Re: Incompleteness of Cantor's enumeration of the rational numbers

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Sujet : Re: Incompleteness of Cantor's enumeration of the rational numbers
De : james.g.burns (at) *nospam* att.net (Jim Burns)
Groupes : sci.math
Date : 06. Nov 2024, 15:22:05
Autres entêtes
Organisation : A noiseless patient Spider
Message-ID : <40ac3ed2-5648-48c0-ac8f-61bdfd1c1e20@att.net>
References : 1 2 3 4 5 6 7
User-Agent : Mozilla Thunderbird
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.

Something of your theory is inconsistent.
Your intuition is disturbed by
an almost.all boundary.
Disturbed intuitions and inconsistencies
are different.

Date Sujet#  Auteur
4 Nov 24 * Re: Incompleteness of Cantor's enumeration of the rational numbers501Jim Burns
4 Nov 24 `* Re: Incompleteness of Cantor's enumeration of the rational numbers500WM
4 Nov 24  `* Re: Incompleteness of Cantor's enumeration of the rational numbers499Jim Burns
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5 Nov 24   i i `* Re: Incompleteness of Cantor's enumeration of the rational numbers (re-Vitali-ized)2Ross Finlayson
6 Nov 24   i i  `- Re: Incompleteness of Cantor's enumeration of the rational numbers (re-Vitali-ized)1Chris M. Thomasson
6 Nov 24   i +* Re: Incompleteness of Cantor's enumeration of the rational numbers463WM
6 Nov 24   i i`* Re: Incompleteness of Cantor's enumeration of the rational numbers462Jim Burns
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6 Nov 24   i i i `* Re: Incompleteness of Cantor's enumeration of the rational numbers457WM
6 Nov 24   i i i  `* Re: Incompleteness of Cantor's enumeration of the rational numbers456Jim Burns
7 Nov 24   i i i   `* Re: Incompleteness of Cantor's enumeration of the rational numbers455WM
7 Nov 24   i i i    +* Re: Incompleteness of Cantor's enumeration of the rational numbers7Jim Burns
7 Nov 24   i i i    i`* Re: Incompleteness of Cantor's enumeration of the rational numbers6WM
7 Nov 24   i i i    i `* Re: Incompleteness of Cantor's enumeration of the rational numbers5Jim Burns
7 Nov 24   i i i    i  `* Re: Incompleteness of Cantor's enumeration of the rational numbers4WM
7 Nov 24   i i i    i   +* Re: Incompleteness of Cantor's enumeration of the rational numbers2Jim Burns
7 Nov 24   i i i    i   i`- Re: Incompleteness of Cantor's enumeration of the rational numbers1WM
7 Nov 24   i i i    i   `- Re: Incompleteness of Cantor's enumeration of the rational numbers1Chris M. Thomasson
7 Nov 24   i i i    `* Re: Incompleteness of Cantor's enumeration of the rational numbers447Jim Burns
7 Nov 24   i i i     `* Re: Incompleteness of Cantor's enumeration of the rational numbers446WM
8 Nov 24   i i i      `* Re: Incompleteness of Cantor's enumeration of the rational numbers445Jim Burns
8 Nov 24   i i i       `* Re: Incompleteness of Cantor's enumeration of the rational numbers444WM
8 Nov 24   i i i        +* Re: Incompleteness of Cantor's enumeration of the rational numbers13Richard Damon
8 Nov 24   i i i        i`* Re: Incompleteness of Cantor's enumeration of the rational numbers12WM
8 Nov 24   i i i        i +* Re: Incompleteness of Cantor's enumeration of the rational numbers2Richard Damon
9 Nov 24   i i i        i i`- Re: Incompleteness of Cantor's enumeration of the rational numbers1WM
8 Nov 24   i i i        i `* Re: Incompleteness of Cantor's enumeration of the rational numbers9joes
8 Nov 24   i i i        i  +* Re: Incompleteness of Cantor's enumeration of the rational numbers7Moebius
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9 Nov 24   i i i        i  i `* Re: Incompleteness of Cantor's enumeration of the rational numbers5WM
9 Nov 24   i i i        i  i  `* Re: Incompleteness of Cantor's enumeration of the rational numbers4Chris M. Thomasson
10 Nov 24   i i i        i  i   `* Re: Incompleteness of Cantor's enumeration of the rational numbers3Moebius
10 Nov 24   i i i        i  i    `* Re: Incompleteness of Cantor's enumeration of the rational numbers2WM
10 Nov 24   i i i        i  i     `- Re: Incompleteness of Cantor's enumeration of the rational numbers1Chris M. Thomasson
9 Nov 24   i i i        i  `- Re: Incompleteness of Cantor's enumeration of the rational numbers1WM
