Path: ...!local-1.nntp.ord.giganews.com!Xl.tags.giganews.com!local-3.nntp.ord.giganews.com!news.giganews.com.POSTED!not-for-mail NNTP-Posting-Date: Thu, 21 Nov 2024 21:19:06 +0000 Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-standard) Newsgroups: sci.math References: <2c04a68c-afea-4843-afdf-ab33609cf710@att.net> <9UidnSm61oVHD6H6nZ2dnZfqnPednZ2d@giganews.com> <3e0339ab-34be-431e-bd07-b457130392cd@att.net> <18ScnXwr0oDIxqD6nZ2dnZfqn_SdnZ2d@giganews.com> <3h-dnYIYqccbw6D6nZ2dnZfqnPednZ2d@giganews.com> <136c5b40-5211-4f9b-9b6b-4da738054a0c@att.net> <6tWdnR1r48t6o6P6nZ2dnZfqnPqdnZ2d@giganews.com> From: Ross Finlayson Date: Thu, 21 Nov 2024 13:19:12 -0800 User-Agent: Mozilla/5.0 (X11; Linux x86_64; rv:38.0) Gecko/20100101 Thunderbird/38.6.0 MIME-Version: 1.0 In-Reply-To: Content-Type: text/plain; charset=utf-8; format=flowed Content-Transfer-Encoding: 8bit Message-ID: Lines: 130 X-Usenet-Provider: http://www.giganews.com X-Trace: sv3-sEq/3gGkZFLsWqWupBluCveva1qM8XQQ4GCjuAq1FXCkINInUTGLpZClRMXQ9IdP6WL52KlzpQK8K3c!n2pTTdd7MvBFiJKeh0McpQ+urbSEDsj2PAoz3tR88OpyNmIl2pRFmDKxpvmLxSnRD7HeYyz8dP2C X-Complaints-To: abuse@giganews.com X-DMCA-Notifications: http://www.giganews.com/info/dmca.html X-Abuse-and-DMCA-Info: Please be sure to forward a copy of ALL headers X-Abuse-and-DMCA-Info: Otherwise we will be unable to process your complaint properly X-Postfilter: 1.3.40 Bytes: 7081 On 11/21/2024 01:10 PM, Ross Finlayson wrote: > On 11/21/2024 12:28 PM, Jim Burns wrote: >> On 11/21/2024 2:46 PM, Ross Finlayson wrote: >>> On 11/21/2024 09:57 AM, Jim Burns wrote: >> >>>> [...] >>> >>> (the existence of a choice function, >>> i.e. a bijection between any set and some ordinal) >> >> A well.ordering of a set is >> a bijection between that set and an ordinal. >> >> A choice function is a function 'choice', >> typically not a bijective function, >> from a collection of non.empty sets S >> to their elements, such that >> for each set S, choice(S) ∈ S >> >> ∀Collection: >> ∃choice: Collection\{∅} -> ⋃Collection: >> ∀S ∈ Collection\{∅}: choice(S) ∈ S >> >>> Yeah, Well-Ordering and Choice (the existence of >>> a choice function, i.e. a bijection between any >>> set and some ordinal) are same. >> >> Well.Ordering and Choice are inter.provable. >> >>> Countable-choice is weak and trivial. >> >> Because we prefer our assumptions weak and trivial, >> that's a good thing. >> >> Countable.choice is sufficient to prove that >> Well.Ordering and Choice are inter.provable. >> Proving they are inter.provable with >> weak and trivial assumptions is a good thing. >> >> > > https://www.youtube.com/watch?v=wmuxeHqF-Vw > > Lecture64||week8||Physico-Mathematical Foundations of the Dynamics of > Nonlinear Processes||by Harsh > > Guy mentions frequency-doubling, a mathematical feature > after continuum mechanics, which cannot be a thing for > those who take the easy way out that shoves itself off. > > > Here it's related to doubling- and halving-spaces, and > measures, real things or rather about real analytical character, > and about models of continuous domains and Vitali and > Hausdorff, great geometers. > > So anyways one thing about that is line-reals and their > doubling-space with regards to taking their integral > and that it doubles itself up, the iota-values that > ran(EF), integrating EF, integrates and equals one. > > Then these are well-ordered, these real-valued members > of a continuous domain. > > Yet, you'll never find one anywhere else with regards > to the complete-ordered-field, because there can't be > an uncountable subset that relates to an uncountable ordinal > where, any subset of a well-ordering the tuples (set, ordinal) > is also a well-ordering a set the tuples (set, ordinal), > there can't be that with uncountably many in their normal > ordering, because, quite directly each pair of those as > read off from the choice function, which is merely the > first element existing according to the mapping of a set > to an ordinal, each pair would have a distinct rational > between them. > > So, "well-order the reals" arrives at "or, you know, > aver that it exists yet don't actually give one, ...", > because it would be contradictory either way. > > Anyways that's come up many times, that "well-order the > reals" never quite works out for retro-thesis hacks > of the quite fully the ordinals and cardinals as sets sort, > then though for example it's built up for line-reals > how a resulting, "set", of them, may be so. > > > So anyways, you don't need any infinity at all for > such usual matters of induction you describe as so > simple, you're welcome to keep it that way, yet then > that's a sort of "finite combinatorics" not mathematics, > per se. > > In Cantor space there are duelling arguments where > according to Borel almost all and according to Combinatorics > almost none, of the members, are a given way, and > then also a third alternative where it's exactly one-half. > > These are a bit independent, say, either ZF minus Infinity > or ZF with Infinity and may even have that there's always > according to Skolem an extension, and according to Mirimanoff > an extra-ordinary, that Russell's retro-thesis an "ordinary", > well-founded infinity is rejected as not-a-thing, instead > that there are either unbounded fragments or extra-ordinary > extensions, in as regards to three definitions of continuous > domains and three definitions (or, perspectives) of Cantor space. > > > Claiming to "make things simple" like "initial ordinal assignment, > a cardinal" or "Dedekind cut, a real", is actually sort of having > conflated separate notions that do not fulfill each other. > > Yeah, it's trivial that the existence of a choice function > and that a subset of the ordered-pairs the tuples a well-ordering > is also a well-ordering, establish each other. > > So, well-order the reals. > > https://www.youtube.com/watch?v=IldqDZklJCg "Lowenheim Skolem Thereom is Explained By Referential Externalism" Here a prototypical continuous domain arrives at a theory with regards to the heno-theory, what's primary in the theory, when continua are primary in the theory.