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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
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From: Ross Finlayson <ross.a.finlayson@gmail.com>
Date: Thu, 21 Nov 2024 13:19:12 -0800
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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.