# Applications of infinite cardinalities in real analysis

What are some topics in real analysis that make use of infinite cardinalities larger than that of the real numbers themselves, preferably [edit: but not necessarily] topics that are widely applied in scientific applications?

Edit: While there may be not any actual infinities in science, the notion of infinity plays an essential part in real analysis which is indeed widely applicable in science, e.g. in the notion of a limit, the Fundamental Theorem of Calculus and the Intermediate Value Theorem . I was just wondering if higher orders of infinities are similarly required.

• "widely applied in scientific applications" ... There are none. – William Sep 22 '14 at 4:27
• The question in the title and the question in the body are different. – Asaf Karagila Sep 22 '14 at 4:28
• If you assume there is a measurable cardinal (which is very large), then analytic and coanalytic subsets of the reals behave more nicely. – William Sep 22 '14 at 4:45
• @William Not sure what you are getting at, but how widely applicable is this notion in science? – Dan Christensen Sep 22 '14 at 4:49
• You are aware that science is finitary, right? Theoretical mathematics dealing with infinite objects don't have direct applications to science, and surely not "widely applicable". – Asaf Karagila Sep 22 '14 at 5:58

The requirement "widely applied in scientific applications" is probably too high of a bar to get anything of interest, but I can think of several examples in real analysis where cardinalities higher than $2^{{\aleph}_0} = c$ are applied.

The standard argument that there exists a Lebesgue measurable set that isn't a Borel set is an example: The Cantor middle thirds set has $2^c$ many subsets, all of which have Lebesgue measure zero, but there are only $c$ many Borel sets.

A similar argument shows there exist Lebesgue measurable functions that are not Borel measurable: There are $2^c$ many characteristic functions of subsets of the Cantor middle thirds set, but there are only $c$ many Borel measurable functions.

Miroslav Chlebík, in this 1991 Proc. AMS paper, showed there exists $2^c$ many symmetrically continuous functions from the reals to the reals, and thus there exist symmetrically continuous functions that are not Borel measurable. When Chlebík's paper appeared, it had been a long unsolved question whether there even exists a symmetrically continuous function that isn't a Baire one function. See also the math StackExchange question Does $\lim_{h\rightarrow 0}\ [f(x+h)-f(x-h)]=0$ imply that $f$ is continuous?.

Because the union of interior of the unit disk in ${\mathbb R}^2$ with any subset of its boundary is a convex set, there exist $2^c$ many convex sets in ${\mathbb R}^{2}.$ Since there are only $c$ many Borel subsets of ${\mathbb R}^{2},$ it follows that there exist convex sets in ${\mathbb R}^2$ that are not Borel. (Note that this is so not true in ${\mathbb R}.)$

• Yes, those are nice applications of cardinals larger than $\frak c$ to real analysis. Their application to "science" remains to be seen. – Asaf Karagila Sep 23 '14 at 20:30
• @Asaf Karagila: And thus the point of my first paragraph... Incidentally, I started writing this as a comment to Nick R's answer, but then decided it was getting too long (and might be of wider interest), so I made it an answer. – Dave L. Renfro Sep 23 '14 at 20:36
• Oh yeah, I am aware of that. I was just explicitly stating this, since the thread seem to have been started by some misguided notion that "widely applicable in science" has a nontrivial intersection with "infinite" (in the sense of cardinality). – Asaf Karagila Sep 23 '14 at 20:40
• @Asaf Karagila: I could have also said (and now I am saying) that I decided to answer the question that should have been asked. As for the nontrivial intersection, this also holds for most areas of real analysis I'm interested in. – Dave L. Renfro Sep 23 '14 at 20:47
• @DaveL.Renfro Thanks Dave (+1). Is it fair to say that, in those topics of real analysis that are widely applied in science and engineering, discussions of orders of infinity are confined to sets being either countable or uncountable? That is my recollection of many years ago. – Dan Christensen Sep 26 '14 at 14:47

My modest understanding of real analysis leads me to believe that infinite cardinals have no role to play here. Indeed, the thrust of the subject's development was to replace the actual infinities of its original formulation with the potential infinities we use today in expressing the notion of convergence and limits.

Regarding scientific applications, the finite nature of our science seems to imply that any role infinity may take can only be the result of our mathematical idealization and abstraction.

• Thanks, Nick. I was just wondering if any such "mathematical idealizations and abstractions" employed in science made use of any use of cardinalities greater than that of the real numbers themselves -- other than, of course, simply admitting the possibility of their existence, e.g. $2^{\mathbb{R}}$. – Dan Christensen Sep 22 '14 at 18:13
• @DanChristensen I'm not smart enough to answer that. I still need to empty by dribble cup every half-hour. My intuition tells me that the set of all functions from $\mathbb R$ to $\mathbb R$ would have cardinality $2^{\mathbb R}$, but I am at a loss to see a role for this set in science. Automorphisms seem more relevant here. – Epsilon Sep 22 '14 at 18:18