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Can you tell me more about this? Why are Cantor's discoveries ignored or how do intuitionism deals with the contradictions that would arise from it?

andrewla 2y ago
Generally unless you can construct an item, it is not really that relevant. Cantor's argument about the sizes of infinities is only interesting if you accept that the definitions make sense -- i.e. that "cardinality" represents a notion of size, and that 1:1 correspondence means that two infinite sets have the same cardinality. So we have the notion of "countable" sets like the integers, and he proceeds to prove that the reals can not be put in 1:1 correspondence, so are a different cardinality. But you can just ignore this and treat infinity in an operational sense rather than try to have a general definition. The problem with the uncountable argument is that not only are the rational numbers countable, but so are the algebraic numbers, and so, in fact, are any numbers that can be constructed from a finitely expressed constructive process. Unfortunately, even after you leave all those subsets behind, you are left with an uncountable number of elements, none of which can be constructed. These phantoms are the "everywhere" in "almost everywhere". The short version, I guess, is that those phantom numbers are artifacts of a formalism that allows you to deduce the existence of something by disproving its non-existence rather than constructing an element. The core of most intuitionist mathematics is that the law of the excluded middle (that something must be either true or false) is disallowed under most circumstances. That pretty much ends up taming most of the crazy counterintuitive junk. Still plenty of unexpected and exciting results, and places where intuition breaks down, but far fewer instances of "this clearly is not true" stuff.

andrewla 2y ago

Generally unless you can construct an item, it is not really that relevant. Cantor's argument about the sizes of infinities is only interesting if you accept that the definitions make sense -- i.e. that "cardinality" represents a notion of size, and that 1:1 correspondence means that two infinite sets have the same cardinality. So we have the notion of "countable" sets like the integers, and he proceeds to prove that the reals can not be put in 1:1 correspondence, so are a different cardinality. But you can just ignore this and treat infinity in an operational sense rather than try to have a general definition. The problem with the uncountable argument is that not only are the rational numbers countable, but so are the algebraic numbers, and so, in fact, are any numbers that can be constructed from a finitely expressed constructive process. Unfortunately, even after you leave all those subsets behind, you are left with an uncountable number of elements, none of which can be constructed. These phantoms are the "everywhere" in "almost everywhere". The short version, I guess, is that those phantom numbers are artifacts of a formalism that allows you to deduce the existence of something by disproving its non-existence rather than constructing an element. The core of most intuitionist mathematics is that the law of the excluded middle (that something must be either true or false) is disallowed under most circumstances. That pretty much ends up taming most of the crazy counterintuitive junk. Still plenty of unexpected and exciting results, and places where intuition breaks down, but far fewer instances of "this clearly is not true" stuff.