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tech / sci.math / Re: Mathematics, science and Abraham Robinson

SubjectAuthor
* Mathematics, science and Abraham RobinsonDavid Petry
+- Re: Mathematics, science and Abraham Robinsonzelos...@gmail.com
+- Re: Mathematics, science and Abraham RobinsonFromTheRafters
+* Re: Mathematics, science and Abraham Robinsonzelos...@gmail.com
|`* Re: Mathematics, science and Abraham RobinsonDavid Petry
| +- Re: Mathematics, science and Abraham Robinsonsergio
| `- Re: Mathematics, science and Abraham Robinsonzelos...@gmail.com
`* Re: Mathematics, science and Abraham Robinsonsobriquet
 `* Re: Mathematics, science and Abraham RobinsonFromTheRafters
  `* Re: Mathematics, science and Abraham RobinsonFredJeffries
   +* Re: Mathematics, science and Abraham Robinsonsergio
   |`* Re: Mathematics, science and Abraham RobinsonFromTheRafters
   | `* Re: Mathematics, science and Abraham Robinsonsergi o
   |  `* Re: Mathematics, science and Abraham RobinsonFredJeffries
   |   +* Re: Mathematics, science and Abraham RobinsonWM
   |   |+* Re: Mathematics, science and Abraham RobinsonFritz Feldhase
   |   ||`* Re: Mathematics, science and Abraham RobinsonWM
   |   || +- Re: Mathematics, science and Abraham RobinsonFromTheRafters
   |   || `- Re: Mathematics, science and Abraham Robinsonsergi o
   |   |`* Re: Mathematics, science and Abraham RobinsonTimothy Golden
   |   | `* Re: Mathematics, science and Abraham RobinsonRoss A. Finlayson
   |   |  `- Re: Mathematics, science and Abraham RobinsonTimothy Golden
   |   `- Re: Mathematics, science and Abraham RobinsonDavid Petry
   `* Re: Mathematics, science and Abraham Robinsonsobriquet
    `* Re: Mathematics, science and Abraham RobinsonWM
     +* Re: Mathematics, science and Abraham Robinsonsobriquet
     |`* Re: Mathematics, science and Abraham RobinsonWM
     | +- Re: Mathematics, science and Abraham RobinsonFromTheRafters
     | +* Re: Mathematics, science and Abraham RobinsonFritz Feldhase
     | |`* Re: Mathematics, science and Abraham RobinsonWM
     | | +- Re: Mathematics, science and Abraham Robinsonsergi o
     | | +* Re: Mathematics, science and Abraham RobinsonJim Burns
     | | |+- Re: Mathematics, science and Abraham RobinsonFritz Feldhase
     | | |`* Re: Mathematics, science and Abraham RobinsonWM
     | | | +- Re: Mathematics, science and Abraham Robinsonsergi o
     | | | +* Re: Mathematics, science and Abraham RobinsonJim Burns
     | | | |`* Re: Mathematics, science and Abraham RobinsonWM
     | | | | +- Re: Mathematics, science and Abraham Robinsonsergi o
     | | | | +* Re: Mathematics, science and Abraham RobinsonJim Burns
     | | | | |`* Re: Mathematics, science and Abraham RobinsonWM
     | | | | | +- Re: Mathematics, science and Abraham Robinsonsergi o
     | | | | | `* Re: Mathematics, science and Abraham RobinsonJim Burns
     | | | | |  +* Re: Mathematics, science and Abraham RobinsonWM
     | | | | |  |+* Re: Mathematics, science and Abraham RobinsonFritz Feldhase
     | | | | |  ||+* Re: Mathematics, science and Abraham RobinsonWM
     | | | | |  |||+- Re: Mathematics, science and Abraham RobinsonJim Burns
     | | | | |  |||`* Re: Mathematics, science and Abraham RobinsonWM
     | | | | |  ||| +* Re: Mathematics, science and Abraham RobinsonFromTheRafters
     | | | | |  ||| |+- Re: Mathematics, science and Abraham RobinsonTimothy Golden
     | | | | |  ||| |+- Re: Mathematics, science and Abraham Robinsonsergi o
     | | | | |  ||| |`* Re: Mathematics, science and Abraham RobinsonWM
     | | | | |  ||| | `- Re: Mathematics, science and Abraham Robinsonsergi o
     | | | | |  ||| `- Re: Mathematics, science and Abraham RobinsonJim Burns
     | | | | |  ||+* Re: Mathematics, science and Abraham RobinsonWM
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     | | | | |  ||+* Re: Mathematics, science and Abraham RobinsonWM
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     | | | | |  ||+* Re: Mathematics, science and Abraham RobinsonWM
     | | | | |  |||`- Re: Mathematics, science and Abraham Robinsonsergi o
     | | | | |  ||`* Re: Mathematics, science and Abraham RobinsonWM
     | | | | |  || `- Re: Mathematics, science and Abraham Robinsonsergi o
     | | | | |  |+* Re: Mathematics, science and Abraham RobinsonJim Burns
     | | | | |  ||+* Re: Mathematics, science and Abraham RobinsonWM
     | | | | |  |||`* Re: Mathematics, science and Abraham RobinsonJim Burns
     | | | | |  ||| `* Re: Mathematics, science and Abraham RobinsonWM
     | | | | |  |||  +- Re: Mathematics, science and Abraham Robinsonsergi o
     | | | | |  |||  `* Re: Mathematics, science and Abraham RobinsonJim Burns
     | | | | |  |||   `* Re: Mathematics, science and Abraham RobinsonJim Burns
     | | | | |  |||    `- Re: Mathematics, science and Abraham RobinsonRoss A. Finlayson
     | | | | |  ||`* Re: Mathematics, science and Abraham RobinsonWM
     | | | | |  || `- Re: Mathematics, science and Abraham Robinsonsergi o
     | | | | |  |+- Re: Mathematics, science and Abraham Robinsonsergi o
     | | | | |  |+- Re: Mathematics, science and Abraham RobinsonFritz Feldhase
     | | | | |  |+- Re: Mathematics, science and Abraham RobinsonFritz Feldhase
     | | | | |  |+* Re: Mathematics, science and Abraham RobinsonRoss A. Finlayson
     | | | | |  ||`- Re: Mathematics, science and Abraham Robinsonsergi o
     | | | | |  |+- Re: Mathematics, science and Abraham Robinsonzelos...@gmail.com
     | | | | |  |+- Re: Mathematics, science and Abraham Robinsonzelos...@gmail.com
     | | | | |  |+- Re: Mathematics, science and Abraham RobinsonFritz Feldhase
     | | | | |  |+- Re: Mathematics, science and Abraham RobinsonFritz Feldhase
     | | | | |  |+* Re: Mathematics, science and Abraham RobinsonFritz Feldhase
     | | | | |  ||`- Re: Mathematics, science and Abraham Robinsonsergi o
     | | | | |  |+- Re: Mathematics, science and Abraham RobinsonFritz Feldhase
     | | | | |  |+- Re: Mathematics, science and Abraham RobinsonFritz Feldhase
     | | | | |  |`- Re: Mathematics, science and Abraham Robinsonzelos...@gmail.com
     | | | | |  `- Re: Mathematics, science and Abraham RobinsonTimothy Golden
     | | | | `- Re: Mathematics, science and Abraham Robinsonsergi o
     | | | `* Re: Mathematics, science and Abraham Robinsonzelos...@gmail.com
     | | |  `* Re: Mathematics, science and Abraham RobinsonWM
     | | |   +- Re: Mathematics, science and Abraham Robinsonsergi o
     | | |   +* Re: Mathematics, science and Abraham RobinsonFritz Feldhase
     | | |   |`* Re: Mathematics, science and Abraham RobinsonWM
     | | |   | +* Re: Mathematics, science and Abraham Robinsonzelos...@gmail.com
     | | |   | |`* Re: Mathematics, science and Abraham RobinsonWM
     | | |   | | +- Re: Mathematics, science and Abraham Robinsonsergi o
     | | |   | | `- Re: Mathematics, science and Abraham Robinsonzelos...@gmail.com
     | | |   | `- Re: Mathematics, science and Abraham Robinsonsergi o
     | | |   `- Re: Mathematics, science and Abraham Robinsonzelos...@gmail.com
     | | `- Re: Mathematics, science and Abraham RobinsonFritz Feldhase
     | +* Re: Mathematics, science and Abraham Robinsonsergi o
     | `- Re: Mathematics, science and Abraham Robinsonzelos...@gmail.com
     `- Re: Mathematics, science and Abraham Robinsonzelos...@gmail.com

