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NNTP-Posting-Date: Tue, 13 Aug 2024 01:06:16 +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 18:06:19 -0700
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On 08/12/2024 05:44 PM, Python wrote:
> Le 13/08/2024 à 02:28, Ross Finlayson a écrit :
>> On 08/12/2024 04:06 PM, Jim Burns wrote:
>>> On 8/12/2024 4:59 PM, Ross Finlayson wrote:
>>>> On 08/11/2024 09:44 PM, Jim Burns wrote:
>>>>> On 8/11/2024 7:39 PM, Ross Finlayson wrote:
>>>
>>>>>> Starting with a theory _without_
>>>>>> the constant introduced named omega,
>>>>>> i.e., finite sets,
>>>
>>>>> 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}
>>>
>>>>>> Then, omega, as you've defined it,
>>>>>
>>>>> ω := {z:𝕆ᶠⁱⁿ(z)}
>>>>>
>>>>>> contains itself,
>>>
>>>>>> 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
>>>
>>> _Where_ though?
>>>
>>>> it's beginning ... ( ... infinitely-many ...) ... end,
>>>> where the upper.end of the finites always exists.
>>>
>>> For ω as I've defined it, no upper.end exists.
>>>
>>> for each k ∈ ω
>>> 𝕆ᶠⁱⁿ(k)
>>> 𝕆ᶠⁱⁿ(k+1)
>>> k+1 ∈ ω
>>> k is not the upper end of ω
>>>
>>> for each k ∉ ω
>>> k is not the upper end of ω
>>>
>>>> Then you claim to have
>>>> an axiom of restriction of comprehension of the finites
>>>
>>> To review:
>>> What I claim is
>>> ⎛ ∃{}
>>> ⎜ ∀x∀y∃z=x∪{y}
>>> ⎝ and extensionality
>>> ⎛ ∃∃xx={z:P(z)}: ∀y: y ∈ {z:P(z)} ⇔ P(y)
>>> ⎝ and extensionality
>>>
>>> ∃∃{z:P(z)} is unrestricted comprehension.
>>> Unless we are no longer uninterested in what words mean.
>>>
>>>> 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.
>>>
>>> Ah.
>>> I've seen this one before.
>>> Your tacit thesis is that
>>> it is preferable to disagree with the Old Ones
>>> even at the cost of being wrong.
>>>
>>> Well, it's a choice.
>>>
>>>
>>
>> Oh, I have the entire canon here along.
>>
>>
>> It's like yesterday, in this thread with the subject
>> of it talking about "infinite in the middle and always
>> with both ends", or, "here...", pointing out that some
>> modern philosophers with their Ph.D.s. resuscitate a
>> metaphysics that Compte and Boole and Russell and Carnap
>> made so nice for Marx and nihilism and extistentialism
>> and the sort of post-modern that begets itself.
>>
>>
>> Now, I confiscate logical positivism from Compte and
>> brush off Boole for De Morgan and point out Russell
>> and for example Whitehead suffer their own arguments
>> and Carnap was quite a pleasant fellow and I like Quine
>> yet I'm not a nominalist fictionalist. So, a stronger
>> logical positivism and the ontological is kept with
>> a strong mathematical platonism and teleological.
>>
>> Talking about "the Old Ones", you still got Zeno
>> shaking his head and pointing at his watch.
>>
>> Furthermore, I'm a constructivist and agree with
>> notions like infinite induction already and as there's
>> already, for example a sort of ubiquitous ordinals,
>> and even a sort of axiomless natural deduction seated
>> in reason.
>>
>> The, "material implication", or, "ex falso quodlibet",
>> has that material implication is neither material nor implication,
>> and ex falso is mistakes or lies.
>>
>> Some kinds of strong constructivists don't accept
>> non-constructive proofs, for example via contradiction,
>> de dicto, at all.
>>
>>
>> Here though it's just "modular: always modular,
>> of integral wholes, infinite in the middle, modular",
>> just so different from "and a 1 and on down and a 2
>> and on down and a 3 and on down and ... an omega and
>> on down", or, you know, not so.
>>
>>
>> See, "modularity" is regular, rulial, in both
>> increment and dispersion.
>>
>>
>> ... Which most have as properties of integers
>> as with regards to associates with magnitudes,
>> or measures.
>>
>>
>>
>> Heh, you brought up "The Old Ones", it's like,
>> what did the librarian or book-keeper say
>> when the paranoiac asked for self-help books,
>> "they're right behind you".
>>
>>
>> So, for example, Anderson's Relevance Logic many
>> have as more relevant than the quasi-modal, which
>> is neither temporal nor modal, like De Morgan's is,
>> with direct implication, there are some fans of Dana Scott
>> and not for his coat-tailing and wall-papering,
>> the theory of types is often attributed to Peirce,
>> the completeness theorems of arithmetic often to Frege,
>> the extra-ordinary of set theory to Mirimanoff and also
>> a bit to Quine about ultimate classes, you keep the
>> Vienna Circle, and, I'll stick with the larger, fuller canon.
>>
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