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Georg Cantor and His Heritage
- az00123 3y agoThe diagonal argument was one of the most mind blowing moments of my maths degree
- A_D_E_P_T 3y agoNow and again, there arise certain trends in science and technology which prove deleterious. Take, for instance, the carbon nanotube. It is, as of 2024, 33 years old, and millions of man-hours have gone into practical nanotube development projects. To say that the reward has not been commensurate with the effort would be far too generous -- just about nothing has come of those millions of hours. In hindsight, this should perhaps have been more obvious; the theoretical benefits of nanotubes hinge on the production of pristine submicron fiber-like (giant-) molecules, and those have always been somewhere over the horizon. I feel that Cantor's theories are much the same way. They have severe logical shortcomings, which were highlighted over 100 years ago by the superior logician Skolem; namely that you can construct an uncountable set out of any countable set, and that every so-called uncountable set has a perfectly isomorphic countable model. Further, the diagonalization argument only works in the limit, with very generous use of ". . .", and the finitists have put together a number of very compelling arguments against it. People claim that Cantor's set theory might be a good foundation for mathematics, but it is at best a foundation made of sand. As with the nanotube, I feel that many researchers have spent countless hours -- millions, perhaps -- following an intellectual/scientific trend, and nothing good has come of it.
- ginnungagap 3y agoLöwenheim-Skolem gives you a countable elementarily equivalent submodel (assuming you're working in a theory in a countable language, otherwise it gives you an elementary substructure of the same cardinality of the language at best), but plenty of interesting properties of familiar mathematical objects cannot be captured by a first-order theory and are not preserved by elementary equivalence, completeness of the reals being the standard example
- A_D_E_P_T 3y agoYet the very notion of countability in ZFC, which is itself a first-order theory, is rendered completely relative by Löwenheim-Skolem. ZFC itself has a countable model.
- ginnungagap 3y agoOf course, but what is your point?
- cubefox 3y agoIf "plenty of interesting properties of familiar mathematical objects cannot be captured by a first-order theory" then that also undermines ZFC, which is a first-order theory.
- nyrikki 3y agoZFC was specifically designed to be immune to the classic paradoxes of naive set theory: Russell's paradox, the Burali-Forti paradox, and Cantor's paradox. You are arguing that the ground moves to perfectly fit the shape of a puddle. Zermelo was one of the first to reference "Cantor's theorem" in his papers.
- nairboon 3y agoNow where did these classical paradoxes originate in? They stem from Cantor's Mengenlehre
- cubefox 3y agoThese paradoxes do not occur in higher-order logic. You don't need ZFC or any first-order set theory for that. (Also, your comment doesn't address the sentence I quoted.)
- nyrikki 3y ago
- openasocket 3y agoCan you elaborate? It all seems really straightforward to me. There is no bijection between a set and its power set, via diagonalization. Thus, there is no bijection from the natural numbers to the power set of natural numbers. By definition, that means the power set of natural numbers is uncountable.
- smokel 3y agoNot the parent, but the argument uses quite a few assumptions (axioms) that may not be intuitive to everyone, but which are quite relevant when studying mathematics at the foundational level. For example, why would one be able to create the diagonal set (those indices of the power set elements that do not contain that index as an element) and the enumeration of the power set (i.e. the entire list of possible sets of numbers) at the same time? The theorem proves that an enumeration of the power set cannot be made. Perhaps some sets cannot be constructed at will just by writing down its properties either? In computer land, one would quickly run into self-referential problems when constructing sets like these. For mathematics of this kind, most people agree that this is all fine, and one can derive interesting things from it. But one can also reject the approach and still do some elementary fun stuff. Then again, I might be completely misunderstanding all of this, and I love to be corrected. Edit: wording
- openasocket 3y agoI’m not sure there’s many axioms used. Given any set A, and a function from A to the power set, P(A), construct the set X = {a in A | a is not in f(a) }. Here all we’re using is the power set axiom to define the power set and the subset axiom schema to construct X. We claim there is no a such that f(a) = X. If there was such an a, is a in X? By construction, a is in X if and only if a is not in X, just by first order logic, which is a contradiction. Thus, X is not in the image of f, so f is not a bijection. Thus, there is no bijection from A to P(A). And that’s it. We don’t even need the axiom of choice
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- Vecr 3y agoInfinities simplify various things in math. Weird multi-sized infinities though don't appear very useful.
- eigenket 3y agoWeird multi-sized infinities pop up in physics in a few places. The number of physical positions you can be in in space is uncountably infinite, the number of protons in the universe appears to be countably infinite. There are interesting physical differences between quantum systems whose spectra are discrete (countably infinite eigenvalues) and continuous (uncountably infinite spectrum) and even combinations of both.
