Re: Replacement of Cardinality (ubiquitous ordinals)

On 7/29/2024 9:31 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.

Is a ubiquitous ordinal a finite ordinal?
I would appreciate a "yes" or a "no" in your response.

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.

Apparently,
what you mean by ubiquitous ordinals are ordinals,
without further qualification.
Ordinals can be represented as sets, and are,
most often by the von Neumann scheme, λ = [0,λ)ᴼʳᵈ
Apparently,
ubiquitous ordinals are
what are represented by the [0,λ)ᴼʳᵈ
Ordinals are well.ordered.
The [0,λ)ᴼʳᵈ are well.ordered.
That is entirely non.accidental.
There isn't much reason to choose between
the von Neumann ordinal.representations and
the raw, unfiltered "ubiquitous" ordinals.
The one advantage which
representations have over the ubiquitous(?) is that
they are are objects in a theory of sets which
we have great confidence isn't contradictory,
and that extends our great confidence to
the non.contradictoriness of theorems about ordinals.
Unless we are considering their existence,
which is to say, their non.contradictoriness,
ordinals.of.unspecified.origin are well.ordered,
and that's an end to their description.

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.

There is no universe in ZFC, don't be saying otherwise.

I will continue not.saying that
there is a universe _IN_ ZFC.
There are universes _OF_ ZFC which are also called domains.
V[κ] is one of the domains of ZFC.
None of the sets described by the theory is
the universal set, holding all sets IN the domain.
Not precisely to your point, but interesting:
Some of the sets IN ZFC satisfy all its axioms,
which make them also domains OF ZFC
Compare that to the way in which
each end.segment of ℕ is also a model of ℕ