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From: Peter Fairbrother <peter@tsto.co.uk>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (extra-ordinary)
Date: Thu, 16 Jan 2025 01:44:44 +0000
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Careful! Next you will have sets with negative cardinality...

On 16/01/2025 01:23, Jim Burns wrote:
> On 1/15/2025 1:17 PM, WM wrote:
>> On 15.01.2025 16:16, Jim Burns wrote:
>>> On 1/14/2025 4:07 AM, WM wrote:
>>>> On 13.01.2025 20:31, Jim Burns wrote:
>>>>> On 1/13/2025 12:17 PM, WM wrote:
> 
>>>>>> therefore creates even numbers.
>>>>>> They do not fit below ω.
>>>>>
>>>>> No.
>>>>> They fit below ω
>>>>
>>>> In completed infinity
>>>> all available places are occupied.
> 
>>> In each of our sets,
>>> each of its elements is in the set,
>>> each available place is occupied.
>>
>> Therefore new numbers are not accepted.
> 
> And all even numbers fit below ω
> 
> None are created.
> 
>>> A potentiallyᵂᴹ infiniteˢᵉᵗ set,
>>> the same as any other set,
>>> has all available places occupied
>>> and is completeᵂᴹ.
>>
>> Potential infinity is growing.
> 
> In each of our sets,
> each element has an available space, and
> only its elements have available spaces.
> 
> A place in a set is occupied by virtue of
> its element being in the set.
> 
> In each of our sets,
> each of its elements is in the set,
> each available place is occupied.
> 
> A potentiallyᵂᴹ infiniteˢᵉᵗ set
> has all available places occupied,
> the same as any other set,
> which is to say,
> it is (has been, will be) completeᵂᴹ.
> 
>> "In analysis we have to deal only
>> with the infinitely small and
>> the infinitely large
>> as a limit-notion,
>> as something becoming, emerging, produced,
>> i.e., as we put it, with the potential infinite.
>> But this is not the proper infinite.
>> That we have for instance
>> when we consider
>> the entirety of the numbers 1, 2, 3, 4, ... itself
>> as a completed unit, or
>> the points of a line as
>> an entirety of things which is completely available.
>> That sort of infinity is named actual infinite."
>> [D. Hilbert: "Über das Unendliche", Mathematische Annalen 95 (1925) p. 
>> 167]
> 
> A finite set has
> emptier.by.one sets which are smaller.
> 
> For each finite set,
> a finite ordinal larger than that set
> exists.
> 
> For the set ℕ of all finite ordinals,
> a finite ordinal larger than ℕ
> doesn't exist.
> 
> Therefore,
> the set ℕ of all finite ordinals
> isn't itself finite,  and,
> unlike a finite set, ℕ doesn't have
> emptier.by.one sets which are smaller.
> 
> ----
> Finite people are able to reason about infinity
> by describing an indefinite one of infinitely.many
> and then supplementing the descriptive claims
> with visibly not.first.false claims.
> 
> As finite people,
> we have not and _cannot_ witness
> the infinitely.many described.
> 
> What we can witness, instead, are
> the finitely.many finite.length claims themselves,
> and witness the correctness of the description of
> that which we are currently discussing,
> and witness the not.first.false.ness of
> the other claims.
> Upon witnessing all that,
> we know the claims are true.
> 
>