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NNTP-Posting-Date: Wed, 06 Nov 2024 17:31:51 +0000
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
 (opinions)
Newsgroups: sci.math
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From: Ross Finlayson <ross.a.finlayson@gmail.com>
Date: Wed, 6 Nov 2024 09:31:44 -0800
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On 11/06/2024 06:22 AM, Jim Burns wrote:
> On 11/6/2024 5:35 AM, WM wrote:
>> On 05.11.2024 18:25, Jim Burns wrote:
>>> On 11/4/2024 12:32 PM, WM wrote:
>
>>>> The intervals together cover a length of less than 3.
>>>> The whole length is infinite.
>>>> Therefore there is plenty of space for
>>>> a point not in contact with any interval.
>>>
>>> ⎛ Assuming the covering intervals are translated
>>> ⎜ to where they are end.to.end.to.end,
>>> ⎜ there is plenty of space for
>>> ⎝ not.in.contact exterior points.
>>
>> This plentiness does not change
>> when the intervals are translated.
>
> ⎛ When the intervals are end.to.end.to.end,
> ⎜ there are exterior points
> ⎝ a distance 10¹⁰⁰⁰⁰⁰ from any interval.
>
> Are there points 10¹⁰⁰⁰⁰⁰ from any interval
> when midpoints of intervals include
> each of {...,-3,-2,-1,0,1,2,3,...} ?
>
> Isn't that a plentiness which changes?
>
>>> I mean 'exterior' in the topological sense.
>>>
>>> For a point x in the boundary ∂A of set A
>>> each open set Oₓ which holds x
>>> holds points in A and points not.in A
>>
>> The intervals are closed with irrational endpoints.
>
> 'Exterior' seems like a good way to say
> 'not in contact'.
>
> It seems to me that you have a better argument
> with open intervals instead of closed,
> but let them be closed, if you like.
>
> Either way,
> there are no points 10¹⁰⁰⁰⁰⁰ from any interval.
>
>>> Each of {...,-3,-2,-1,0,1,2,3,...} is
>>> the midpoint of an interval.
>>> There can't be any exterior point
>>> a distance 1 from any interval.
>>>
>>> There can't be any exterior point
>>> a distance ⅟2 from any interval.
>>> Nor ⅟3. Nor ⅟4. Nor any positive distance.
>>
>> Nice try.
>> But there are points outside of intervals,
>
> Are any of these points.outside
>   ⅟2 from any interval? ⅟3? ⅟4?
>
>> and they are closer to interval ends
>> than to the interior, independent of
>> the configuration of the intervals.
>
> Shouldn't I be pointing that out
> to you?
>
> If there is no point with more.than.⅟2
> between it and any midpoint,
> shouldn't there be fewer.than.no points
> with more.than.⅟2 between it and
> any closer endpoint?
>
>> Note that
>> only 3/oo of the points are inside.
>
> Yes, less than 2³ᐟ²⋅ε
>
> If the intervals were open,
> all of that would be "inside"
> in the interior of their union.
>
> Of the rest,
> none of it is more.than.⅟2 from any interval.
>
>>> An exterior point which is not
>>> a positive distance from any interval
>>> is not an exterior point.
>>
>> Positive is what you can define,
>
> Positive ℕ⁺ holds countable.to from.1
> Positive ℚ⁺ holds ratios of elements of ℕ⁺
> Positive ℝ⁺ holds points.between.splits of Q⁺
>
>> but there is much more in smaller distance.
>
> Distances are positive or zero.
> Two distinct points are a positive distance apart.
>
>> Remember the infinitely many unit fractions
>> within every eps > 0 that you can define.
>
> For each of the infinitely.many unit fractions
> there is no point a distance of that unit fraction
> or more from any interval.
>
>>> Therefore,
>>> in what is _almost_ your conclusion,
>>> there are no exterior points.
>>
>> There are 3/oo of all points exterior.
>
> Did you intend to write "interior"?
>
> An exterior point is in
> an open interval holding no rational.
>
> There are no
> open intervals holding no rational.
>
> There are no exterior points.
>
> However,
> there are boundary points.
> All but 2³ᐟ²⋅ε are boundary points.
>
>>> Instead, there are boundary points.
>>>   For each x not.in the intervals,
>>>   each open set Oₓ which holds x
>>>   holds points in the intervals and
>>>   points not.in the intervals.
>>> x is a boundary point.
>>
>> The intervals are closed
>
> We are only told
> that Oₓ is an open set holding x
> not that Oₓ is one of the ε.cover of ℚ
> The question is whether x is a boundary point.
>
>>>> The rationals are dense
>>>
>>> Yes.
>>> Each multi.point interval [x,x′] holds
>>> rationals.
>>>
>>>> but the intervals are not.
>>>
>>> No.
>>> Each multi.point interval [x,x′] holds
>>> ε.cover intervals.
>>
>> Therefore not all rationals are enumerated.
>
> Explain why.
>
> ⎛ i/j ↦ kᵢⱼ = (i+j-1)(i+j-2)/2+i
> ⎜ k ↦ iₖ+jₖ = ⌈(2⋅k+¼)¹ᐟ²+½⌉
> ⎜  iₖ = k-(iₖ+jₖ-1)(iₖ+jₖ-2)/2
> ⎜  jₖ = (iₖ+jₖ)-iₖ
> ⎝  (iₖ+jₖ-1)(iₖ+jₖ-2)/2+iₖ = k
>>> proves that
>>> the rationals are countable.
>>
>> Contradiction.
>
> It contradicts a non.empty exterior.
> It doesn't contradict an almost.all boundary.
>
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