Re: Replacement of Cardinality (infinite middle)

On 8/11/2024 7:39 PM, Ross Finlayson wrote:

On 08/11/2024 02:38 PM, Jim Burns wrote:

[...]

>
Starting with a theory _without_
the constant introduced named omega,
i.e., finite sets,

If you're referring to a theory of only finite sets,
let us say, a theory of
von Neumann's V[ω] the hereditarily finite sets,
it literally can't say anything about ω
On the other hand,
here's St+PQ which can talk about ω
It has two kinds of existential claims.
Boolos's ST
⎛ ∃{}
⎜ ∀x∀y∃z=x∪{y}
⎝ and extensionality
with pluralities of sets
⎛ ∃∃xx={z:P(z)}: ∀y: y ∈ {z:P(z)} ⇔ P(y)
⎝ and extensionality
https://en.wikipedia.org/wiki/General_set_theory
https://en.wikipedia.org/wiki/Plural_quantification
For P(z), use a description 𝕆ᶠⁱⁿ(z) of a finite ordinal,
and ω := {z:𝕆ᶠⁱⁿ(z)} exists
For example, use
𝕆ᶠⁱⁿ(z)  ⇔
(z ∋ {} ∧ ∀y ∈ z+1: y≠{} ⇒ ∃x∈z: x+1=y)
∨ (z = {})
z+1 = z∪{z}

given that there's axiomatized well-foundedness
when otherwise
simple comprehension would make the "omega" into
an extra-ordinary or non-well-founded or
inconsistent-multiplicity of a set,
starting _without_ omega,
the finite sets like ordinals, are, exactly
those sets that don't contain themselves.

Above, ST+PQ has not axiomatized well.foundedness.
There are no axioms at all saying which sets DON'T exist.
ω is what ω is, and what ω is isn't
non.well.founded or inconsistent.multiplicity.
An ordinal is
a well.founded transitive set of transitive sets.
It's well.foundedness is accomplished by
being {} or holding {}
Things which aren't well.founded aren't ω
The finite sets like ordinals don't contain themselves.
They aren't _exactly_
those sets that don't contain themselves
because
some sets that don't contain themselves
aren't ordinals.

Then, omega, as you've defined it,

ω := {z:𝕆ᶠⁱⁿ(z)}

contains itself,

ω doesn't contain itself.
Moreover,
anything which contains itself isn't an ordinal.

again just quantifying over
the specification of what omega purports to be,

ω isn't anything other than "what ω  purports to be"
That's how definitions work.
ω might not exist.
ω doesn't exist in V[ω], but
neither is ω anything else in V[ω]

I'm curious, now that you have
a beginning and an end of
the finite, or 0 and omega in ZF,

ω is the least.upper.bound of the finites.
ω is not a finite.
ω is not the upper.end of the finites.
The upper.end of the finites doesn't exist.