# Cardinality and Concrete Mathematics

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.

• I strongly disagree with your characterization of number theory and analysis, and do not understand your distinction between concrete and abstract mathematics. – Travis Willse Apr 30 '15 at 11:40
• Related regarding cardinality of set used in analysis: mathoverflow.net/questions/44705/… – Todd Wilcox Apr 30 '15 at 12:00
• I am fairly sure that this was asked and discussed before, at least once. Unfortunately I am quite in a hurry so I don't have time to search for the duplicate. – Asaf Karagila Apr 30 '15 at 12:01
• There is a whole branch of combinatorial number theory that makes use of $\beta\Bbb N$, whose cardinality is $2^{|\Bbb R|}$. – Brian M. Scott Apr 30 '15 at 20:52

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".

• Not quite; since the continuous functions, or the Borel functions, or the differentiable functions, these only make sets of size $2^{\aleph_0}$. Considering all the functions is barely interesting. It is however true that the set of all Lebesgue measurable sets is of size $2^{2^{\aleph_0}}$. – Asaf Karagila Apr 30 '15 at 12:01
• @AsafKaragila There are interesting non trivial results about arbitrary real functions. Two classical examples: (1) [Blumberg] Every real function is continuous when restricted to some dense set of reals. (2) [Davies] There is a real function countably many rotations of whose graph cover the plane. Many modern results here use set-theoretic tools, so the area is aptly called set-theoretic real analysis. – hot_queen Apr 30 '15 at 21:49
• @hot_queen: I didn't know these things. But they sound pretty cool. As for set theoretic real analysis, at least tell me that there is no set theoretic PDE or set theoretic differential geometry... :-) – Asaf Karagila Apr 30 '15 at 21:53
• Most problems in fields like PDE have very low level complexity so it is hard to see how one could apply set theory there (Shoenfield's absoluteness). But here's an example due to Donald Martin where you can use forcing to prove a ZFC result: Every compact subset of plane of positive area contains a rectangle $A \times B$ where $A, B$ are perfect set of reals and A has positive length. This is a $\Sigma^1_2$ sentence but Martin used forcing to prove it! I am only saying that mathematics is what mathematicians do and there is no reason why interesting results cannot expand an area. – hot_queen Apr 30 '15 at 22:03
• @Asaf Karagila: Some reasonably natural sets with cardinality greater than $c = 2^{{\aleph}_0}$ are: (a) there are $2^c$ many functions from $[0,1]$ to $\mathbb R$ that are Riemann integrable; (b) on $\mathbb R$ there are $2^c$ many complete Borel measures and there are $2^c$ many $\sigma$-finite measures; (c) there are $2^c$ many convex sets in ${\mathbb R}^{2};$ (d) I think there are $2^{2^c}$ many $\sigma$-algebras on ${\mathbb R}.$ – Dave L. Renfro May 4 '15 at 19:02

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 $\aleph_\omega$.

• What? $\aleph_1\leq2^{\aleph_0}$? What sort of talk is that to a man spending his time studying determinacy? :-) – Asaf Karagila Apr 30 '15 at 18:03
• @Asaf Ha :-) Probably I should have said "subfields that study surjective images of the reals probably don't count" to be more general (and also more correct, if we're not assuming $\mathsf{AC}$.) – Trevor Wilson Apr 30 '15 at 18:08