The continuum Hypothesis says:
There is no set whose cardinality is strictly between that of the integers and the real numbers
and the generalised Continuum Hypothesis, additionally says:
if an infinite set's cardinality lies between that of an infinite set S and that of the power set of S, then it either has the same cardinality as the set S or the same cardinality as the power set of S.
The hypothesis is actually named after the continuum of the real numbers in the first statement. But one can additionally see this in a different light by thinking of it as asking about the continuum of infinite cardinals.
Assuming the Continuum hypothesis one gets a sequence of cardinalities like we get the sequence of integers. But of course it has proved a very useful kind of fiction to assume that between these integers lay the real numbers.
So, first question: In a similar kind of way, if we deny the continuum hypthesis, what kind of other infinite numbers do we get to fill out the infinite cardinal line?
It might be worth pointing out that the Continuum Hypothesis is undecided in ZFC so one can choose either to add or deny it. But quite possibly once might say that to add this hypothesis when it is undecided from more natural axioms means that it is an exercise in aribitrariness.
In this connection, it might also be worth pointing out, that paraconsistent set theory does settle the continuum hypothesis in the negative.
Second question: Are there any other set theories whose axioms settles the continuum hypothesis, which doesn't assume it or its denial in a thinly assumed manner?
Apparently 2nd-order logic settles the issue - but I haven't found a reference to say in which direction.
One useful consequence of assuming the hypothesis is that the hyper-reals become unique (upto isomorphism).