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Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (extra-ordinary)
Newsgroups: sci.logic
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From: Ross Finlayson <ross.a.finlayson@gmail.com>
Date: Tue, 5 Nov 2024 18:48:44 -0800
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On 11/05/2024 02:29 AM, Mikko wrote:
> On 2024-11-04 18:12:55 +0000, WM said:
>
>> On 04.11.2024 18:49, Mikko wrote:
>>> On 2024-11-04 10:47:19 +0000, WM said:
>>>
>>>> On 04.11.2024 11:31, Mikko wrote:
>>>>> On 2024-11-04 09:55:24 +0000, WM said:
>>>>>
>>>>>> On 03.11.2024 23:18, Jim Burns wrote:
>>>>>>
>>>>>>> There aren't any neighboring intervals.
>>>>>>> Any two intervals have intervals between them.
>>>>>>
>>>>>> That is wrong. The measure outside of the intervals is infinite.
>>>>>> Hence there exists a point outside. This point has two nearest
>>>>>> intervals
>>>>>
>>>>> No, it hasn't.
>>>>
>>>> In geometry it has.
>>>
>>> This discussion is about numbers, not geometry.
>>
>> Geometry is only another language for the same thing.
>
> Another language is an unnecessary complication that only reeasls
> an intent to deceive.
>
>>>>> Between that point an an interval there are rational
>>>>> numbers and therefore other intervals
>>>>
>>>> I said the nearest one. There is no interval nearer than the nearest
>>>> one.
>>>
>>> There is no nearesst one. There is always a nearer one.
>>
>> Nonsense.
>
> No, the meaning is clear. Of course, because some intevals overlap,
> you should have specified what exacly you mean by "nearer". But as
> ε shriks the overlappings disappear and the distance between any
> two intevals approaches the distance between their centers we may
> define distance between the intervals as the distance between their
> endpoints even wne ε > 0.
>
>>>>> Therefore the
>>>>> point has no nearest interval.
>>>>
>>>> That is an unfounded assertions and therefore not accepted.
>>>
>>> It is not unfounded.
>>
>> Of course it is. It is the purest nonsense.
>
> That you don't even try to support your clam to support your claim
> indicates that you don't really believe it. Cantor's results are
> conclusions of proofs and you have not shown any error in the proofs.
> You are free to deny one of more of the assumptions that constitue
> the foudations of the results but you havn't. Even if you will that
> will not make the results unfounded. It only means that you want to
> use a different foundation. Whether you can find one that you like
> is your problem.
>

Here what's considered an "opinion" of ZF is any axiom,
of the theory, what results "restriction of comprehension",
for example the Axiom of Regularity, or, the Axiom of (Regular)
Infinity.

Somebody like Mirimanoff, who introduced the plain "extra-ordinary",
then saw that as soon as Mirimanoff brought that up, then
ZF set theory had an axiom of regular/ordinary infinity added to it,
thus that Russell's "paradox" was put away, then for some
relatively simple things or the establishment of an inequality
the uncountability, to so follow.

The anti-diagonal argument as discovered by du Bois Reymond,
and nested intervals known since forever, the m-w proof as
is one of the number-theoretic proofs of uncountability,
another bit for continued fractions, these are the number-theoretic
results for uncountability, then there's the set-theoretic
bit or the powerset result.

So, the idea of providing an example to uncountability,
would be a 1-1 and onto function a bijection, between
countable domain and here most succinctly, the unit interval,
each of the points of the unit interval. This would be
with regards to the "number-theoretic" arguments.

Then, there would also need be a "set-theoretic" counter-example.


Here's that's provided by the "natural/unit equivalency function",
which falls out of the number-theoretic results un-contradicted,
and then some "ubiquitous ordinals" between ordering-theory and
set-theory, not unlike Cohen's forcing establishing the ndependence
of the Continuum Hypothesis, which one can also see as forestalling
what's a contradiction after ZF, since ordinals either would or
wouldn't live between cardinals with or without CH.

So, providing a counterexample and noting that
the "restrictions of comprehension" are _stipulations_
and thusly _non-logical_, makes for an inclusive take
on a foundation beneath _ordinary_ set theory: _extra-ordinary_
set theory. ("A theory of one relation: elt.")

In this way we can have extra-ordinary theory and plain
simple classical logical theory and plain ordinary regular
set theory, all quite thoroughly logical.