Path: ...!eternal-september.org!feeder2.eternal-september.org!news.eternal-september.org!.POSTED!not-for-mail From: "Chris M. Thomasson" Newsgroups: sci.math Subject: Re: How many ... (Sparse, Signal, Square Cantor "SPACE") Date: Sun, 3 Nov 2024 21:11:20 -0800 Organization: A noiseless patient Spider Lines: 120 Message-ID: References: <30dffbdf129483f7b61e3284d1e7bf2ad2e5ea16@i2pn2.org> <9ca97f4a24ae1e3041583265125cf860d2fada11@i2pn2.org> <7KednW4Tm59PP7v6nZ2dnZfqn_ednZ2d@giganews.com> MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 8bit Injection-Date: Mon, 04 Nov 2024 06:11:22 +0100 (CET) Injection-Info: dont-email.me; posting-host="a3560c29bf30eac6bd9d64bfc512bec5"; logging-data="855502"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX19S5p6s6sc5Ul8vNbxAFmQkFo4/gfnluiw=" User-Agent: Mozilla Thunderbird Cancel-Lock: sha1:NULcNWGyjou71CfLOnWp54yRmnI= Content-Language: en-US In-Reply-To: Bytes: 6432 On 11/3/2024 4:41 PM, Ross Finlayson wrote: > On 11/03/2024 02:53 PM, Chris M. Thomasson wrote: >> On 11/3/2024 9:09 AM, Ross Finlayson wrote: >>> On 11/02/2024 08:41 PM, Chris M. Thomasson wrote: >>>> On 11/2/2024 3:08 PM, Ross Finlayson wrote: >>>>> On 11/02/2024 01:22 PM, Chris M. Thomasson wrote: >>>>>> On 11/2/2024 11:03 AM, Ross Finlayson wrote: >>>>>>> On 11/02/2024 06:54 AM, FromTheRafters wrote: >>>>>>>> WM was thinking very hard : >>>>>>>>> On 01.11.2024 22:53, FromTheRafters wrote: >>>>>>>>>> WM explained on 11/1/2024 : >>>>>>>>>>> On 01.11.2024 19:39, FromTheRafters wrote: >>>>>>>>>>>> WM formulated the question : >>>>>>>>>>> >>>>>>>>>>>>>> Infinite subsets don't do that for you, even if you wish >>>>>>>>>>>>>> really >>>>>>>>>>>>>> hard. >>>>>>>>>>>>> >>>>>>>>>>>>> They cannot evade if they are invariable. >>>>>>>>>>>> >>>>>>>>>>>> Sets don't change. >>>>>>>>>>> >>>>>>>>>>> Therefore the elements do not depend on us and our knowledge. >>>>>>>>>>> "If I >>>>>>>>>>> find x, then I can find x + 1" is not relevant. "For every x >>>>>>>>>>> (that I >>>>>>>>>>> find) there is x + 1" is no relevant. All elements are there, >>>>>>>>>>> independent of what we know or do. Therefore the first and the >>>>>>>>>>> last >>>>>>>>>>> are also there independent of us. If they weren't, their >>>>>>>>>>> existence >>>>>>>>>>> would depend on some circumstances and could change. >>>>>>>>>> >>>>>>>>>> Circumstances like "there is no last element"? >>>>>>>>> >>>>>>>>> That means, there is always another element. Potential infinity. >>>>>>>> >>>>>>>> Sets don't change. Forget about amplifying 'not finite' with >>>>>>>> such as >>>>>>>> 'actual' and potential' -- infinite simply means not finite and >>>>>>>> 'actual/potential' is a distinction without a difference. A useless >>>>>>>> concept outside of math philosophy. >>>>>>>> >>>>>>>>>> the set of denominators have no largest element to 'start' with. >>>>>>>>> >>>>>>>>> If all unit fractions are existing, then a smallest unit >>>>>>>>> fraction is >>>>>>>>> existing. If NUF(x) has grown to ℵ₀ at x₀, then ℵ₀ unit fractions >>>>>>>>> must >>>>>>>>> be between 0 and x₀. Hence at least ℵ₀ points with ℵ₀ intervals of >>>>>>>>> uncountably many points must be between 0 and x₀. That cannot >>>>>>>>> happen >>>>>>>>> at x₀ = 0. >>>>>>>>> >>>>>>>>> Is that too hard to understand? >>>>>>>> >>>>>>>> Apparently, for you. >>>>>>> >>>>>>> Au contraire, there are multiple law(s) of large numbers, >>>>>> >>>>>> What about this sucker (42^999429994299942) * 2 ? >>>>>> >>>>>> Oh, don't forget about ((42^999429994299942) * 2) + 1   damn it! >>>>>> >>>>>> >>>>>> >>>>>>> and in mathematics like emergence after convergence, >>>>>>> the potential / practical / effective / actual distinction, >>>>>>> of "infinity", is a thing. >>>>>>> >>>>>>> You know who discovered mathematics? Philosophers. >>>>>>> >>>>>>> >>>>>> >>>>> >>>>> Well, I frame it about Cantor space, because, there are >>>>> some facts deducible from the asymptotic density of zero's >>>>> and one's, in the rows and columns Cantor space, even if >>>>> there are infinitely or trans-finitely many. >>>> [...] >>>> >>>> Cantor Pairing is fun. Mapping to and fro. Nothing is lost. WM >>>> disagrees >>>> here, but shit happens. >>>> >>>> >>>>> >>>> >>> >>> Here's a way to think about it. >> [...] >> >> Cantor pairing works, and nothing is lost. That's that. >> > > Numbering and counting ("pairing") are two different things, > much like order theory and set theory are two different things, > Cartesian and non-Cartesian functions are two different things, > ordinals and cardinals are two different things, > and non-Cartesian orderings in numberings are not pairings, > and won't be lost. Well, we can turn two naturals into a single and unique natural, then convert that back to the two original naturals. They were always there to begin with because the naturals have no end... ;^) > > Don't get me wrong, I've lots experience with pairing, ..., > yet it works out that counting is not numbering, > and that counting does not suffice all cases of numbering. > > > Or, "Eudoxus/Cauchy/Dedekind is not sufficient", > though it's great for what it is, > it's not sufficient for a theory of everything, > and it's _not_ complete there. > >