Georg Cantor and His Heritage

https://arxiv.org/abs/math/0209244

by hackandthink • 2 years ago

92 131 2 years ago

az00123• 2 years ago

The diagonal argument was one of the most mind blowing moments of my maths degree

A_D_E_P_T2• 2 years ago

Now 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.

ginnungagap4• 2 years ago

Löwenheim-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

openasocket7• 2 years ago

Can 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.

Vecr4• 2 years ago

Infinities simplify various things in math. Weird multi-sized infinities though don't appear very useful.

superb-owl• 2 years ago

I 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...

woopsn• 2 years ago

Great 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.

prvc2• 2 years ago

Does anyone actually understand this?

hackandthink• 2 years 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

prvc2• 2 years ago

None of the 84 extant comments in this thread address the (putative) substance of the article. Maybe next generation, then?

xtiansimon• 2 years ago

I 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• 2 years ago

Interval Arguments: Two Refutations of Cantor’s 1874 and 1878 Arguments:

https://www.academia.edu/93528167/Interval_Arguments_Two_Ref...

scapp• 2 years ago

It'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• 2 years ago

the infinite we can do right away. the finite takes longer. -SU

ludston20• 2 years ago

Nah, 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• 2 years ago

I like this. Nicely put!

xanderlewis4• 2 years ago

Why 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.

nairboon9• 2 years ago

You 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.

eigenket15• 2 years ago

I don't understand what you mean by this. What would it mean for an infinity or any number to "exist", does the number 2 "exist" somewhere in a way that infinities do not?

In any case our best physical models right now are full of infinities. Space looks like it is infinitely big to an absurd degree of precision. The spectrum of the hydrogen atom contains two different infinities! A countable infinity of bound states and a continuum of free electron states at higher energies.

rolisz5• 2 years ago

Well, I have two apples on my desk. I can't have an infinity of anything. In that sense 2 exists more tangibly than infinity.

Our best physical models are just that - models. All models are wrong, some are useful. Infinity in physics is usually a shorthand for "it's so big I don't have to care about the edges"

dlkf2• 2 years ago

Your critique of the use of ”bigger” in your strawman applies equally to your own use of ”longer” - neither loop will ever terminate. So in what sense is either one longer?

The diagonal proof is arguing that there is no bijection from N to {0, 1}∞ , despite the fact that both are infinite. The sense in which the latter is ”bigger” is that there are always elements left over that are covered by N.

ludston20• 2 years ago

So in what sense is either one longer?

Neither. That's my point. Literally any definition of something infinite can always be reduced to a procedure that recursively transforms or observes some prior state. To say that one of these functions can produce more distinct states than another is pointless, because the procedure that produces the most states will always be the one that you ran the most times.

There is nothing observably infinite, since it would take infinite time to observe that any given thing was infinite. The only possible proof of infinity would be a machine that runs infinitely quickly. e.g. https://qntm.org/responsibility

empath-nirvana5• 2 years ago

Mathematics is just an exercise in symbol manipulation according to a particular rule set. Mathematics is a _game_. Park Place exists in the context of a game of Monopoly and also exists in Atlantic City. The tokens in a game of Reversi don't correspond to anything in the real world. You might be playing it on a physical board of a particular size, but you could just as easily play it on an imaginary board, or on a piece of paper, or a computer, or in some imaginary board of infinite size. Whether or not either exists in "reality" doesn't matter in the context of the game, though.

If the rules of mathematics allow for infinities, then they exist in the context of mathematics. Whether they exist in reality doesn't matter. You can still say things that are true about these imaginary objects in the context of the game that everyone is playing with these symbols, just like you can say things that are true about the game of Reversi, even one played entirely in the minds of two players on an imaginary, infinite board.

ludston20• 2 years ago

If you use "exist" to mean "is a symbol defined within some system" rather than the typical "has isomorphism with reality" then you can say literally anything exists. Unicorns exist because English has a word for them.

So be it, but it is a useless definition of exists then isn't it?

benrutter• 2 years ago

I 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• 2 years ago

Of 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 :)

nairboon9• 2 years ago

There is no actual infinity. The Cantorians forgot this, and so have fallen into contradiction. (Henri Poincaré, 1906)

Georg Cantor and His Heritage

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