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.
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
Yet 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.
If "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.
ZFC 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.
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.
Not 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.
I’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
You can't prove such a thing in ZFC [1]. ZFC's "power set axiom" is a misnomer. It doesn't imply the existence of infinite power sets. It just says if any set S exists and is a subset of an infinite set A, then S is element of a set P (the supposed "power set"). But the axiom doesn't imply the existence of "all" subsets of A. There is no way for first-order logic to state "every possible combination of elements of A forms a set". And Cantor's theorem stating that there is no mapping f from A onto P merely means that the mapping f itself can't exist inside a model of ZFC. [2]
[1] Or any other theory of first-order logic.
[2] This is explained in more detail in Stewart Shapiro's book "Foundation without Foundationalism" p. 114f.
I’m uncertain what your point is. Like I said, in ZFC, there is no bijection from A to its power set. I don’t think anything you’ve stated contradicts that. Are there alternative axiom systems in which you can construct such a bijection?
Weird 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.
> Actually, doesn't the notion of Planck length / Planck time
Nope, we have absolutely no evidence of that. Space-time may be either discrete or continuous. Based on our current understanding we have no evidence either way.
Aleph 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.
It'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.
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.