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NNTP-Posting-Date: Sun, 17 Nov 2024 05:30:07 +0000
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (extra-standard)
Newsgroups: sci.math
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From: Ross Finlayson <ross.a.finlayson@gmail.com>
Date: Sat, 16 Nov 2024 21:30:19 -0800
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On 11/16/2024 08:57 PM, Jim Burns wrote:
> On 11/16/2024 11:35 PM, Ross Finlayson wrote:
>> On 11/16/2024 08:18 PM, Jim Burns wrote:
>>> On 11/16/2024 7:17 PM, Ross Finlayson wrote:
>>>> On 11/16/2024 02:46 PM, Jim Burns wrote:
>>>>> On 11/16/2024 5:31 PM, Ross Finlayson wrote:
>>>>>> On 11/16/2024 12:29 PM, Jim Burns wrote:
>>>>>>> On 11/16/2024 12:07 PM, Ross Finlayson wrote:
>>>>>>>> On 11/16/2024 08:58 AM, Ross Finlayson wrote:
>>>>>>>>> On 11/16/2024 02:22 AM, Jim Burns wrote:
>>>>>>>>>> On 11/15/2024 9:52 PM, Ross Finlayson wrote:
>>>>>>>>>>> On 11/15/2024 02:37 PM, Jim Burns wrote:
>>>>>>>>>>>> On 11/15/2024 4:32 PM, Ross Finlayson wrote:
>
>>>>>>>>>>>>> Ah, yet according to Mirimanoff,
>>>>>>>>>>>>> there do not exist standard models of integers,
>>>>>>>>>>>>
>>>>>>>>>>>> If it is true that
>>>>>>>>>>>> our domain of discourse is a model of ST+PQ
>>>>>>>>>>>> then it is true that
>>>>>>>>>>>> our domain of discourse holds a standard integer.model.
>>>>>>>>>>>> What is Mirimanoff's argument that
>>>>>>>>>>>> it doesn't exist?
>>>>>>>>>>>
>>>>>>>>>>> Mirimanoff's? Russell's Paradox.
>>>>>>>>>>
>>>>>>>>>> ST+PQ does not suffer from claiming
>>>>>>>>>> that the set of all non.self.membered sets
>>>>>>>>>> is self.membered or claiming it isn't.
>>>>>>>
>>>>>>>>>> I don't say "infinity" is an axiom
>>>>>>>>>> primarily because
>>>>>>>>>> "infinity" is not an axiom of ST+PQ
>>>>>>>>>> ST+PQ:
>>>>>>>>>> ⎛ set {} exists
>>>>>>>>>> ⎜ set x∪{y} exists
>>>>>>>>>> ⎜ set.extensionality
>>>>>>>>>> ⎜ plurality ⦃z:P(z)⦄ exists
>>>>>>>>>> ⎝ plurality.extensionality
>>>>>>>>>>
>>>>>>>>>> "Infinity exists" ==
>>>>>>>>>> "the minimal inductive plurality exists"
>>>>>>>>>> is a theorem of those axioms.
>>>>>>>
>>>>>>>>> ^- Fragment
>>>>>>>
>>>>>>> No.
>>>>>>> The minimal inductive plurality is
>>>>>>> a standard model of the integers.
>>>>>>>
>>>>>>>> Let's recall an example geometrically of what's
>>>>>>>> so inductively and not so in the limit.
>>>>>>>>
>>>>>>>> Take a circle and draw a diameter, then bisect
>>>>>>>> the diameter resulting diameters of common circles,
>>>>>>>> all sharing a common diameter, vertical, say.
>>>>>>>>
>>>>>>>> Then, notice the length of the circle, is
>>>>>>>> same, as the sum of the lengths of the half-diameter
>>>>>>>> circles, their sum.
>>>>>>>>
>>>>>>>> So, repeat his dividing ad infinitum. In the limit,
>>>>>>>> the length is that of the diameter, not the perimeter,
>>>>>>>> while inductively, it's the diameter.
>>>>>>>>
>>>>>>>> Thusly, a clear example "not.first.false" being
>>>>>>>> "ultimately.untrue".
>>>>>>>
>>>>>>> A finite sequence of claims, each claim of which
>>>>>>> is true.or.not.first.false  is
>>>>>>> a finite sequence of claims, each claim of which
>>>>>>> is true.
>>>>>>>
>>>>>>> The reason that that's true is that
>>>>>>> THE SEQUENCE OF CLAIMS is finite.
>>>>>>>
>>>>>>> Whatever those CLAIMS refer to,
>>>>>>> none of those CLAIMS are first.false.
>>>>>>> (They're each not.first.false.)
>>>>>>>
>>>>>>> Since none of those CLAIMS are first.false,
>>>>>>> none of those CLAIMS are false.
>>>>>>> (That sequence is finite.)
>>>>>>>
>>>>>>> What those claims are ABOUT doesn't affect that.
>>>>>>> For example,
>>>>>>> being ABOUT an indefinite one of infinitely.many
>>>>>>> doesn't affect that.
>>>>>>>
>>>>>>> Discovering
>>>>>>> a finite sequence of claims, each claim of which
>>>>>>> is true.or.not.first.false
>>>>>>> in which there IS an untrue claim
>>>>>>> is akin to
>>>>>>> counting the eggs in a carton and
>>>>>>> discovering that, there, in that carton,
>>>>>>> 7 is NOT between 6 and 8.
>>>>>>> There is a problem, but not with mathematics.
>>>>>>>
>>>>>>>> Then, with regards to your fragment,
>>>>>>>
>>>>>>> ...the minimal inductive plurality...
>>>>>>>
>>>>>>>> congratulations,
>>>>>>>
>>>>>>> Thank you.
>>>>>>>
>>>>>>>> you have ignored Russell his paradox and so on
>>>>>>>
>>>>>>> No.
>>>>>>>
>>>>>>> Selecting axioms which do not suffer from claiming
>>>>>>>   that the set of all non.self.membered sets
>>>>>>>   is self.membered or claiming it isn't
>>>>>>> is not ignoring Russel,
>>>>>>> it is responding to Russell.
>>>>>>>
>>>>>>> Russell points out that
>>>>>>> _we do not want_ to claim
>>>>>>> that the set of all non.self.membered sets
>>>>>>>   is self.membered or claiming it isn't.
>>>>>>>
>>>>>>> We respond: Okay, we'll stop doing that.
>>>>>>>
>>>>>>>> and quite fully revived Frege and given yourself
>>>>>>>> a complete theory and consistent as it may be, and
>>>>>>>> can entirely ignore all of 20'th century mathematics.
>>>>>>>>
>>>>>>>> It's small, .... Fragment
>>>>>>>
>>>>>>> The minimal inductive plurality.
>>>>>>> Big or small, that's the thing,
>>>>>>> the whole thing, and nothing but the thing.
>>>>>>
>>>>>> Bzzt, flake-out.
>>>>>> It's not pretty the act of making lies.
>>>>>
>>>>> Tell me what you think is a lie:
>>>>
>>>
>>> Quote what I wrote which you think is a lie:
>>>
>>
>> Well, it's among what you clipped because it was un-answerable,
>
> I didn't clip any quote of me by you.
> Anyway, do you need help finding earlier posts in a thread?
>
> ⎛ Noun
> ⎜ lie (plural lies)
> ⎜
> ⎜ 1. An intentionally false statement; an intentional falsehood.
> ⎜ 2. A statement intended to deceive, even if literally true.
> ⎝ 3. (by extension) Anything that misleads or disappoints.
>
> https://en.wiktionary.org/wiki/lie
>
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