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Re: Mathematics, science and Abraham Robinson

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Subject: Re: Mathematics, science and Abraham Robinson
From: askaske...@gmail.com (WM)
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 by: WM - Sun, 22 May 2022 13:42 UTC

Jim Burns schrieb am Samstag, 21. Mai 2022 um 19:06:27 UTC+2:
> On 5/21/2022 11:30 AM, WM wrote:

> > That is why your k are irrelevant as a
> > vanishing minority.
> The relevance of this whole argument is that
> the k are a vanishing minority which are
> nonetheless enough to match the whole matrix.

The k do not cover the matrix but only one column at the start. During the infinite operation their share cannot grow .
>
> Your counter-argument, if one calls it that, is that
> this obviously can't be true, and, therefore,
> it isn't true.
>
> Your judgment of obvious-falsehood is rooted in
> your experience of _some_ collections.

My judgement is rooted in the fact that an infinite sequence of +1, -1, +1, -1, +1, -1, ... can never, at no finite step and not in the limit either, lead to value larger than 10.

> That sheep-pebbles-sand-stars experience is not
> representative of all collections. Some collections
> can be matched by a vanishing minority. This is
> the point being made by k = (m+n-1)*(m+n-2)/2+m

For that assumption you have to claim that the sequence +1, -1, +1, -1, +1, -1, ... will acumulate infinitely many units. I will not join you. Hardly will any mathematician unless he is told that otherwise set theory would be wrong.

Regards, WM

Re: Mathematics, science and Abraham Robinson

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Subject: Re: Mathematics, science and Abraham Robinson
From: franz.fr...@gmail.com (Fritz Feldhase)
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 by: Fritz Feldhase - Sun, 22 May 2022 15:12 UTC

On Sunday, May 22, 2022 at 3:32:46 PM UTC+2, WM wrote:
> Fritz Feldhase schrieb am Samstag, 21. Mai 2022 um 18:25:13 UTC+2:
> > On Saturday, May 21, 2022 at 5:30:17 PM UTC+2, WM wrote:
> > > Jim Burns schrieb am Freitag, 20. Mai 2022 um 20:22:18 UTC+2:
> > > >
> > > > k = (m + n - 1)(m + n - 2)/2 + m
> > > > there are no _not-indexed_ fractions.
> > > >
> > > But these k don't cover all IN because they have aleph_0 successors in IN.
> >
> > I see. So let's consider the/all k with k e IN.
> >
> > So "these k don't cover all IN because they have aleph_0 successors in IN."
> >
> > (Actually, "these k don't cover all IN because _each and every_ k has aleph_0 successors in IN.")

No reaction/answer?

Ok. So let's consider the following:

> If the fractions m/n are enumerated by the natural numbers k according to Cantor's function
> k = (m + n - 1)(m + n - 2)/2 + m
> then all the fractions of the sequence
> 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, ...
> are enumerated.
>
> [I]f the natural numbers [...] are in bijection with the integer fractions of the first column of the matrix
>
> 1/1, 1/2, 1/3, 1/4, ...
> 2/1, 2/2, 2/3, 2/4, ...
> 3/1, 3/2, 3/3, 3/4, ...
> 4/1, 4/2, 4/3, 4/4, ...
> 5/1, 5/2, 5/3, 5/4, ...
> ...
>
> then they [can] be distributed over the matrix such that no fraction remains without index. That means, there is a [distribution] such that the X of the first column
>
> XOOOO...
> XOOOO...
> XOOOO...
> XOOOO...
> XOOOO...
> ...
>
> [...] cover all matrix positions.

Indeed!

There's a simple distribution of the X to achieve this goal:

> 1/1, 1/2, 1/4, 1/7, ...
> 1/3, 1/5, 1/8, ...
> 1/6, 1/9, ...
> 1/10, ...
> ...

Well done, Mückenheim!

