural numbers, but still have a cardinality less than that of the real numbers.

Ultimately, the exact construction of such a set would depend on the specific axioms and theories that we are working with. But regardless of the approach, it is clear that such a set would have a unique and complex structure that would challenge our understanding of the mathematical universe.