Timeline for Who discovered the difference between the infinities?
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 28, 2017 at 14:41 | comment | added | Franz Kurz | What Bolzano applies is simple mathematics: If for every finite set of ellipses the ratio of foci to centres is $2$, this can never change. "Never" is same as "in the infinite". In analysis we apply the limit: If we have the sequence $2, 2, 2, ...$, then its limit is $2$. Cantor offends against analysis. For every interval $(0, n)$ the ratio of natural numbers and not enumerated fractions of that interval is $0$. To obtain another limit is impossible in analysis. And there is no argument $\omega$ among natural numbers. | |
| Jun 28, 2017 at 11:45 | comment | added | Michael Bächtold | Thanks for clarifying. I'd have to read the original sources to see if I can make sense of Bolzano. (I don't understand how one can deny bijections to compare infinities, and at the same time compare integers with even integers and conclude that that are twice as many.) In any case, I conclude from you remarks, that Bolzanos notion of different infinities is not the same as Cantors. | |
| Jun 28, 2017 at 11:11 | comment | added | Franz Kurz | @Michael Bächtold: There is nothing "uncountable" in Bolzano's world. He does explicitly deny a bijection as a meaningful measure of infinities. (And this does not make him wrong.) There are merely different infinities. There are twices as many integers as even integers. There are infinitely many cubes but a sixfold of cube areas and an eightfold of cube corners (my example). There are infinitely many natural numbers and infinitely many circles. And there are infinitely many more rational or real numbers between them and infinitely many more diameters of the circles. | |
| Jun 28, 2017 at 6:51 | comment | added | Michael Bächtold | @Claus I can't make any sense of that argument by Bolzano. Among other things, there is an uncountable number of different circles, so how can they represent the natural numbers? | |
| Jun 26, 2017 at 16:09 | comment | added | Franz Kurz | Bolzano argues: Every ellipse has two focal points but one centre. Every circle has one circumference (or centre) but infinitely many diameters. The diameters include arbitrary real angles between 0 and the full angle, say 1. There is an infinite number of circles (at least in imagination). So they represent all natural numbers, and the diameters represent all reals between them. | |
| Jun 26, 2017 at 15:40 | comment | added | Michael Bächtold | I don't know hot to interpret the statement that there are infinitely more diameters than centres in order to deduce the other result from that. If you can expand I would definitely upvote. | |
| Jun 26, 2017 at 15:18 | comment | added | Franz Kurz | @Michael Bächtold: The discovery that there are infinitely many more diameters of circles than centres of circles implies the discovery that there are infinitely many more real numbers than natural numbers. Probably Bolzano has mentioned this in his collected works, but since they cover more than one meter in the book shelf I am not eager to search for. | |
| Jun 26, 2017 at 14:43 | comment | added | Michael Bächtold | I did not downvote, but I can understand the downvote: the question asks explicitly about the discovery that naturals and reals have different cardinalities. I don't see how your answer addresses that. | |
| Jun 26, 2017 at 11:12 | comment | added | Franz Kurz | @DukeZhou: The reason is simply the sentence "Cantor merely devised a certain, rather arbitrary, tool". There are many people who adore Cantor so much that they consider this sentence a sacrilege. If someone even dares to say that this is not only an arbitrary but a useless tool because "for every n in N: n belongs to a finite initial segment which is followed by infinitely many natural numbers such that general quantification fails for infinite sets and equicardinality does not prove anything about same number of elements", the statemenet will be heavily downvotes or even deleted. | |
| Jun 25, 2017 at 22:06 | comment | added | DukeZhou | This was a very interesting answer. I wonder why it got downvoted. | |
| Jun 24, 2017 at 22:12 | history | edited | Franz Kurz | CC BY-SA 3.0 |
added 348 characters in body
|
| Jun 24, 2017 at 14:18 | history | answered | Franz Kurz | CC BY-SA 3.0 |