Path: ...!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!eternal-september.org!.POSTED!not-for-mail From: Peter Fairbrother Newsgroups: sci.math Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary) Date: Thu, 16 Jan 2025 01:44:44 +0000 Organization: A noiseless patient Spider Lines: 107 Message-ID: References: <417ff6da-86ee-4b3a-b07a-9c6a8eb31368@att.net> <07258ab9-eee1-4aae-902a-ba39247d5942@att.net> <1ebbc233d6bab7878b69cae3eda48c7bbfd07f88@i2pn2.org> <4c89380adaad983f24d5d6a75842aaabbd1adced@i2pn2.org> <494bfd3b-3c70-4d8d-9c70-ce917c15fc22@att.net> <72142d82-0d71-460a-a1be-cadadf78c048@att.net> <812e64b1-c85c-48ac-a58c-e8955bc02f8c@att.net> <22b74adc-bf38-4aa4-a44f-622f0a2a5c41@att.net> MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 8bit Injection-Date: Thu, 16 Jan 2025 02:44:45 +0100 (CET) Injection-Info: dont-email.me; posting-host="5a00a0e104fbe60b6569f92bbc11fae2"; logging-data="3366344"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX1+651XWU+RNt/Gqmuw7Ln5NNTuEaTuVDqk=" User-Agent: Mozilla Thunderbird Cancel-Lock: sha1:JaWV2XbFhLlLcY8yqOsnCv/7qxc= Content-Language: en-GB In-Reply-To: Bytes: 5111 Careful! Next you will have sets with negative cardinality... On 16/01/2025 01:23, Jim Burns wrote: > On 1/15/2025 1:17 PM, WM wrote: >> On 15.01.2025 16:16, Jim Burns wrote: >>> On 1/14/2025 4:07 AM, WM wrote: >>>> On 13.01.2025 20:31, Jim Burns wrote: >>>>> On 1/13/2025 12:17 PM, WM wrote: > >>>>>> therefore creates even numbers. >>>>>> They do not fit below ω. >>>>> >>>>> No. >>>>> They fit below ω >>>> >>>> In completed infinity >>>> all available places are occupied. > >>> In each of our sets, >>> each of its elements is in the set, >>> each available place is occupied. >> >> Therefore new numbers are not accepted. > > And all even numbers fit below ω > > None are created. > >>> A potentiallyᵂᴹ infiniteˢᵉᵗ set, >>> the same as any other set, >>> has all available places occupied >>> and is completeᵂᴹ. >> >> Potential infinity is growing. > > In each of our sets, > each element has an available space, and > only its elements have available spaces. > > A place in a set is occupied by virtue of > its element being in the set. > > In each of our sets, > each of its elements is in the set, > each available place is occupied. > > A potentiallyᵂᴹ infiniteˢᵉᵗ set > has all available places occupied, > the same as any other set, > which is to say, > it is (has been, will be) completeᵂᴹ. > >> "In analysis we have to deal only >> with the infinitely small and >> the infinitely large >> as a limit-notion, >> as something becoming, emerging, produced, >> i.e., as we put it, with the potential infinite. >> But this is not the proper infinite. >> That we have for instance >> when we consider >> the entirety of the numbers 1, 2, 3, 4, ... itself >> as a completed unit, or >> the points of a line as >> an entirety of things which is completely available. >> That sort of infinity is named actual infinite." >> [D. Hilbert: "Über das Unendliche", Mathematische Annalen 95 (1925) p. >> 167] > > A finite set has > emptier.by.one sets which are smaller. > > For each finite set, > a finite ordinal larger than that set > exists. > > For the set ℕ of all finite ordinals, > a finite ordinal larger than ℕ > doesn't exist. > > Therefore, > the set ℕ of all finite ordinals > isn't itself finite,  and, > unlike a finite set, ℕ doesn't have > emptier.by.one sets which are smaller. > > ---- > Finite people are able to reason about infinity > by describing an indefinite one of infinitely.many > and then supplementing the descriptive claims > with visibly not.first.false claims. > > As finite people, > we have not and _cannot_ witness > the infinitely.many described. > > What we can witness, instead, are > the finitely.many finite.length claims themselves, > and witness the correctness of the description of > that which we are currently discussing, > and witness the not.first.false.ness of > the other claims. > Upon witnessing all that, > we know the claims are true. > >