Re: Mathematics, science and Abraham Robinson

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Subject: Re: Mathematics, science and Abraham Robinson
From: timbandt...@gmail.com (Timothy Golden)
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 by: Timothy Golden - Sun, 22 May 2022 16:16 UTC

On Saturday, May 21, 2022 at 5:18:00 PM UTC-4, Ross A. Finlayson wrote:
> On Wednesday, May 18, 2022 at 11:54:33 AM UTC-7, timba...@gmail.com wrote:
> > On Wednesday, May 18, 2022 at 2:32:16 PM UTC-4, WM wrote:
> > > FredJeffries schrieb am Mittwoch, 18. Mai 2022 um 18:14:16 UTC+2:
> > > >
> > > > <quote>
> > > > Robinson emphasized two factors in rejecting his earlier Platonism in favor of a formalist position:
> > > >
> > > > (i) Infinite totalities do not exist in any sense of the word (i.e., either really or ideally). More precisely, any mention, or purported mention, of infinite totalities is, literally, meaningless.
> > > >
> > > > (ii) Nevertheless, we should continue the business of Mathematics 'as usual,' i.e. we should act as if infinite totalities really existed.
> > > > </quote>
> > > Let's continue with the origianl source:
> > >
> > > "I feel quite unable to grasp the idea of an actual infinite totality.. To me there appears to exist an unbridgeable gulf between sets or structures of one, or two, or five elements, on one hand, and infinite structures on the other hand [...] I must regard a theory which refers to an infinite totality as meaningless in the sense that its terms and sentences cannot posses the direct interpretation in an actual structure that we should expect them to have by analogy with concrete (e.g., empirical) situations." [A. Robinson: "Formalism 64", North-Holland, Amsterdam, p. 230f]
> > >
> > > Regards, WM
> > I wonder to what degree those concerned with natural valued infinity are possibly entertaining the birth of the continuum?
> > As you start plopping down positions on a line and they keep going at some large value the discernment of the smaller values seems imperceptible. Particularly thinking in terms of large n a relative position abstraction with arbitrarily fine granularity ensues.
> >
> > This bears out as we consider the decimal value as a natural value. For instance as 1/3= 0.333... then dropping the decimal point we are dealing in a natural value 333.... Even a value such as 2/5 in perfection will yield 0.4000... which again as a natural value is 4000...
> > These sorts of infinite precision values are in denial of epsilon/delta theory, whereas 2/5=0.4000 is a finite precision instance. In other words this is a computationally valid instance and if we did wish to work in greater precision we could. Dropping the decimal from this instance we see 2/5 as 4000 and of course its data can be recovered by building up a structure xe where x is the natural value and e is the decimal place. The 'e' portion is arguably natural valued, but it is of a different meaning. It's position is a matter of placing a secondary unity upon the otherwise purely natural value.
> >
> > In this way the continuum can be built by having a high regard for the natural value; augmenting it with a new sense of unity; and as well keeping a regard for epsilon/delta theory and its adjustable precision. In effect those presuming perfection in their rational values have been working with an infinite form of the natural value and never saw it as ambiguous. Could these be the infinite concerns of the natural valued philosophy? I do credit in part WM's dark number with helping me develop the gray number that is the continuum; the more truly 'real' value. That the interpretation could come back then onto the naturals in this way is an interesting turn. So then are values such as 333... dark values? Hah! I have managed to instantiate them!
>
> "Double or nothing"
>
> It is like physics with the relativist and absolutist again,
> simply mathematics with the regular iteration and regular increment, again.
>
> Then, it really is at least _some_ matters of philosophy,
> that the primary objects of mathematics are defined in
> a usual language of a philosophy of mathematics, because
> mathematical objects fulfill the definition of being mathematical objects..

Yes, well here I think that as you distinguish mathematical objects from philosophy you are implying the existence of some pure form with a mechanism that can be detailed which deserves attention. As to when such a thing becomes 'mathematical'; isn't it a matter of ambiguity? To achieve 'mathematical' implies purity; free of ambiguity. So long as we discover ambiguity within the constructions then this terminology suffers.

I don't generally care to dabble with infinity as a touchable thing. If it doesn't work at ten thousand and work better at a hundred thousand it's probably not going to work at all.

Then too there was a time when these branches were better joined; when philosophy and mathematics and physics all mattered to one mind. This classical mindset is nearly gone. Lopping these branches off from one another is suicidal. It would be wiser for the quantum people to admit that they seek a new philosophy rather than claim immunity from philosophy altogether. Will there be any consequence when the quantum computer never arrives? Never say never... The matter of time shows that we are engaged in a progression but also in the extreme accumulation that we are facing the hope of great simplification should be kept alive. Otherwise we are buried.

>
> So, while there are paradoxes like Cantor's paradox or Kunen's paradox,
> or Russell and Burali-Forti, again here are not (paradoxes) , being explained
> with mathematical logic, why there is the extra-ordinary, and it's quite
> well-phrased already in the usual most technical philosophy,
> for the usual fundamental canon (or inventing one).
>
> Robinson's "infinitesimals" are basically just only the neighborhoods
> about a point. The "hyper-integers" or "hyper-reals", are not much more
> than that after Skolem, having a "non-standard, countable ... model" or
> "standard, countable middle", the integers, those are Skolem's also.

Not so well versed in these people, but how about Dedekinds irrational system applied to all on the continuum?
This is the gray number. This is the engineer's number. The physicists number. Sad that I cannot say that this is the real number.

>
>
> Peano for example has ideas like "these standard infinitesimals are
> like Leibniz' differentials, as far as I'm concerned" - correctly framed.

Re: Mathematics, science and Abraham Robinson

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From: james.g....@att.net (Jim Burns)
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Subject: Re: Mathematics, science and Abraham Robinson
Date: Sun, 22 May 2022 12:45:27 -0400
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 by: Jim Burns - Sun, 22 May 2022 16:45 UTC

On 5/22/2022 9:42 AM, WM wrote:
> Jim Burns schrieb
> am Samstag, 21. Mai 2022 um 19:06:27 UTC+2:

>> That sheep-pebbles-sand-stars experience is not
>> representative of all collections. Some collections
>> can be matched by a vanishing minority. This is
>> the point being made by k = (m+n-1)*(m+n-2)/2+m
>
> For that assumption

I describe a fraction m/n.
That's the assumption:
That the fractions we are talking about are
as I describe them.

_From that description_ I show that k/1
is a vanishing minority of those fractions.
That part is NOT an assumption.

