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/).
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/).