- Vecr 3y agoSure, those infinities. I meant the huge stack of new ones that Cantor started on in his book. What's Aleph 1 actually good for?
- eigenket 3y agoAleph 1 turns up when you try to (very) formally deal with probability theory, specifically when dealing with probability measures over the reals. This is sort of useful for physics for example when trying to be very careful about operators like position and momentum in quantum mechanics but it isn't really central. Its sort of nice to know you can do this stuff "properly" but physicists don't care much.
- Vecr 3y agoIt's arguable that there's no such thing as a probability measure over the reals, because Solomonoff induction only works over computable programs, and the reals (in the sense needed) are not computable.
- eigenket 3y agoI think such an argument would need quite a lot more work, the lack of Solomonoff induction doesn't mean we don't have probability theory.
- superb-owl 3y agoI wrote a bit about Cantor's quest to get people to take infinity seriously here: https://superbowl.substack.com/p/church-of-reality-cantor-on-infinity https://superbowl.substack.com/p/church-of-reality-cantor-on...
- woopsn 3y agoGreat series. It really humanizes Cantor for me to see him trying to walk back a claim in those letters. I did my senior project on the continuum hypothesis, up through Paul Cohen's proof of it's independence (he wrote a short book about it that is very clear and accessible). Thanks for sharing, was very interesting to learn a little more about Cantor's motivations and the contemporary reaction to his ideas outside mathematics.
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- prvc 3y agoDoes anyone actually understand this?
- hackandthink 3y ago"several thousand in each generation" I don't understand most of Yuri Manin's mathematics, but I still find some of it interesting. "Manin: I think that people engaged in research in mathematics today are doing so the same way it was done 200 years ago. This is partly because we don’t choose mathematics as our profession, but rather it chooses us. And it chooses a certain type of person, of which there are no more than several thousand in each generation, worldwide. And they all carry the stamp of those sorts of people mathematics has chosen." https://www.ams.org/notices/200910/rtx091001268p.pdf https://www.ams.org/notices/200910/rtx091001268p.pdf
- prvc 3y agoNone of the 84 extant comments in this thread address the (putative) substance of the article. Maybe next generation, then?
- xtiansimon 3y agoI learned about Cantor from this readable book: Aczel, Amir D. "The Mystery of the Aleph Mathematics, the Kabbalah, and the Search for Infinity" Simon & Schuster (2001)
- jabowery 3y agoInterval Arguments: Two Refutations of Cantor’s 1874 and 1878 Arguments: https://www.academia.edu/93528167/Interval_Arguments_Two_Refutations_of_Cantors_1874_and_1878_1_Arguments https://www.academia.edu/93528167/Interval_Arguments_Two_Ref...
- scapp 3y agoIt's always fun to make fun of cranks. Thanks for linking that. The author really needs to find the right statement of what they call the Nested Interval Theorem. I cracked up at the complete misuse of it in the "Interval Argument for the Rationals" section
- johnthescott 3y agothe infinite we can do right away. the finite takes longer. -SU
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- ludston 3y agoNah, infinities don't exist. There are just recursive procedures that emit ever-increasing numbers of digits with no bounds checking. The diagonal proof is just arguing that two nested while(true) loops will run for longer than one. (And then we define this as "bigger" just to confuse undergraduates)
- snthpy 3y agoI like this. Nicely put!
- xanderlewis 3y agoWhy is “longer” any less confusing than “bigger”? Also, what has nested loops got to do with it? You can use one loop to generate the natural numbers, and a pair of nested loops to generate pairs of natural numbers (if you like, the rationals). But the diagonal proof doesn’t show that these will have ‘different’ run times — they’re in fact in bijective correspondence.
- nairboon 3y agoYou need another loop for each real number. A "proper" real number is in itself not finite. If you want to actually physically create a real number, for almost all of them, the process will never terminate.
- xanderlewis 3y agoYou need to define your terms more precisely to be able to make these kind of assertions. What do you mean by physically create? What do you mean by process? The answers depend on these very strongly.
- nairboon 3y agoCan you write down the complete decimal expansion of let's say sqrt(5) ?
- benrutter 3y agoI learned about Cantor in university, and it blew my mind. I've always wondered if anyone's used his work in an applied field yet (like how non-euclidean geometry was just crazy maths for a while and now underpins most physics). Was hoping this paper would tell me, but from what I've read its more if a (very nicely written) summary of his core work. Anyone know if there's any applications yet?
- andrepd 3y agoOf course. Theory of computation and formal logic are two examples of math built on top of Cantor. And theory of computation has one or two real-world applications or so I've heard :)
- nairboon 3y agoThere is no actual infinity. The Cantorians forgot this, and so have fallen into contradiction. (Henri Poincaré, 1906)