soft question - What happens after the cardinality $mathfrak{c}$?

While having measure theory this year the following came in my mind:

• 
  When we go from finite objects to infinite we "lose" a lot of properties. For example
  the summation isn't well defined anymore, when the sum doesn't converge, and operations
  which are abelian in the finite case are not abelian in the infinite case (think of conditional convergent series).


• When we go from countable to uncountable we have things like that too. Measures are most time countable additive but not uncountable additive (at least the lebesgue measure). Sums of uncountable indexing sets only can converge when they are in fact countable (meaning nearly all are 0).

So my question is after Arthur Comments slightly rephrased

What happens when you go from the cardinality $\mathfrak{c}$ to a higher cardinality? Are there more intermediate steps, meaning having a cardinality strictly greater than $\mathfrak{c}$ but not "to big"?

Answer

Allow me to share a small piece of my philosophical about the uncountable. It may not actually answer your question, but in a way it does.

Mathematics is created by humans. Humans are essentially finite beings, and they communicate in a finite matter. This is why we base things on finite intuition, which is the leading cause of confusion when people first deal with infinite sets.

Countable sets are sets that can be enumerated, and approximated by finite means. This is exactly the idea behind Cauchy's definition of the limit (and even more so if we only use rational numbers). We can list the rational numbers, for example, in a way that is assuring us that every number is met at a finite stage.

For this reason separability is a useful property. It allows us to approximate every element of the space using countable means. This is visible in metric space more than it is in non-metrizable spaces, but it's still true. That's the idea behind separability. So in some sense we may have an uncountable set, but we can approximate it using its structure and finite means.

But the uncountable, the truly uncountable, is not describable by such means. It's too big for us to assure that finite approximations can eventually swallow the set whole. We can't describe its initial segments in a finite fashion. We can't even assure that we can order the set nicely. This is why the axiom of choice is so important to us in the modern era of mathematics. We deal with abstract objects and we usually don't pose much limitations on their size (even though we usually care for objects quite small), and the axiom of choice gives us reasonable means to handle their sizes and structure.

But the uncountable is still uncountable. Note that even if we assume that $2^{\aleph_0}=\aleph_2$, without further assumptions such as Martin's Axiom, then we don't even know what's going on at the $\aleph_1$ level. All we know is that we can find a subset of the real numbers which has this cardinality. This is why the continuum hypothesis - in my opinion - is unprovable. The uncountable is indeed too large for us to comprehend (and manage in full) using our essentially finite means.

So what does happen beyond the continuum? A lot that we can't get a good grip on. The Lowenheim-Skolem theorem tells us that given a first-order structure which has an infinite model, it has an infinite model of every cardinality. This means that everything that happens at one infinite cardinality happens at the others, at least first-order-wise (e.g. there aren't many Archimedean fields, but there are real-closed fields). So we have a lot of fields, rings, measure algebras, Banach spaces, ordered sets, and so on.

What properties do they have? The answer often depends on set theoretic assumptions now. Much like many things at the countable level can fail without some axiom of choice (which assures the transition from finite to countable is smooth), many more can fail at the uncountable level. Assumptions such as $\lozenge_\kappa$, Martin's Axiom, cardinal arithmetics (e.g. $\kappa^{<\kappa}=\kappa$), and many many more set theoretical assumptions can - and will - tell you about the behavior of very large objects, but you will need to know about them first.