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NNTP-Posting-Date: Tue, 07 Jan 2025 20:23:44 +0000
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (extra-ordinary)
Newsgroups: sci.math
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From: Ross Finlayson <ross.a.finlayson@gmail.com>
Date: Tue, 7 Jan 2025 12:23:16 -0800
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On 01/07/2025 12:06 PM, FromTheRafters wrote:
> Ross Finlayson used his keyboard to write :
>> On 01/07/2025 01:40 AM, FromTheRafters wrote:
>>> WM has brought this to us :
>>>> On 06.01.2025 23:43, Jim Burns wrote:
>>>>> On 1/5/2025 1:14 PM, WM wrote:
>>>>
>>>>>> ℕ cannot be covered by FISONs,
>>>>>> neither by many nor by their union.
>>>>>> If ℕ could be covered by FISONs
>>>>>> then one would be sufficient.
>>>>>
>>>>> ℕ is the set of finite.ordinals.
>>>>> ℕ holds each finite ordinal.
>>>>> ℕ holds only finite.ordinals.
>>>>
>>>>> ℕ| is by definition the smallest transfinite number, larger than every
>>>> natural number by every visible factor.
>>>>>
>>>>> k ∈ ℕ  ⇒  k+1 ∈ ℕ
>>>>> is true for both the darkᵂᴹ and the visibleᵂᴹ.
>>>>
>>>> One exception exists: ω-1.
>>>
>>> Which remains undefined.
>>
>> That's _under_-defined, with regards to
>> its term, omega, 1, and subtraction,
>> or just omega and decrement,
>> or just constant and increment.
>>
>> Well, now it's sort of defined, ....
>>
>>
>> The "counting backward from infinity"
>> is one of Cantor's great ideas,
>> you saying he can't just define it?
>
> No, I'm saying that omega is the first infinite ordinal and there is no
> predecessor function for it just as there isn't for zero.

Yeah, I know about that. This would be
a bit different, obviously.

The, "axiomatic set theory", to
make, "descriptive set theory,
is an exercise, in
"definition and model theory",
what by structures of models
that model "relates",
makes for "proofs in proof theory",
which is equi-interpretable model theory,
as to that whatever structures exist, exist.

So, the "infinite-middle" as its put
here, is simple, and there's a model
of a model of integers bounded above and below,
for example each the finite ones.

That that that that, ....