Path: ...!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!.POSTED!not-for-mail From: "Chris M. Thomasson" Newsgroups: sci.math Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary) Date: Wed, 4 Dec 2024 11:51:56 -0800 Organization: A noiseless patient Spider Lines: 91 Message-ID: References: <87frn50zjp.fsf@bsb.me.uk> <87a5db1zui.fsf@bsb.me.uk> MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 8bit Injection-Date: Wed, 04 Dec 2024 20:51:56 +0100 (CET) Injection-Info: dont-email.me; posting-host="18f532fff7cb0dbc37a1aea16bf9bfbd"; logging-data="1127389"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX184zl3AnORDS8qeFdUDAZn6X+uQgoAf9DM=" User-Agent: Mozilla Thunderbird Cancel-Lock: sha1:aB+Tbp0HeiOIGOep72kHTVIjxvE= In-Reply-To: <87a5db1zui.fsf@bsb.me.uk> Content-Language: en-US Bytes: 5281 On 12/4/2024 2:56 AM, Ben Bacarisse wrote: > "Chris M. Thomasson" writes: > >> On 12/3/2024 3:35 AM, Ben Bacarisse wrote: >>> "Chris M. Thomasson" writes: >>> >>>> On 12/2/2024 4:00 PM, Chris M. Thomasson wrote: >>>>> On 12/2/2024 3:59 PM, Moebius wrote: >>>>>> Am 03.12.2024 um 00:58 schrieb Chris M. Thomasson: >>>>>>> On 12/2/2024 3:56 PM, Moebius wrote: >>>>>>>> Am 03.12.2024 um 00:51 schrieb Chris M. Thomasson: >>>>>>>>> On 12/1/2024 9:50 PM, Moebius wrote: >>>>>>>>>> Am 02.12.2024 um 00:11 schrieb Chris M. Thomasson: >>>>>>>>>>> On 11/30/2024 3:12 AM, WM wrote: >>>>>>>>>> >>>>>>>>>>>> Finite initial segment[s]: F(n) = {1, 2, 3, ..., n}    (n e IN). >>>>>>>>> [...] >>>>>>>>> >>>>>>>>> When WM writes: >>>>>>>>> >>>>>>>>> {1, 2, 3, ..., n} >>>>>>>>> >>>>>>>>> I think he might mean that n is somehow a largest natural number? >>>>>>>> >>>>>>>> Nope, he just means some n e IN. >>>>>>> >>>>>>> So if n = 5, the FISON is: >>>>>>> >>>>>>> { 1, 2, 3, 4, 5 } >>>>>>> >>>>>>> n = 3 >>>>>>> >>>>>>> { 1, 2, 3 } >>>>>>> >>>>>>> Right? >>>>>> >>>>>> Right. >>>>> Thank you Moebius. :^) >>>> >>>> So, i n = all_of_the_naturals, then >>> You are in danger of falling into one of WM's traps here. Above, you >>> had n = 3 and n = 5. 3 and 5 are naturals. Switching to n = >>> all_of_the_naturals is something else. It's not wrong because there are >>> models of the naturals in which they are all sets, but it's open to >>> confusing interpretations and being unclear about definition is the key >>> to WM's endless posts. >>> >>>> { 1, 2, 3, ... } >>>> >>>> Aka, there is no largest natural number and they are not limited. Aka, no >>>> limit? >>> The sequence of FISONs has a limit. Indeed that's one way to define N >>> as the least upper bound of the sequence >>> {1}, {1, 2}, {1, 2, 3}, ... >>> although the all terms involved need to be carefully defined. >>> >>>> Right? >>> The numerical sequence 1, 2, 3, ... has no conventional numerical limit, >>> but, again, if the symbols 1, 2, 3 etc stand for sets (as in, say, Von >>> Neumann's model for the naturals) then the set sequence >>> 1, 2, 3, ... >>> does have a set-theoretical limit: N. >> >> However, there is no largest natural number, > > Yes, there is no largest natural. Let's not loose sight of that. > >> when I think of that I see no >> limit to the naturals. I must be missing something here? ;^o > > It's just that there are lots of kinds of limit, and a limit is not > always in the set in question. Very often, limits take us outside of > the set in question. R (the reals) can be defined as the "smallest" set > closed under the taking of certain limits -- the limits of Cauchy > sequences, the elements of which are simply rationals. So, the limit of the natural numbers is _outside_ of the set of all natural numbers? > Even if we don't consider FISONs, we can define a limit (technically a > least upper bound) for the sequence 1, 2, 3, ... It won't be a natural > number. We will have to expand our ideas of "number" and "size" to get > the smallest "thing", larger than all naturals. This is how the study > of infinite ordinals starts. > Okay... I need to ponder on that. Can I just, sort of, make up a "symbol" and say this is larger than any natural number, however it is definitely _not_ a natural number in and of itself? Damn. Still missing something here. Thanks for your patience... ;^o