For each BEFORE and AFTER =< m
and each BEFORE and AFTER =< n,
some j ends BEFORE and j+1 begins AFTER.

From that description of each BEFORE and AFTER,
we know that,
if
there are _any_ m and n for which
it's false that,
for each BEFORE and AFTER =< m+n,
some j ends BEFORE and j+1 begins AFTER,
then
there is a _first_ n₁+1 =< n for which
it's false that,
for each BEFORE and AFTER =< m+(n₁+1),
some j ends BEFORE and j+1 begins AFTER.

Because n₁+1 is _first_
we know that
it's true that,
for each BEFORE and AFTER =< m+n₁,
some j ends BEFORE and j+1 begins AFTER.

That leads to a contradiction.
Therefore,
there is no _first_ n₁+1 for which that's false.
Therefore,
there is no n _at all_ for which that's false.
Therefore,
for each BEFORE and AFTER =< m+n,
some j ends BEFORE and j+1 begins AFTER.

And so on. The same argument gets repeated with
minor variations.

> For that assumption you have to claim that the
> sequence +1, -1, +1, -1, +1, -1, ... will acumulate
> infinitely many units. I will not join you. Hardly
> will any mathematician unless he is told that
> otherwise set theory would be wrong.

If,
for each BEFORE and AFTER =< m
and each BEFORE and AFTER =< n,
some j ends BEFORE and j+1 begins AFTER,
and
k = (m+n-1)*(m+n-2)/2+m
then
for each BEFORE and AFTER =< k,
some j ends BEFORE and j+1 begins AFTER.

Re: Mathematics, science and Abraham Robinson

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Subject: Re: Mathematics, science and Abraham Robinson
From: franz.fr...@gmail.com (Fritz Feldhase)
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 by: Fritz Feldhase - Sun, 22 May 2022 17:02 UTC

On Sunday, May 22, 2022 at 5:12:21 PM UTC+2, Fritz Feldhase wrote:
> On Sunday, May 22, 2022 at 3:32:46 PM UTC+2, WM wrote:
> > Fritz Feldhase schrieb am Samstag, 21. Mai 2022 um 18:25:13 UTC+2:
> > > On Saturday, May 21, 2022 at 5:30:17 PM UTC+2, WM wrote:
> > > > Jim Burns schrieb am Freitag, 20. Mai 2022 um 20:22:18 UTC+2:
> > > > >
> > > > > k = (m + n - 1)(m + n - 2)/2 + m
> > > > > there are no _not-indexed_ fractions.
> > > > >
> > > > But these k don't cover all IN because they have aleph_0 successors in IN.
> > >
> > > I see. So let's consider the/all k with k e IN.
> > >
> > > So "these k don't cover all IN because they have aleph_0 successors in IN."
> > >
> > > (Actually, "these k don't cover all IN because _each and every_ k has aleph_0 successors in IN.")
> No reaction/answer?
>
> Ok. So let's consider the following:
> > If the fractions m/n are enumerated by the natural numbers k according to Cantor's function
> > k = (m + n - 1)(m + n - 2)/2 + m
> > then all the fractions of the sequence
> > 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, ...
> > are enumerated.
> >
> > [I]f the natural numbers [...] are in bijection with the integer fractions of the first column of the matrix
> >
> > 1/1, 1/2, 1/3, 1/4, ...
> > 2/1, 2/2, 2/3, 2/4, ...
> > 3/1, 3/2, 3/3, 3/4, ...
> > 4/1, 4/2, 4/3, 4/4, ...
> > 5/1, 5/2, 5/3, 5/4, ...
> > ...
> >
> > then they [can] be distributed over the matrix such that no fraction remains without index. That means, there is a [distribution] such that the X of the first column
> >
> > XOOOO...
> > XOOOO...
> > XOOOO...
> > XOOOO...
> > XOOOO...
> > ...
> >
> > [...] cover all matrix positions.
>
> Indeed!
>
> There's a simple distribution of the X to achieve this goal:
>
> > 1/1, 1/2, 1/4, 1/7, ...
> > 1/3, 1/5, 1/8, ...
> > 1/6, 1/9, ...
> > 1/10, ...
> > ...

Ooops, I meant

> > 1/1, 2/1, 4/1, 7/1, ...
> > 3/1, 5/1, 8/1, ...
> > 6/1, 9/1, ...
> > 10/1, ...
> > ...

> Well done, Mückenheim!

Re: Mathematics, science and Abraham Robinson

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From: james.g....@att.net (Jim Burns)
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Subject: Re: Mathematics, science and Abraham Robinson
Date: Sun, 22 May 2022 14:49:43 -0400
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 by: Jim Burns - Sun, 22 May 2022 18:49 UTC

On 5/22/2022 9:32 AM, WM wrote:
> Fritz Feldhase schrieb
> am Samstag, 21. Mai 2022 um 18:25:13 UTC+2:
>> [...]

> If the fractions m/n are enumerated by
> the natural numbers k according to Cantor's function
> k = (m + n - 1)(m + n - 2)/2 + m
> then all the fractions of the sequence
> 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, ...
> are enumerated.
>
> But if the natural numbers first are in bijection with
> the integer fractions of the first column of the matrix

.... of m/n:k/1

1/1:1/1 1/2:2/1 1/3:4/1 1/4:7/1 ...
2/1:3/1 2/2:5/1 2/3:8/1 2/4:12/1 ...
3/1:6/1 3/2:9/1 3/3:13/1 3/4:18/1 ...
4/1:10/1 4/2:14/1 4/3:19/1 4/4:25/1 ...
5/1:15/1 5/2:20/1 5/3:26/1 5/4:33/1 ...
6/1:21/1 6/2:27/1 6/3:34/1 6/4:42/1 ...
7/1:28/1 7/2:35/1 7/3:43/1 7/4:52/1 ...
8/1:36/1 8/2:44/1 8/3:53/1 8/4:63/1 ...
9/1:45/1 9/2:54/1 9/3:64/1 9/4:75/1 ...
... ... ... ...

> then they must be distributed over the matrix
> such that no fraction remains without index.

Note that no fraction is without index.

> That means, there is a permutation such that
> the X of the first column
> XOOOO...
> XOOOO...
> XOOOO...
> XOOOO...
> XOOOO...
> ...
>
> by being exchanged with the O's cover all
> matrix positions.

