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NNTP-Posting-Date: Mon, 12 Aug 2024 21:00:07 +0000
Subject: Re: Replacement of Cardinality (infinite middle)
Newsgroups: sci.logic,sci.math
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From: Ross Finlayson <ross.a.finlayson@gmail.com>
Date: Mon, 12 Aug 2024 13:59:58 -0700
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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:
>
>>> [...]
>>
>> 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.
>
>

Here though it's beginning ... ( ... infinitely-many ...) ... end,
where the upper.end of the finites always exists.

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.