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Interesting, thanks for taking the time to type that. So you really would have to change the basis of logic to get rid of the strangeness of Cantor's sets, huh

Interesting, thanks for taking the time to type that. So you really would have to change the basis of logic to get rid of the strangeness of Cantor's sets, huh? That's quite a leap, I wonder if that would lose other properties of mathematics as well, some that would be nice to actually have? I mean if you insisted infinities don't exist, you'd get rid of a lot of funky stuff, but lose analytic derivatives and integrals, which definitely is not a good trade-off in my opinion. Some people actually advocate for this, all working on discrete math of course.

andrewla 2y ago
> change the basis of logic to get rid of the strangeness Short answer is yes. Long answer is that the accepted framework of the "basis of logic", i.e. ZF set theory, is a direct result of Cantor's program -- Cantor was not working from an axiomatic basis, he was creating the formalism for set theory. Where things went wrong were not so much that he was an idiot or anything; clearly he is a tremendous genius and saw implications of his programme that led to incredibly strange and counterintuitive spaces. But instead of revisiting the basis, he found himself drawn to this verdant landscape. Intuitionists (and its various offshoots and cousins) don't reject the notion of infinity per se; even finitists, the most extreme class, still accept that there is an infinity in the form of a repeated process -- that the positive integers are "infinite" in the sense that you can always produce a larger one than any proposed maximum. Integration and differentiation still exist, but are much easier to formalize, because, essentially, the behavior of any constructible function is completely defined by its behavior on rational numbers (or any other constructive version of dense number systems, like binary or decimal expansions). In re: "different infinities", this is the big red herring of Cantor's work. This requires that you accept that a 1:1 correspondence of infinite sets yields a class of sets that you can group by into "cardinality", and those cardinality classes have an interesting meaning. But this defies the operational use of infinity -- there are more natural numbers than even numbers if we're thinking about strict subsets, but when we're summing series, they are effectively the same size. So it's not necessary to choose some definition of the "size" of an infinite set; you can just choose what operational characteristic you are looking at in the context you're working in.

andrewla 2y ago

> change the basis of logic to get rid of the strangeness Short answer is yes. Long answer is that the accepted framework of the "basis of logic", i.e. ZF set theory, is a direct result of Cantor's program -- Cantor was not working from an axiomatic basis, he was creating the formalism for set theory. Where things went wrong were not so much that he was an idiot or anything; clearly he is a tremendous genius and saw implications of his programme that led to incredibly strange and counterintuitive spaces. But instead of revisiting the basis, he found himself drawn to this verdant landscape. Intuitionists (and its various offshoots and cousins) don't reject the notion of infinity per se; even finitists, the most extreme class, still accept that there is an infinity in the form of a repeated process -- that the positive integers are "infinite" in the sense that you can always produce a larger one than any proposed maximum. Integration and differentiation still exist, but are much easier to formalize, because, essentially, the behavior of any constructible function is completely defined by its behavior on rational numbers (or any other constructive version of dense number systems, like binary or decimal expansions). In re: "different infinities", this is the big red herring of Cantor's work. This requires that you accept that a 1:1 correspondence of infinite sets yields a class of sets that you can group by into "cardinality", and those cardinality classes have an interesting meaning. But this defies the operational use of infinity -- there are more natural numbers than even numbers if we're thinking about strict subsets, but when we're summing series, they are effectively the same size. So it's not necessary to choose some definition of the "size" of an infinite set; you can just choose what operational characteristic you are looking at in the context you're working in.