Path: ...!weretis.net!feeder9.news.weretis.net!news.nk.ca!rocksolid2!i2pn2.org!.POSTED!not-for-mail From: joes Newsgroups: sci.math Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary) Date: Sat, 11 Jan 2025 13:58:28 -0000 (UTC) Organization: i2pn2 (i2pn.org) Message-ID: References: <66868399-5c4b-4816-9a0c-369aaa824553@att.net> <417ff6da-86ee-4b3a-b07a-9c6a8eb31368@att.net> <07258ab9-eee1-4aae-902a-ba39247d5942@att.net> <9f6faf842a0202b345f3912fe352044dfabfc56d@i2pn2.org> <5e1c366f8e2cfe7f5439f3d2bb65d4695d17c6b9@i2pn2.org> <2c1e0afa5652b5f139abcda8822ac14ccc939f2b@i2pn2.org> <14c40a8944eb8e3ffc72ceed400106b988340891@i2pn2.org> <2734a9c30b5cddf6be417bb317be9b28e6236e66@i2pn2.org> MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Injection-Date: Sat, 11 Jan 2025 13:58:28 -0000 (UTC) Injection-Info: i2pn2.org; logging-data="3102326"; mail-complaints-to="usenet@i2pn2.org"; posting-account="nS1KMHaUuWOnF/ukOJzx6Ssd8y16q9UPs1GZ+I3D0CM"; User-Agent: Pan/0.145 (Duplicitous mercenary valetism; d7e168a git.gnome.org/pan2) X-Spam-Checker-Version: SpamAssassin 4.0.0 Bytes: 6069 Lines: 87 Am Sat, 11 Jan 2025 12:44:59 +0100 schrieb WM: > On 11.01.2025 10:41, joes wrote: >> Am Sat, 11 Jan 2025 09:50:02 +0100 schrieb WM: >>> On 10.01.2025 22:51, joes wrote: >>>> Am Fri, 10 Jan 2025 22:38:51 +0100 schrieb WM: >>>>> On 10.01.2025 21:06, joes wrote: >>>>>> Am Fri, 10 Jan 2025 20:17:37 +0100 schrieb WM: >>>>>>> On 10.01.2025 19:28, joes wrote: >>>>>>>> Am Fri, 10 Jan 2025 18:21:43 +0100 schrieb WM: >>>>>>> >>>>>>>>> I have no expectations about cardinality. I know that for every >>>>>>>>> finite initial segment the even numbers are about half of the >>>>>>>>> natural numbers. >>>>>>>>> This does not change anywhere. It is true up to every natural >>>>>>>>> number. >>>>>>>> You wrongly expect this to hold in the infinite. >>>>>>> No, I expect it is true for all natural numbers, none of which is >>>>>>> infinite. >>>>>> But it is true for every natural >>>>> Of course. Otherwise you would have to find a counterexample. >>>> Good. It is not true for the infinite sets. >>> The natural numbers are an infinite set. For all of them it is true, >> But not for omega, which is not a natural. > Therefore it is irrelevant. No bijection from ℕ contains it. Then don't claim that some sentence held for omega. >>>>>> (if you formalise it correctly)! >>>>> Irrelevant. >>>> Mathematics is all about formalising. >>> No, that is only a habit of the last century. >> Informal reasoning gets you nowhere, see the centuries before that. > There mathematics has flourished. Now mainly nonsense is produced. Crawl back into your cave and marvel about infinity. >>>>> ∀n ∈ ℕ: |{1, 2, 3, ..., 2n}|/|{2, 4, 6, ..., 2n}| = 2. >>>> Those are not N and E. >>> Find an element of N or E that is not covered by the equation. >> Not what I said. Every natural is finite, and so are the starting >> segments of N and E. > And which are not? See: >> The whole sets (which can be seen as the limits) are not finite. > My claim holds for all numbers only. That is mathematics. [?] Your claim does not hold for the sets. >>>>>> That doesn't make it true for N and G. >>>>> I am not interested in these letters but only in all natural >>>>> numbers. >>>>> All natural numbers are twice as many as all even natural numbers. >>>>> If your N and G denote all natural numbers and all even numbers, >>>>> then 2 is true also for them. >>>> No. For n->oo, >>> Every n is finite. >> The *set* of all of them isn't. > Irrelevant. My claim holds for all natnumbers only. That's what I'm saying. >>>> E is both the set {2, 4, ..., 2n} and {2, 4, ..., 42n}; >>>> indeed, {2, 4, ..., 2kn} for every k e N. >>> And all of them can be denoted by n. >> All what? > All natnumbers which Cantor uses in bijections: "such that every element > of the set stands at a definite position of this sequence". If this has > been accomplished, and then more numbers are created, the bijection > fails. This must not happen. No numbers are "created" (I guess you mean the image is a subset of the domain?). >>> "thus we get the epitome (ω) of all real algebraic numbers [...] and >>> with respect to this order we can talk about the th algebraic number >>> where not a single one of this epitome () has been forgotten." [E. >>> Zermelo: "Georg Cantor – Gesammelte Abhandlungen mathematischen und >>> philosophischen Inhalts", Springer, Berlin (1932) p. 116] >>> Afterwards no extension by 42 is allowed. >> There is no "after" an infinity. > Cantor maps all natural numbers to a set. Afterwards these natural > numbers can be multiplied by 2. Not all remain those which Cantor has > applied. Cantor bijectively maps the naturals to the algebraic numbers. You can also biject the naturals and the evens. Call the first one f and the second g; then you can compose g(f(n)): E->(N->)A, a bijection from the even to the algebraic numbers. The function {(k, f(k) for all k e N} is also bijective, but A->N: {(f(k), k) for all k e E} of course isn't, if I parsed your last sentence correctly. -- Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math: It is not guaranteed that n+1 exists for every n.