Yes.

> All O's will vanish.

Yes.

> This is obviously impossible

No. See above.

> This is obviously impossible because exchanging
> cannot reduce them.

Exchanging does not reduce them.
They are not reduced.
They are the same size (cardinality) as
one of their proper subsets.
See above.

Some collections can be the same size (cardinality)
as one of their proper subsets.
For example, see above.

> The number of not indexed fractions,

No fraction is without index.
See above.

> The number of not indexed fractions,
> represented by O's, will remain constant forever,
> in infinity.

Re: Mathematics, science and Abraham Robinson

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 by: zelos...@gmail.com - Mon, 23 May 2022 05:06 UTC

torsdag 19 maj 2022 kl. 15:44:32 UTC+2 skrev WM:
> Jim Burns schrieb am Donnerstag, 19. Mai 2022 um 02:30:59 UTC+2:
> > On 5/18/2022 4:48 PM, WM wrote:
>
> > > Between every number subject to induction and omega
> > > there are aleph_0 natural numbers which are not
> > > subject to induction because they do never vanish.
> > You are facing in the wrong direction.
> So it appears to someone who never faced in the right direction. What is wrong with my direction? (Except that it disproves your beliefs.)
> > Stand on a number k finitely-many '+1" from 0.
> > Turn away from omega and look at 0.
> >
> > k is subject to induction *because*
> > k is finitely-many '+1' from 0.
> That does not exclude aleph_0 dark numbers immediately before omega.

All of which you fail to define in any meaningful way.

> >
> > Infinity enters the story because there are
> > infinitely-many numbers which are finitely-many '+1'
> > from 0.
> Impossible if actual infinity is meant, that is *more* than any finite number of +1.
>
> Every finite number of +1 is contained in a potentially infinite set which is followed by an infinite set - if infinity is actual.

there is no "potential infinite set", that is not a defined thing in mathematics.

>
> Regards, WM

Re: Mathematics, science and Abraham Robinson

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Subject: Re: Mathematics, science and Abraham Robinson
From: askaske...@gmail.com (WM)
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 by: WM - Mon, 23 May 2022 12:13 UTC

Fritz Feldhase schrieb am Sonntag, 22. Mai 2022 um 17:12:21 UTC+2:
> On Sunday, May 22, 2022 at 3:32:46 PM UTC+2, WM wrote:
That means, there is a [distribution] such that the X of the first column
> >
> > XOOOO...
> > XOOOO...
> > XOOOO...
> > XOOOO...
> > XOOOO...
> > ...
> >
> > [...] cover all matrix positions.
>
> Indeed!
>
> There's a simple distribution of the X to achieve this goal:
>
> > 1/1, 1/2, 1/4, 1/7, ...
> > 1/3, 1/5, 1/8, ...
> > 1/6, 1/9, ...
> > 1/10, ...
> > ...
>
Please explain why the sequence -1 + 1 - 1 + 1 - 1 + 1 -+ has sum oo or write down the complete matrix in order to prove your claim.

Regards, WM

Re: Mathematics, science and Abraham Robinson

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Subject: Re: Mathematics, science and Abraham Robinson
From: askaske...@gmail.com (WM)
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 by: WM - Mon, 23 May 2022 12:16 UTC

Fritz Feldhase schrieb am Sonntag, 22. Mai 2022 um 17:19:30 UTC+2:
> On Sunday, May 22, 2022 at 3:42:20 PM UTC+2, WM wrote:
>
> > My judgement is rooted in the fact that an infinite sequence of +1, -1, +1, -1, +1, -1, ... can never, at no finite step and not in the limit either

> Since all terms in this sequence are either 1 or -1, no term is > 10. WOW!
>
This sequence yields the share of fractions enumerated in addition to the outset.

Regards, WM

Re: Mathematics, science and Abraham Robinson

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Subject: Re: Mathematics, science and Abraham Robinson
From: askaske...@gmail.com (WM)
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 by: WM - Mon, 23 May 2022 12:20 UTC

Jim Burns schrieb am Sonntag, 22. Mai 2022 um 18:45:38 UTC+2:
> On 5/22/2022 9:42 AM, WM wrote:

> And so on. The same argument gets repeated with
> minor variations.
> > For that assumption you have to claim that the
> > sequence +1, -1, +1, -1, +1, -1, ... will acumulate
> > infinitely many units. I will not join you. Hardly
> > will any mathematician unless he is told that
> > otherwise set theory would be wrong.
> If,
> for each BEFORE and AFTER =< m

There is no relevance of BEFORE and AFTER. There is relevance only of the fact that every indexed fraction can get an index only from a previously indexd fraction. And previously most fractions were not indexed.

Regards, WM

Re: Mathematics, science and Abraham Robinson

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Subject: Re: Mathematics, science and Abraham Robinson
From: askaske...@gmail.com (WM)
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 by: WM - Mon, 23 May 2022 12:23 UTC

Jim Burns schrieb am Sonntag, 22. Mai 2022 um 20:49:56 UTC+2:
> On 5/22/2022 9:32 AM, WM wrote:

> > That means, there is a permutation such that
> > the X of the first column
> > XOOOO...
> > XOOOO...
> > XOOOO...
> > XOOOO...
> > XOOOO...
> > ...
> >
> > by being exchanged with the O's cover all
> > matrix positions.
> Yes.

By exchanging X and O the shares will remain constant.
I am not willing to discuss with stubborn stupidity.
Come back when you will have recovered.
EOD.

Regards, WM

Re: Mathematics, science and Abraham Robinson

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Subject: Re: Mathematics, science and Abraham Robinson
From: askaske...@gmail.com (WM)
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 by: WM - Mon, 23 May 2022 12:25 UTC

zelos...@gmail.com schrieb am Montag, 23. Mai 2022 um 07:06:09 UTC+2:
> torsdag 19 maj 2022 kl. 15:44:32 UTC+2 skrev WM:

> > > k is subject to induction *because*
> > > k is finitely-many '+1' from 0.
> > That does not exclude aleph_0 dark numbers immediately before omega.
> All of which you fail to define in any meaningful way.

All natnumbers which you can define have an infinite distance from omega. This distance is existing but undefinable.