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8 Nov 24   i i i        `* Re: Incompleteness of Cantor's enumeration of the rational numbers428Jim Burns
9 Nov 24   i i i         `* Re: Incompleteness of Cantor's enumeration of the rational numbers427WM
10 Nov 24   i i i          `* Re: Incompleteness of Cantor's enumeration of the rational numbers426Jim Burns
10 Nov 24   i i i           `* Re: Incompleteness of Cantor's enumeration of the rational numbers425WM
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11 Nov 24   i i i            i`* Re: Incompleteness of Cantor's enumeration of the rational numbers386WM
11 Nov 24   i i i            i `* Re: Incompleteness of Cantor's enumeration of the rational numbers385Jim Burns
11 Nov 24   i i i            i  `* Re: Incompleteness of Cantor's enumeration of the rational numbers384WM
11 Nov 24   i i i            i   +* Re: Incompleteness of Cantor's enumeration of the rational numbers5FromTheRafters
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12 Nov 24   i i i            i   i  `- Re: Incompleteness of Cantor's enumeration of the rational numbers1WM
12 Nov 24   i i i            i   +* Re: Incompleteness of Cantor's enumeration of the rational numbers2Jim Burns
12 Nov 24   i i i            i   i`- Re: Incompleteness of Cantor's enumeration of the rational numbers1WM
12 Nov 24   i i i            i   `* Re: Incompleteness of Cantor's enumeration of the rational numbers376Jim Burns
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16 Nov 24   i i i            i             i    i  +* Re: Incompleteness of Cantor's enumeration of the rational numbers2Chris M. Thomasson
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16 Nov 24   i i i            i             i    i  +* Re: Incompleteness of Cantor's enumeration of the rational numbers13FromTheRafters
16 Nov 24   i i i            i             i    i  i`* Re: Incompleteness of Cantor's enumeration of the rational numbers12Chris M. Thomasson
16 Nov 24   i i i            i             i    i  i +* Re: Incompleteness of Cantor's enumeration of the rational numbers2Moebius
16 Nov 24   i i i            i             i    i  i i`- Re: Incompleteness of Cantor's enumeration of the rational numbers1Moebius
17 Nov 24   i i i            i             i    i  i +* Re: Incompleteness of Cantor's enumeration of the rational numbers7Moebius
17 Nov 24   i i i            i             i    i  i i`* Re: Incompleteness of Cantor's enumeration of the rational numbers6Chris M. Thomasson
17 Nov 24   i i i            i             i    i  i i `* Re: Incompleteness of Cantor's enumeration of the rational numbers5Moebius
17 Nov 24   i i i            i             i    i  i i  +* Re: Incompleteness of Cantor's enumeration of the rational numbers2Moebius
17 Nov 24   i i i            i             i    i  i i  i`- Re: Incompleteness of Cantor's enumeration of the rational numbers1WM
17 Nov 24   i i i            i             i    i  i i  `* Re: Incompleteness of Cantor's enumeration of the rational numbers2Chris M. Thomasson
17 Nov 24   i i i            i             i    i  i `* Re: Incompleteness of Cantor's enumeration of the rational numbers2FromTheRafters
16 Nov 24   i i i            i             i    i  `* Re: Incompleteness of Cantor's enumeration of the rational numbers257Moebius
16 Nov 24   i i i            i             i    +- Re: Incompleteness of Cantor's enumeration of the rational numbers1Moebius
16 Nov 24   i i i            i             i    `* Re: Incompleteness of Cantor's enumeration of the rational numbers2Moebius
14 Nov 24   i i i            i             `* Re: Incompleteness of Cantor's enumeration of the rational numbers69Jim Burns
10 Nov 24   i i i            `* Re: Incompleteness of Cantor's enumeration of the rational numbers36Chris M. Thomasson
6 Nov 24   i i `* Re: Incompleteness of Cantor's enumeration of the rational numbers (opinions)2Ross Finlayson
6 Nov 24   i `* Re: Incompleteness of Cantor's enumeration of the rational numbers5WM
4 Nov 24   `* Re: Incompleteness of Cantor's enumeration of the rational numbers24Chris M. Thomasson

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