What if you state that the very fundamentals of chemistry, physics, and mathematics are in fact based on entirely different principles…like 1x1=2 ? 
For me the AC is incredibly, “well duh” obvious.
At first take.
Then I start considering some situations I would want to use it. Then it becomes, “Well, maybe not.”
And I started really digging into it and I end up “This is one really big, unfounded premise.”
As opposed to CH where Cantor should have just declared Aleph-1 to be the cardinality of the Reals and moved on. Left the “in between” question to the mega basement dwelling Math nerds.
Is it even proven that there is an Aleph-1? That is to say, a smallest cardinal that’s strictly larger than Aleph-0? Maybe, not only are there cardinals between Aleph-0 and 2^Aleph-0, but there are are an infinite number of them, and they’re not well-ordered.
I have to think about it more rigorously, but I think (without resorting to any axiom of choice) you can define Aleph-1 as the cardinality of the least ordinal such that there is no injective map from it into Aleph-0. It will really be a successor cardinal, that is, a minimal cardinal greater than Aleph-0.
But without AC, I don’t see why there should be such a least ordinal. I am so used to AC that I find it hard to think of set theory without it (although I know many models of ZF that lack AC).
Mega bottom dwelling math nerd
Any ordinal, and the class of all ordinals, is well-ordered with respect to set membership… I think that does not depend on the axiom of choice? Without AC one does need to be careful though.
The key here seems to be Hartogs’s Lemma: if X is any set, then \{\,\alpha\in\mathrm{Ord} \mid \alpha \le X\,\} is a set, [where A \le B here means there exists an injection of A into B], which is an ordinal, and is the minimal ordinal with no injection into X.
As a corollary, if \kappa is a well-ordered cardinal, we can define its successor \kappa^+ as \{\,\alpha\in\mathrm{Ord} \mid \alpha\le\kappa\,\}; then \kappa^+ is an ordinal, \kappa^+ is a cardinal, \kappa<\kappa^+, and \kappa^+\not\le\kappa (and there is no smaller ordinal with those properties!).
Point is, even without AC it makes sense to define \aleph_1 as “the smallest uncountable cardinal”, and there is nothing strictly in-between \aleph_0 and \aleph_1.
Y’all deserve this oldie but baddie:
Q. What’s yellow and equivalent to the Axiom of Choice?
A. Zorn’s Lemon.
What’s purple and commutes?
An Abelian grape
I’ve heard he’s working from home now, though.
So I just watched a YouTube video that had a cool example of this. They have come up with a Turing Machine that only halts if and only if the Reimann Hypothesis is true:
Quanta Magazine.
A great many unsolved mathematical problems can be reduced to halting problems. Which is a good intuitive demonstration of the fact that the halting problem is very difficult (of course, “known to be impossible” is a special case of “very difficult”).