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"practically useful"

The Mathematician blinks several times, looking at you as if you'd just grown a third elbow. :-O :-)

The problem is that there's nothing arbitrary about Cantor's work. Now, you could (along with Aristotle, IIRC) just declare "there are no actual infinities, only potential infinities" and thus that the whole investigation is verboten. But that's not very mathy.

Another alternative would be to redefine the fundamentals to outlaw actual infinite sets, but if you do that then you have redo everything, because some of the things that you might consider useful and related to other things go out the window. (Offhand, I think the result would be constructive mathematics. Putting on my computer scientist hat, I'm perfectly happy with constructive math. But most mathematicians aren't really.)

But if you allow infinite sets, then the properties of them, including infinite sets of different sizes, are derived directly from basic set theory. They are as "true" as anything else in math. (I'm a formalist---we're just playing a game that has a specific set of rules. :-))

"Sedenion neural networks provide a means of efficient and compact expression in machine learning applications and were used in solving multiple time series forecasting problems." "Metacognitive Sedenion-Valued Neural Network and its Learning Algorithm" (https://ieeexplore.ieee.org/document/9160921/).

> While with differently-sized infinite sets, the point is that they aren't (to the best of my knowledge) either 1) directly useful or 2) particularly interrelated with anything else.

The distinction between countable and uncountable is very practically useful. You can always encode the elements of a countable set such that every element is represented in the computer by a finite-length bit string, but you can never design an encoding that does this for every member of an uncountable set. No exceptions.

In addition, this notion of distinguishing countability from uncountability can be applied to show that the set of Turing machines is countable (up to isomorphism). This immediately informs one of the limits of computation, and why, e.g. trying to design a computer or program that performs exact arithmetic operations on real numbers in general, or say finding the limit of a series in real numbers in general, is as achievable as squaring the circle.

And make no mistake, computational modeling that today already permeates every aspect of society would look very different if exact operations on real numbers were computable and efficient. (and for the general programmer, no need to worry about the imprecision of floats and especially with financial stuff)

> While with infinite sets, the point is that they aren't (to the best of my knowledge) either 1) directly useful or 2) particularly interrelated with anything else.

Formal treatments of calculus are based around infinitesimals. Infinitesimals are the same set of ideas about infinite sets, but applied to the spaces between arbitrarily close points.

Specifically formal treatments of calculus work because between any two points there are an infinite number of points and the size of that infinity is the same size as the entire set of real numbers.

At least this is my undergrad level of comprehension. I'm certain there are more exotic treatments, but this is the one that you have to do all the proofs for.

> Formal treatments of calculus are based around infinitesimals

formal treatments of calculus don't use infinitesimals, and analysis using infinitesimals is explicitly separate from traditional analysis

You're absolutely right. I don't know what I was thinking when I wrote this last night.

I meant to say that formal treatments are based around being able to find arbitrarily small neighborhoods around a point, and that this only works when you have something the size of the reals, for which you can find a real-sized space between any two points.

It's this infinite self-similarity at any arbitrarily small scale that is at the heart of calculus.

This is not true. The usual calculus that people learn (and usually learn with rigor in a real analysis course) does not touch infinitesimals.

The approach via infinitesimals is even called "nonstandard analysis".

https://en.wikipedia.org/wiki/Nonstandard_analysis

You're absolutely right. I was going for a different concept. See explanation above. I should stop writing after 02:00.

The problem here is that you're basing your conception of what is "good," "interesting," and "useful" based off of what you might discuss in an undergraduate class in Real or Complex Analysis.

The notion of cardinality and how we can have different "sizes of infinity" is an enormously important and interesting idea that comes up all the time in mathematics.

How can you say that infinities of different sizes are not useful? All of analysis is built on the fact that R is the unique complete ordered field, and therefore uncountable.

And if you want every infinite set to have size continuum, then N (and so Z and Q) break horribly

The field of constructive analysis would beg to differ with you.

An amazing amount of real analysis can be done with absolutely no reference to the excluded middle, completeness axiom, or accepting the existence of uncountable sets.