Re: Replacement of Cardinality (ubiquitous ordinals)

On 07/30/2024 11:18 AM, Jim Burns wrote:

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:

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

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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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Apparently,
what you mean by ubiquitous ordinals are ordinals,
without further qualification.
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Ordinals can be represented as sets, and are,
most often by the von Neumann scheme, λ = [0,λ)ᴼʳᵈ
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Apparently,
ubiquitous ordinals are
what are represented by the [0,λ)ᴼʳᵈ
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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.
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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.
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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.
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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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There is no universe in ZFC, don't be saying otherwise.

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I will continue not.saying that
there is a universe _IN_ ZFC.
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There are universes _OF_ ZFC which are also called domains.
V[κ] is one of the domains of ZFC.
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None of the sets described by the theory is
the universal set, holding all sets IN the domain.
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Not precisely to your point, but interesting:
Some of the sets IN ZFC satisfy all its axioms,
which make them also domains OF ZFC
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Compare that to the way in which
each end.segment of ℕ is also a model of ℕ
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Well, thanks for asking, I guess.
Starting with the axioms of ZFC before getting
to an actual axiom of infinity, then the embodiment
of Russell's paradox is those finite sets you've found,
all the sets that don't contain themselves, then you
invoke the the restriction of comprehension of the
axiom of infinity (you mean it doesn't say there's
an infinite, it says there's not an extra-ordinary infinite?)
those fulfilling Russell's retro-thesis for him.
"Because it would immediately lead to a contradiction."
If infinite end-segments are models of N,
they're also models of co-finite with non-empty
complements.
I'm not sure you're aware of usual operations "direct sum"
and "direct product", of sets, and as with regards to that the
direct sum of infinitely many copies of a set like as
modeled by finite natural integers, is a matter of definition
and it sticks out that according to the usual definition,
it's one way, and according to the usual imposed definition
on the case of the infinite, it's the opposite. This is
about the direct sum of the infinitely-many copies of N
and whether it's empty or full, that the imposed definition
is the other way than how it is for the finite sets.
"Because it would immediately lead to a contradiction."
So, these are examples, in ZF set theory,
some that you know you have and have to remember
to invoke or as result usual sorts fundamental theorems
that are always assumed, and more examples that you
don't know you have, about consequences of comprehension
and the fact that it's natural that the infinite is extra-ordinary
and then when there's an Integer Continuum then the model of
words is on a substrate of "ubiquitous ordinals".
Then, for talk about the universe and the constructible universe
and V = L and with regards to whether V = L, according to Goedel
it's not, that a gist of his incompleteness theorems, which of
course follow his more naive though entirely ordinary completeness
theorems, for which you can thank Frege.
That V = L has there's a universe already and it constructs.
Then, your talk of domains, like "the equivalence class of
singletons the cardinal 1", and so on, and don't ask why
they don't just call that a set, in as regards to whether
cardinals are even sets at all, is another example that
there's a theory where domains are primary and elements
and it's called "domain theory".
The idea that there's one theory for all this theory,
has that otherwise there isn't and you're not talking
about any of them.