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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: Wed, 6 Nov 2024 13:48:59 -0800
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On 11/05/2024 06:48 PM, Ross Finlayson wrote:
> 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.
>
>

See, it's possible to make a thorough, reasoned, deconstructive
account of ordinary set theory, with regards to many sorts
under-served yet perfectly apropos models of continuity,
with regards to infinity.