Cardinality and Concrete Mathematics

Question

First, let me distinguish Concrete Mathematics ( Mathematic branches that studies fixed structures ) from Abstract Mathematics ( branches that study classes of structures, such as algebra , topology, etc ).

Number theory studies fixed set(s) whose cardinality is at most of the reals ( but mainly of the naturals ).

Analysis studies fixed set(s) whose cardinality is at most the cardinality of the reals ( but mainly of the reals ).

Is there any actual concrete mathematics that is done under fixed set(s) of cardinality greater than the reals, in other words, is there any set with cardinality greater than the reals that is important enough to have its own discipline ?
Does anyone know why that's the case ?
P.S : Obviously, i'm not considering set theory as a possible option, since the sets there are not concrete, but abstract.

Thanks in advance

Answer 1

The set of all functions $\mathbb R\to\mathbb R$ has cardinality $2^{2^{\aleph_0}}$ which is greater than the cardinality of the reals. Those are often studied in analysis. Proving your characterisation of analysis is wrong, and providing an example of a set larger than the reals that is often studied. I have no idea (and frankly don't care) if that fits your weird definition of "concrete mathematics".

Answer 2

Outside of set theory and closely related fields, one usually considers sets that are finite, cardinality $\aleph_0$ (countably infinite) cardinality $2^{\aleph_0}$ (the cardinality of the continuum) or of arbitrary cardinality.

So I think any "yes" answer would have to be a subfield of set theory. (I'm going to ignore your instruction to "not consider set theory as a possible option, since the sets there are not concrete, but abstract." Under your definition of "concrete" versus "abstract" mathematics, I think set theory is partly concrete and partly abstract.)

Subfields that study objects of cardinality $\aleph_1$ probably don't count because $\aleph_1 \le 2^{\aleph_0}$. Studying objects of cardinality $\kappa$ where $\kappa$ is a large cardinal probably doesn't count either because $\kappa$ is not "fixed".

I think the best candidate would be singular cardinal combinatorics. Although its results are often phrased in terms of arbitrary singular cardinals, the case of $\aleph_\omega$ is the most important special case and many problems in the field are specifically about $\aleph_\omega$.