Re: Replacement of Cardinality

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.