Regards, WM

Re: Mathematics, science and Abraham Robinson

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 by: FromTheRafters - Mon, 23 May 2022 12:29 UTC

WM was thinking very hard :
> Jim Burns schrieb am Sonntag, 22. Mai 2022 um 20:49:56 UTC+2:
>> On 5/22/2022 9:32 AM, WM wrote:
>
>>> That means, there is a permutation such that
>>> the X of the first column
>>> XOOOO...
>>> XOOOO...
>>> XOOOO...
>>> XOOOO...
>>> XOOOO...
>>> ...
>>>
>>> by being exchanged with the O's cover all
>>> matrix positions.
>> Yes.
>
> By exchanging X and O the shares will remain constant.
> I am not willing to discuss with stubborn stupidity.

Decades prove otherwise.

Re: Mathematics, science and Abraham Robinson

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Subject: Re: Mathematics, science and Abraham Robinson
From: timbandt...@gmail.com (Timothy Golden)
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 by: Timothy Golden - Mon, 23 May 2022 13:47 UTC

On Monday, May 23, 2022 at 8:29:49 AM UTC-4, FromTheRafters wrote:
> WM was thinking very hard :
> > Jim Burns schrieb am Sonntag, 22. Mai 2022 um 20:49:56 UTC+2:
> >> On 5/22/2022 9:32 AM, WM wrote:
> >
> >>> That means, there is a permutation such that
> >>> the X of the first column
> >>> XOOOO...
> >>> XOOOO...
> >>> XOOOO...
> >>> XOOOO...
> >>> XOOOO...
> >>> ...
> >>>
> >>> by being exchanged with the O's cover all
> >>> matrix positions.
> >> Yes.
> >
> > By exchanging X and O the shares will remain constant.
> > I am not willing to discuss with stubborn stupidity.
> Decades prove otherwise.

Doesn't the algebraic index convert back to a one dimensional system?
Is there some consequence to requiring a two dimensional format?
Meanwhile we all must admit that there is a clean ordering of these supposed continuous values that fits them on a line, and this perfect order cannot be attained by your works. This is because the finest gradation is at large value (denominator). Where we do discover clean order is simply choosing a large denominator and exposing how many of these early values can be covered by it. The willingness to play around with infinities here suggests you are comfortable specifying this value. Is it merely a matter of specifying this odd algorithm and enlisting induction through it? So for any d you choose I can fit it and all predecessors on some large base b. Spec the algorithm, and invoke induction as d+1?

Nah. I've got a better way. Same old method used on some past thread. Each line is worked modulo n so the first fraction line becomes
0.1, 0.1, 0.1, 0.1, 0.1, ...
where the first is modulo-one, the second is modulo-two, and so forth. The next line is just twice the first line
Each line is merely the sum of this first course with the previous course. The nth line is simply the sum from 1 to n of 0.1 with some subscript say to denote the modulo nature, but we now have a simplified expression. In effect the entire construction collapses to a constant notation.
Sigma of 0.1 basically.

No idea how this format is convincing to anybody. What matters to me is to expose that the decimal value as in with the usage of the decimal point is in a class of its own. The claim that it is a rational value is a fraud that is gotten by building the rational value first and calling it fundamental.. It is not. It contains an operator and therefor is not fundamental. Upon computation we generally land with the decimal point usage, and it's structure stands out as a new format of number. The disregard of the floating point value by the mathematicians is probably the second worst misnomer that status quo mathematics sits upon; the first being sign.

As the rational value in general is exposed as a reradixed system you all have to be labelled as dirty reradixers. That epsilon/delta theory has something to say here, and particularly when applied universally to the floating point value, then a different interpretation of number ensues. The most stubborn point in it all is that the magic and supremacy of your natural value as fundamental actually returns into the continuum. It is just that the continuum is gray, whereas the naturals are black on white. The en masse delusion is secure obviously and has been far longer than formalizations of the real number have been around. Indeed, as to which came first arguably it was the continuum; and here maybe lays some deep physical correspondence that we don't have yet. In effect mathematical theory does ask for spacetime correspondence, for all of our symbolic works have been carried out in a medium and via techniques whose side-effects may be consequential. It is this draw down to a blank slate and its early beginnings where simplicity lays. She barely awoke and the accumulation began. It might as well be true that our intelligences are actually diminishing as modern society has polluted us with lead and so forth. Perhaps the coming generation will be better off; some of them hopefully. Too many though are coming out troubled as I see it. Be glad if you missed out on the fossil fuel pigging, if you were so lucky. You likely have more skills and strength than the men who push hydraulic control levers and get things done around here. Inhale those diesel fumes and you know something is wrong. How hidden away could it be? Could this explain the need to control so much? I can't help but feel that something is this deeply wrong with our culture. The level of absurdity that we've come to has to break. Skepticism and gullibility have to be refreshed as the integrity of the system takes a dive so deep.

Re: Mathematics, science and Abraham Robinson

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 by: sergi o - Mon, 23 May 2022 14:21 UTC

On 5/23/2022 7:29 AM, FromTheRafters wrote:
> WM was thinking very hard :
>> Jim Burns schrieb am Sonntag, 22. Mai 2022 um 20:49:56 UTC+2:
>>> On 5/22/2022 9:32 AM, WM wrote:
>>
>>>> That means, there is a permutation such that the X of the first column XOOOO... XOOOO... XOOOO... XOOOO... XOOOO... ...
>>>> by being exchanged with the O's cover all matrix positions.
>>> Yes.
>>
>> By exchanging X and O the shares will remain constant.
>> I am not willing to discuss with stubborn stupidity.
>
> Decades prove otherwise.

agree

Re: Mathematics, science and Abraham Robinson

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Subject: Re: Mathematics, science and Abraham Robinson
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 by: sergi o - Mon, 23 May 2022 14:22 UTC

On 5/23/2022 7:16 AM, WM wrote:
> Fritz Feldhase schrieb am Sonntag, 22. Mai 2022 um 17:19:30 UTC+2:
>> On Sunday, May 22, 2022 at 3:42:20 PM UTC+2, WM wrote:
>>
>>> My judgement is rooted in the fact that an infinite sequence of +1, -1, +1, -1, +1, -1, ... can never, at no finite step and not in the limit either
>
>> Since all terms in this sequence are either 1 or -1, no term is > 10. WOW!
>>
> This sequence yields the share of fractions enumerated in addition to the outset.
>
> Regards, WM

no, it does not. you are stubbornly stupid.

