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> The work of Cantor and beyond regarding infinite sets is generally taught as mathematical "fact", as fundamental as 2+2=4, but unlike 2+2=4 is there any pract
by jyounker 5y ago
> The work of Cantor and beyond regarding infinite sets is generally taught as mathematical "fact", as fundamental as 2+2=4, but unlike 2+2=4 is there any practical constraints or applications around it? It seems more like an almost arbitrary puzzle than "fact".
The field of numerical analysis depends on formal treatments of calculus. Formal treatments of calculus don't work in sets smaller than the reals.
- crazygringo 5y agoWhen I studied real analysis, I don't recall anything regarding real numbers that seemed to depend on assigning different sizes to infinite sets. Is there something I'm missing, something that the formal treatment of calculus depends on them for?
- dnautics 5y agoDedekind cuts (which are necessary for epsilon-delta proofs) or alternatively Bolzano Weierstrass theorem (if you use sequential convergence definition) imply multiple sizes of infinity, so the logical arrow is in the other direction, roughly: "real analysis implies assigning different sizes to sets", not "assigning different sizes to sets implies real analysis". It's been a while but iirc there are other constructions (such as p-adic numbers) which create multiple sizes of infinities out of the rationals which aren't real analysis.
- SAI_Peregrinus 5y agoThe Surreal Numbers are a particularly intuitive sort that create different sizes of infinities. All the ordinals, really. They're isomorphic (in NBG) to a maximal-class hyperreal field of nonstandard analysis.