Before you get yourself too exercised here, be aware that there are two vastly different mathematical concepts that could take this moniker when given the status of a basic assumption of a logical system.
In the late 1800's, Cantor's theory of transfinite sets ignited fierce controversy. The mathematicians supporting him tended to be younger and more flexible thinking, and those opposed tended towards being of the old guard. As a result, Cantor found it impossible to get a job more prestigious than a low-paying lecturer at the Univerity of Halle. This frustrated Cantor to no end, and he became obsessed with proving one concept that he thought would vindicate his ideas, that the (infinite) cardinality of the real numbers had the next higher (infinite) cardinality from the integers.
This became known as the Continuum Hypothesis. As mathematicians are wont to do, the concept was eventually abandoned in favor of a broader, more general concept, that the cardinality of any transfinite set's power set was the next higher cardinality from the set itself.
It's easy to show that the power set is bigger, not easy to show that there's nothing in between or off to the side.
Well, actually, it's impossible. Mercifully, perhaps, Cantor did not live until 1963, when Paul Cohen showed that the Generalized Continuum Hypothesis was independent of the "lesser" axioms of set theory. This means that the GCH can be "true" if you want it to be true. It can also be "false" if you want it to be.
When such a situation arises, it's often useful to assume one or the other and see what the consequences are. Unless you're an intuitionist and consider the whole idea as so much wasted breath. For the rest of us, the GCH is one of the concepts that we might apotheosize into an axiom.
But there's a much more restricted idea, which is to simply assert that
the real numbers exist
as a set, or equivalently, that all of the sets of natural numbers can be collected into a set (the "power set" of the natural numbers). But the axiom only lets you do this with the natural numbers; not any other sets.
By necessity, this set has a larger (infinite) cardinality than the natural numbers, but this has nothing to do with the Continuum Hypothesis; we don't care if there larger cardinals, ones in between, or even any off to the side.
This is an optional axiom introduced by Paul Bernays in his expansion of Gödel-Von Neumann set theory. He labels it "AC", which is unfortunately easy to confuse with the axiom of choice. So we'll call it "ACo".
GCH and ACo have different ideas at their base, and interoperate with the other "exotic" axioms in different ways:
- GCH requires the axiom of infinity to be true and implies ACo, the potency axiom, and the axiom of choice.
- ACo implies the axiom of infinity. Although it's independent from the axiom of choice and the potency axiom, the Axiom of Choice is required for the full spectrum of analysis, and with the Axiom of Choice it implies the potency axiom and hence GCH.
The biggest consequence of ACo is the necessary existence of non-computable real numbers. With the axiom of infinity but not ACo, you are left with the rational numbers and computable irrational numbers. But you have lots of problems working with the convergence of arbitrary sets of real numbers. Which suits the intuitionists just fine.