Re: Mathematics, science and Abraham Robinson

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 by: sergi o - Mon, 23 May 2022 14:23 UTC

On 5/23/2022 7:20 AM, WM wrote:
> Jim Burns schrieb am Sonntag, 22. Mai 2022 um 18:45:38 UTC+2:
>> On 5/22/2022 9:42 AM, WM wrote:
>
>> And so on. The same argument gets repeated with
>> minor variations.
>>> For that assumption you have to claim that the
>>> sequence +1, -1, +1, -1, +1, -1, ... will acumulate
>>> infinitely many units. I will not join you. Hardly
>>> will any mathematician unless he is told that
>>> otherwise set theory would be wrong.
>> If,
>> for each BEFORE and AFTER =< m
>
> There is no relevance of BEFORE and AFTER. There is relevance only of the fact that every indexed fraction can get an index only from a previously indexd fraction.

liar.

And previously most fractions were not indexed.

liar.

>
> Regards, WM

Re: Mathematics, science and Abraham Robinson

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 by: sergi o - Mon, 23 May 2022 14:24 UTC

On 5/23/2022 7:25 AM, WM wrote:
> zelos...@gmail.com schrieb am Montag, 23. Mai 2022 um 07:06:09 UTC+2:
>> torsdag 19 maj 2022 kl. 15:44:32 UTC+2 skrev WM:
>
>>>> k is subject to induction *because*
>>>> k is finitely-many '+1' from 0.
>>> That does not exclude aleph_0 dark numbers immediately before omega.
>> All of which you fail to define in any meaningful way.
>
> All natnumbers which you can define have an infinite distance from omega. This distance is existing but undefinable.
>
> Regards, WM

diversion.

Re: Mathematics, science and Abraham Robinson

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Subject: Re: Mathematics, science and Abraham Robinson
From: franz.fr...@gmail.com (Fritz Feldhase)
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 by: Fritz Feldhase - Mon, 23 May 2022 14:57 UTC

On Monday, May 23, 2022 at 2:26:04 PM UTC+2, WM wrote:
> zelos...@gmail.com schrieb am Montag, 23. Mai 2022 um 07:06:09 UTC+2:
>
> All natnumbers [...] have an infinite distance from omega. This distance is existing but undefinable.

Oh, it is "undefinable" but you are still talking about "it"? (Since "it" is existing - somehow?)

Fascinating - not!

Hau ab, Mückenheim!

Re: Mathematics, science and Abraham Robinson

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Subject: Re: Mathematics, science and Abraham Robinson
From: ross.fin...@gmail.com (Ross A. Finlayson)
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 by: Ross A. Finlayson - Mon, 23 May 2022 16:30 UTC

On Sunday, May 22, 2022 at 6:32:46 AM UTC-7, WM wrote:
> Fritz Feldhase schrieb am Samstag, 21. Mai 2022 um 18:25:13 UTC+2:
> > On Saturday, May 21, 2022 at 5:30:17 PM UTC+2, WM wrote:
> > > Jim Burns schrieb am Freitag, 20. Mai 2022 um 20:22:18 UTC+2:
> > > >
> > > > k = (m + n - 1)(m + n - 2)/2 + m
> > > > there are no _not-indexed_ fractions.
> > > >
> > > But these k don't cover all IN because they have aleph_0 successors in IN.
> >
> > I see. So let's consider the/all k with k e IN.
> >
> > So "these k don't cover all IN because they have aleph_0 successors in IN."
> >
> > (Actually, "these k don't cover all IN because _each and every_ k has aleph_0 successors in IN.")
> Here you can learn what the k c an do:
>
> If the fractions m/n are enumerated by the natural numbers k according to Cantor's function
> k = (m + n - 1)(m + n - 2)/2 + m
> then all the fractions of the sequence
> 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, ...
> are enumerated.
>
> But if the natural numbers first are in bijection with the integer fractions of the first column of the matrix
> 1/1, 1/2, 1/3, 1/4, ...
> 2/1, 2/2, 2/3, 2/4, ...
> 3/1, 3/2, 3/3, 3/4, ...
> 4/1, 4/2, 4/3, 4/4, ...
> 5/1, 5/2, 5/3, 5/4, ...
> ...
> then they must be distributed over the matrix such that no fraction remains without index. That means, there is a permutation such that the X of the first column
>
> XOOOO...
> XOOOO...
> XOOOO...
> XOOOO...
> XOOOO...
> ...
>
> by being exchanged with the O's cover all matrix positions. All O's will vanish. This is obviously impossible because exchanging cannot reduce them. The number of not indexed fractions, represented by O's, will remain constant forever, in infinity.
>
> Regards, WM

It's all "resources and rates".

Whatever resources there are
make an automatic machine - ....

Numerical resources, ..., reading and writing them.

Concrete mathematics and finite combinatorics, and
here rates in growing terms, yes make for examples
like "these terms evolve in these rates" why the truncating
part for example, assumes constant-time sorting.

I.e. WM's terms left sitting about have their usual meaning.

Which conscientious mathematicians are expected to "know".

Re: Mathematics, science and Abraham Robinson

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From: inva...@invalid.com (sergi o)
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Subject: Re: Mathematics, science and Abraham Robinson
Date: Mon, 23 May 2022 12:34:16 -0500
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 by: sergi o - Mon, 23 May 2022 17:34 UTC

