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NNTP-Posting-Date: Tue, 30 Jul 2024 01:31:43 +0000
Subject: Re: Replacement of Cardinality (ubiquitous ordinals)
Newsgroups: sci.logic,sci.math
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From: Ross Finlayson <ross.a.finlayson@gmail.com>
Date: Mon, 29 Jul 2024 18:32:02 -0700
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On 07/29/2024 06:31 PM, Ross Finlayson wrote:
> On 07/29/2024 02:12 PM, Jim Burns wrote:
>> On 7/29/2024 3:44 PM, Ross Finlayson wrote:
>>> On 07/29/2024 05:32 AM, Jim Burns wrote:
>>>> On 7/28/2024 7:42 PM, Ross Finlayson wrote:
>>
>>>>> about ubiquitous ordinals
>>>>
>>>> What are ubiquitous ordinal?
>>>
>>> Well, you know that ORD, is, the order type of ordinals,
>>> and so it's an ordinal, of all the ordinals.
>>
>> If a ubiquitous ordinal is an ordinal,
>> then I recommend referring to as an ordinal.
>>
>>> The "ubiquitous ordinals", sort of recalls Kronecker's
>>> "G-d made the integers, the rest is the work of Man",
>>> that the Integer Continuum, is the model and ground
>>> model, of any sort of language of finite words,
>>> like set theory.
>>
>> If a ubiquitous ordinal is
>>   a finite ordinal ==
>>   a natural number ==
>>   a non.negative integer,
>> then
>> (I bet you see where I'm headed here)
>> I recommend that you refer to it as
>>   a finite ordinal or
>>   a natural number or
>>   a non.negative integer.
>>
>>> It's like the universe of set theory,
>>
>> Do you and I mean the same by "universe of set theory"?
>>
>> I am most familiar with theories of
>>   well.founded sets without urelements.
>>
>> In the von Neumann hierarchy of hereditary well.founded sets
>> V[0] = {}
>> V[β+1] = 𝒫(V[β])
>> V[γ] = ⋃[β<γ] V[β]
>>
>> V[ω] is the universe of hereditarily finite sets.
>>
>> For the first inaccessible ordinal κ
>> V[κ] is a model of ZF+Choice.
>>
>> For first inaccessible ordinal κ
>> [0,κ) holds an uncountable ordinal and
>>   is closed under cardinal arithmetic.
>>
>>> then as that there's _always_ an arithmetization, or
>>> as with regards to ordering and numbering
>>> as a bit weaker property than collecting and counting,
>>> so that "ubiquitous ordinals" is
>>> what you get from a discrete world.
>>
>> Is a ubiquitous ordinal a finite ordinal?
>> I would appreciate a "yes" or a "no" in your response.
>>
>>> Then there's that
>>> according to the set-theoretic Powerset theorem of Cantor,
>>> that when the putative function is successor,
>>> in ubiquitous ordinals
>>> where order type is powerset is successor,
>>> then there's no missing element.
>>>
>>> So, "ubiquitous ordinals" is exactly what it says.
>>
>> I find it concerning that you (Ross Finlayson) think that
>> "what it says" answers "What does it say?" in any useful way.
>>
>>
>
>
> The ubiquitous ordinals are, for example, a theory where
> the primary elements are ordinals, for ordering theory,
> and numbering theory, which may be more fundamental,
> than set theory, with regards to a theory of one relation.
>
> The "template theory" is as of a Comenius language, a
> language of only truisms its elements, one of which is
> the Liar paradox, only a truism as prototyping a fallacy,
> here relating that to ORD, that ORD exists is a truism
> in the theory.
>
> The "universals" in language invoke "the universals",
> that's exactly what it says, that's its name.
>
>
> Usual ordinary well-founded set theory is a nice little theory.
>
> Giving it an axiom of "ordinary infinity" is a sort of
> restriction of comprehension, sort of like it's a false axiom,
> or rather, hypocritical, in the sense that "Russell's retro-thesis",
> rejects what's otherwise a matter of deductive resolution of
> the paradoxes of the quantifier ambiguity and impredicativity,
> that you claim Russell claims don't exist.
>
> What it says is what's its name is what it is.
>
> There is no universe in ZFC, don't be saying otherwise.
> It simply doesn't exist and isn't available. Then, if
> you get into class/set distinction and these kinds of things,
> then it's automatically extra-ordinary and your meta-theory
> doesn't admit Russell's retro-thesis except as a conditional
> property of a sort of sub-class of propositions in tertium non datur.
>
> ZFC with classes, ..., "proper" classes, or "ultimate" as Quine puts it.
>
>
> Of course, some have integers as primary, while others have
> continuity as primary, that the integers and modular just fall
> out in the middle, sort of downward-Kronecker invoking Skolem and
> Louwenheim and Levy, if you've heard of them.
>
>
> So, if you've heard of Skolem and Louwenheim and Levy,
> then you've probably heard about the generic extension,
> and if you've heard of the Independence of the Continuum Hypothesis
> you've probably heard about the generic extension, and collapse,
> to either a higher or lower cardinality an ordinal still keeping
> a model, like, ..., a model of ubiquitous ordinals.
>
> Add it up, it's where the numbers come from, not what you make of them.
>
>

And von Neumann was a belligerent drunk.