Re: Replacement of Cardinality (ubiquitous ordinals)

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:

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about ubiquitous ordinals

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What are ubiquitous ordinal?

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Well, you know that ORD, is, the order type of ordinals,
and so it's an ordinal, of all the ordinals.

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If a ubiquitous ordinal is an ordinal,
then I recommend referring to as an ordinal.
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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.

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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.
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It's like the universe of set theory,

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Do you and I mean the same by "universe of set theory"?
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I am most familiar with theories of
  well.founded sets without urelements.
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In the von Neumann hierarchy of hereditary well.founded sets
V[0] = {}
V[β+1] = 𝒫(V[β])
V[γ] = ⋃[β<γ] V[β]
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V[ω] is the universe of hereditarily finite sets.
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For the first inaccessible ordinal κ
V[κ] is a model of ZF+Choice.
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For first inaccessible ordinal κ
[0,κ) holds an uncountable ordinal and
  is closed under cardinal arithmetic.
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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.

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Is a ubiquitous ordinal a finite ordinal?
I would appreciate a "yes" or a "no" in your response.
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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.
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So, "ubiquitous ordinals" is exactly what it says.

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I find it concerning that you (Ross Finlayson) think that
"what it says" answers "What does it say?" in any useful way.
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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.
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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.
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The "universals" in language invoke "the universals",
that's exactly what it says, that's its name.
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Usual ordinary well-founded set theory is a nice little theory.
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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.
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What it says is what's its name is what it is.
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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.
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ZFC with classes, ..., "proper" classes, or "ultimate" as Quine puts it.
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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.
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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.
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Add it up, it's where the numbers come from, not what you make of them.
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