I’ll explain my intuition of the difference between c and aleph1. Understand, of course, this is only my intuition; none of the crux of it can be traced back to any ZFC axioms.
I explained in this thread (post 4) how to define aleph1 in terms of the order-structure distinct ways of well ordering a countable set (by the way, my definition of a “well ordering” in that thread should include the fact that the set must also be linearly ordered).
c, on the other hand, can be defined as the number of different permutations of the natural numbers. What I mean is: Look at all the ways of ordering the natural numbers by the ordinal omega:
is one way, of course.
is a completely different one. Notice that this is distinct from post 4 in the previously mentioned thread; here, the different labelings make the different permutations distinct.
We also have:
as another example.
(To be clear, by “permutation” or “well ordering by omega”, I simply mean all the different labelings of the natural numbers that are still order isomorphic to the standard ordering of the natural numbers, just with different labelings).
Anyway, this is where the non-axiomatic intuition comes into it for me. A well ordering of a set appeals to me as being a very structurally precise ordering. There are “relatively few” of these, to my intuition
On the other hand, permutations seem to me to be a much “less structured” ordering than a well ordering. In particular, the orderings
are order isomorphically the same, but different permutations. Not to mention all the uncountably many other examples that are still order isomorphic to these, yet permutationally distinct.
Based on this idea, it tends to be my intuition that there are many more distinct permutations of a countable set than there are distinct well orderings.