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NNTP-Posting-Date: Fri, 30 Aug 2024 00:28:45 +0000
Subject: Re: Replacement of Cardinality (infinite middle)
Newsgroups: sci.logic,sci.math
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From: Ross Finlayson <ross.a.finlayson@gmail.com>
Date: Thu, 29 Aug 2024 17:28:54 -0700
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On 08/29/2024 05:12 PM, Ross Finlayson wrote:
> On 08/29/2024 04:32 PM, Jim Burns wrote:
>> On 8/29/2024 6:46 PM, Ross Finlayson wrote:
>>> On 08/19/2024 12:27 PM, Ross Finlayson wrote:
>>>> On 08/18/2024 09:56 PM, Jim Burns wrote:
>>
>>>>>>>>> Definition.
>>>>>>>>> ⎛ An order ⟨B,<⟩ of B is finiteᵖᵍˢˢ  iff
>>>>>>>>> ⎜ each non.empty subset S ⊆ B holds
>>>>>>>>> ⎝ both min[<].S and max[<].S
>>
>>>> Why is it you think that Stackel's definition of finite
>>>> and "not Dedekind's definition of countably infinite"
>>>> don't agree?
>>
>> I don't think they disagree, normally.
>>
>> Note: If you mean Dedekind's definition of infinite,
>> it isn't limited to countably.infinite.
>>
>>>> The entire idea here that there's a particular _regularity_
>>>> due dispersion and modularity only courtesy division down
>>>> from a fixed-point, that "Peano's axioms" don't give integers,
>>>> they only give increments, i.e. not necessarily constant increments,
>>>> that there's more than one _regularity_, REQUIRED, is another
>>>> little fact of mathematics missing from your neat little hedgerow.
>>>
>>> ..., REQUIRED, ....
>>
>> Things missing from my neat little hedgerow are
>> missing because I intend for them to be missing.
>> My neat little hedgerow has no weeds.
>> It has not had and will not have weeds.
>> And weeds would not be an improvement.
>>
>> My neat little hedgerow is well.ordered;
>> each non.empty subset holds a minimum.
>>
>> In my neat little hedgerow,
>> each Little Bunny Foo Foo has a successor,
>> scooping up the field mice and bopping them on the head,
>> and is a successor, except the first, named 0.
>>
>> Successors are non.0 non.doppelgänger non.final.
>>
>> You are welcome to talk about something else, Ross.
>> Note, though, that,
>> if you are talking about something else,
>> then you are talking about something else.
>> Non.triangles are not counter.examples to triangles.
>> Non.Bunny.Foo.Foos are not counter.examples to Bunny.Foo.Foos.
>>
>> Have a nice day.
>>
>>
>
> The other day I read or leafed through and enjoyed
> this pretty good little book called "Us & Them: The
> Science of Identity", by a D. Berreby. Now, I don't
> necessarily adhere to any same opinions, yet it's
> rather didactic and establishes a sort of discourse
> about what is so and considered so and what's not
> and considered not.
>
> Then, the idea that that sort of reflexivity is or isn't
> symmetrical, about the usual notions of conservation
> and symmetry in this sort of world, is explored as
> for matters of Berreby's opinion and lens about
> how science that isn't physics or "mathematical",
> i.e. that it's "non-logical", at all, isn't science.
>
> So, for nominalist fictionalists of the formalist
> sort, while there may be strong mathematical
> platonists who are also formalist constructivists,
> it's suggested that a reading of Berreby might
> result them being non-logical and fundamentally
> as of matters of mere opinion and not of relevance,
> here as with respect to the Relephant, since at least
> times when flying rainbow sparkle ponies were
> putative models of continuous domains or "sets
> of reals", and various ones at that.
>
>
>
> Huntington's postulates are mentioned again,
> quite all about universals. (A president of the MAA.)
>
> Peter of Spain's appositve and suppositive and
> about use/mention distinction making it so that
> "terms" in some "universal particulars" are
> REQUIRED their context, helps explain why
> theories like universal ordinals for any model
> of an integer continuum and the duBois-Reymond
> long-line of all real expressions, which has a larger
> cardinal than c and is on the same line already,
> all make one milieu, and it's logical.
>
> No-one's trying to take away your triangles,
> nor anything else that's mathematical for
> that matter, it is though pointed out that
> this wider world of a strong mathematical
> platonist's universal criteria _always exists_,
> basically pointing out that you can't wish that away.
>
> Often this is mentioned, "that is like the pot,
> one of the implements in the fire along with
> the kettle, who are both blackened by the fire,
> that is like the pot, calling the kettle black,
> when indeed the pot and the kettle are both
> quite black", yet it's not relevant here, because,
> the issue is that for all your reasonable and correct
> criticisms of perceived and demonstrated formal
> incorrectness according to formal constructions,
> then you claim ignorance of "theories with universes",
> for example, without which there isn't one, or,
> this simple "only diagonal" after you've spent
> an entire course establishing why the non-constructive
> "anti-diagonal" makes your system of inequalities
> giving measure after least-upper-bound (axiomatized)
> and measure 1.0 (axiomatized), why the one is so yet
> the other with pretty much the exact same form
> is not: it demonstrates that a hedgerow without
> it would be a mathematical absurdity, and thus
> not mathematical.
>
> Or, you know, trivial, which is acceptable for itself,
> a "fragment", of a, "the mathematics", this though
> is about a "the mathematics" for _all_ the objects
> of the universe of mathematical objects, including
> itself.
>
> For example, at one point it was brought out that
> in the theories about relations of triangles, that
> sine and cosine and the Pythagorean has another
> way to make it, where the Pythagoarean triples
> are basically the end result or the completions
> instead of the other way around, demonstrating
> that lines don't make points and points don't make
> lines, according to induction, yet they do, according
> to deduction, hence/whence/thence they do.
> Then for example the Phythian, more or less
> does the same for uniqueness of Fourier-style series,
> liberating what are some "uniqueness" results to
> "distinctness" results, and making more "repleteness"
> of this "completeness", "re-pletion".
>
> 2500 years later, ....
>
> So anyways, there's basically the "only diagonal" bit
> that sets up there's a non-Cartesian function so that
> there's a model of a countable continuous domain,
> with least-upper-bound and measure according to
> there being sigma algebras established, then you
> get both or none.
>
> Of course you needn't _apply_ such definitions,
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