Re: Replacement of Cardinality (infinite middle)

On 08/12/2024 01:59 PM, Ross Finlayson wrote:

On 08/11/2024 09:44 PM, Jim Burns wrote:

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

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

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[...]

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

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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
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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
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https://en.wikipedia.org/wiki/General_set_theory
https://en.wikipedia.org/wiki/Plural_quantification
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For P(z), use a description 𝕆ᶠⁱⁿ(z) of a finite ordinal,
and ω := {z:𝕆ᶠⁱⁿ(z)} exists
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For example, use
𝕆ᶠⁱⁿ(z)  ⇔
(z ∋ {} ∧ ∀y ∈ z+1: y≠{} ⇒ ∃x∈z: x+1=y)
∨ (z = {})
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z+1 = z∪{z}
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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.

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Above, ST+PQ has not axiomatized well.foundedness.
There are no axioms at all saying which sets DON'T exist.
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ω is what ω is, and what ω is isn't
non.well.founded or inconsistent.multiplicity.
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An ordinal is
a well.founded transitive set of transitive sets.
It's well.foundedness is accomplished by
being {} or holding {}
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Things which aren't well.founded aren't ω
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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.
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Then, omega, as you've defined it,

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ω := {z:𝕆ᶠⁱⁿ(z)}
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contains itself,

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ω doesn't contain itself.
Moreover,
anything which contains itself isn't an ordinal.
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again just quantifying over
the specification of what omega purports to be,

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

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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.
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Here though it's beginning ... ( ... infinitely-many ...) ... end,
where the upper.end of the finites always exists.
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Then you claim to have an axiom of restriction of comprehension
of the finites unless Russell grants you a dispensation of
Russell's retro-thesis, and say it's always so for others, too,
congratulations, you claim to have invented a mathematics
where you = Russell.
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See, one of Russell's better-known aphorisms is that
he employs "ex falso quodlibet" or "EFQ" that from a
contradiction, anything can be said, to establish that
according to Russell that 0 = 1 and "Russell = The Pope".
Neither of which are true, ....
Here we have a principle of "ex falso nihilum" as where
when there's arrived at a contradiction, you're doing it wrong.
One time in third grade my work was in pencil and my
pencil had an eraser and when I discovered several errors
I used my eraser to erase the mistakes and correctly
completed the work.  Then, my teacher, and I'll remember
this, and I'm glad I do, said that was un-acceptable, and
that I was to strike-through my mistakes and write the
corrections aside, because there's no going back.
I was lucky in the lower grades the teacher often
had me standing next to them in the photos.
By second grade I'd already filled in all the third grade work.
When the teacher handed out assignments then she
would give me a larger assignment, if quietly.
Correctly, ....
So anyways that's where I learned
"ex falso quodlibet is improper".
We didn't have tee-ball back then
yet everyone could play kick-ball alright.