[HN Gopher] Georg Cantor and His Heritage
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Georg Cantor and His Heritage
Author : hackandthink
Score : 53 points
Date : 2024-03-16 17:09 UTC (5 hours ago)
(HTM) web link (arxiv.org)
(TXT) w3m dump (arxiv.org)
| az00123 wrote:
| The diagonal argument was one of the most mind blowing moments of
| my maths degree
| A_D_E_P_T wrote:
| 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.
| ginnungagap wrote:
| Lowenheim-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 wrote:
| Yet the very notion of countability in ZFC, which is itself
| a first-order theory, is rendered completely relative by
| Lowenheim-Skolem. ZFC itself has a countable model.
| ginnungagap wrote:
| Of course, but what is your point?
| openasocket wrote:
| 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.
| smokel wrote:
| 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.
|
| Edit: wording
| openasocket wrote:
| 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
| Vecr wrote:
| Infinities simplify various things in math. Weird multi-sized
| infinities though don't appear very useful.
| superb-owl wrote:
| 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...
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