On 5/23/2022 11:30 AM, Ross A. Finlayson wrote:
> On Sunday, May 22, 2022 at 6:32:46 AM UTC-7, WM wrote:
>> Fritz Feldhase schrieb am Samstag, 21. Mai 2022 um 18:25:13 UTC+2:
>>> On Saturday, May 21, 2022 at 5:30:17 PM UTC+2, WM wrote:
>>>> Jim Burns schrieb am Freitag, 20. Mai 2022 um 20:22:18 UTC+2:
>>>>>
>>>>> k = (m + n - 1)(m + n - 2)/2 + m
>>>>> there are no _not-indexed_ fractions.
>>>>>
>>>> But these k don't cover all IN because they have aleph_0 successors in IN.
>>>
>>> I see. So let's consider the/all k with k e IN.
>>>
>>> So "these k don't cover all IN because they have aleph_0 successors in IN."
>>>
>>> (Actually, "these k don't cover all IN because _each and every_ k has aleph_0 successors in IN.")
>> Here you can learn what the k c an do:
>>
>> If the fractions m/n are enumerated by the natural numbers k according to Cantor's function
>> k = (m + n - 1)(m + n - 2)/2 + m
>> then all the fractions of the sequence
>> 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, ...
>> are enumerated.
>>
>> But if the natural numbers first are in bijection with the integer fractions of the first column of the matrix
>> 1/1, 1/2, 1/3, 1/4, ...
>> 2/1, 2/2, 2/3, 2/4, ...
>> 3/1, 3/2, 3/3, 3/4, ...
>> 4/1, 4/2, 4/3, 4/4, ...
>> 5/1, 5/2, 5/3, 5/4, ...
>> ...
>> then they must be distributed over the matrix such that no fraction remains without index. That means, there is a permutation such that the X of the first column
>>
>> XOOOO...
>> XOOOO...
>> XOOOO...
>> XOOOO...
>> XOOOO...
>> ...
>>
>> by being exchanged with the O's cover all matrix positions. All O's will vanish. This is obviously impossible because exchanging cannot reduce them. The number of not indexed fractions, represented by O's, will remain constant forever, in infinity.
>>
>> Regards, WM
>
> It's all "resources and rates".
>
> Whatever resources there are
> make an automatic machine - ....
>
> Numerical resources, ..., reading and writing them.
>
> Concrete mathematics and finite combinatorics, and
> here rates in growing terms, yes make for examples
> like "these terms evolve in these rates" why the truncating
> part for example, assumes constant-time sorting.
>
> I.e. WM's terms left sitting about have their usual meaning.
>
> Which conscientious mathematicians are expected to "know".

your vagueness is exceeded by lack of resources.

Re: Mathematics, science and Abraham Robinson

<3fa7fdab-0510-c9dd-cf43-674bae5eaf74@att.net>

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From: james.g....@att.net (Jim Burns)
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Subject: Re: Mathematics, science and Abraham Robinson
Date: Mon, 23 May 2022 14:01:39 -0400
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 by: Jim Burns - Mon, 23 May 2022 18:01 UTC

On 5/23/2022 8:20 AM, WM wrote:
> Jim Burns schrieb
> am Sonntag, 22. Mai 2022 um 18:45:38 UTC+2:

>>> For that assumption you have to claim that the
>>> sequence +1, -1, +1, -1, +1, -1, ... will acumulate
>>> infinitely many units. I will not join you. Hardly
>>> will any mathematician unless he is told that
>>> otherwise set theory would be wrong.
>>
>> If,
>> for each BEFORE and AFTER =< m
>
> There is no relevance of BEFORE and AFTER.

For each entry in sequence 1,2,3,4...
for each split BEFORE and AFTER =< that entry,
some j ends BEFORE and j+1 ends AFTER.

That describes each of the entries, rows, and columns
of a matrix in which the first column of X's
can match the whole matrix of X's and O's

X:X O:X O:X O:X ...
X:X O:X O:X O:X ...
X:X O:X O:X O:X ...
X:X O:X O:X O:X ...
X:X O:X O:X O:X ...
X:X O:X O:X O:X ...
X:X O:X O:X O:X ...
X:X O:X O:X O:X ...
X:X O:X O:X O:X ...
.... ... ... ...

This is how they match:

1/1:1/1 1/2:2/1 1/3:4/1 1/4:7/1 ...
2/1:3/1 2/2:5/1 2/3:8/1 2/4:12/1 ...
3/1:6/1 3/2:9/1 3/3:13/1 3/4:18/1 ...
4/1:10/1 4/2:14/1 4/3:19/1 4/4:25/1 ...
5/1:15/1 5/2:20/1 5/3:26/1 5/4:33/1 ...
6/1:21/1 6/2:27/1 6/3:34/1 6/4:42/1 ...
7/1:28/1 7/2:35/1 7/3:43/1 7/4:52/1 ...
8/1:36/1 8/2:44/1 8/3:53/1 8/4:63/1 ...
9/1:45/1 9/2:54/1 9/3:64/1 9/4:75/1 ...
.... ... ... ...

An entry is m/n:k/1 such that
k = (m+n-1)*(m+n-2)/2+m

For each m and n in 1,2,3,4,...
k is in 1,2,3,4,...

This is a consequence of the description I gave of
being in 1,2,3,4,...

For each entry in sequence 1,2,3,4...
for each split BEFORE and AFTER =< that entry,
some j ends BEFORE and j+1 ends AFTER.

> There is relevance only of the fact that
> every indexed fraction can get an index
> only from a previously indexd fraction.

Each indexed fraction gets its index from
the first column.

> And previously most fractions were not indexed.

Each fraction is indexed.

Not all collections have Bob-conservation.

Re: Mathematics, science and Abraham Robinson

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Subject: Re: Mathematics, science and Abraham Robinson
Date: Mon, 23 May 2022 14:13:57 -0400
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 by: Jim Burns - Mon, 23 May 2022 18:13 UTC

On 5/23/2022 8:23 AM, WM wrote:
> Jim Burns schrieb
> am Sonntag, 22. Mai 2022 um 20:49:56 UTC+2:
>> On 5/22/2022 9:32 AM, WM wrote:

>>> That means, there is a permutation such that
>>> the X of the first column

X:X O:X O:X O:X ...
X:X O:X O:X O:X ...
X:X O:X O:X O:X ...
X:X O:X O:X O:X ...
X:X O:X O:X O:X ...
X:X O:X O:X O:X ...
X:X O:X O:X O:X ...
X:X O:X O:X O:X ...
X:X O:X O:X O:X ...
.... ... ... ...

>>> by being exchanged with the O's cover all
>>> matrix positions.
>>
>> Yes.
>
> By exchanging X and O the shares will remain constant.

Not all collections have Bob-conservation.

1/1:1/1 1/2:2/1 1/3:4/1 1/4:7/1 ...
2/1:3/1 2/2:5/1 2/3:8/1 2/4:12/1 ...
3/1:6/1 3/2:9/1 3/3:13/1 3/4:18/1 ...
4/1:10/1 4/2:14/1 4/3:19/1 4/4:25/1 ...
5/1:15/1 5/2:20/1 5/3:26/1 5/4:33/1 ...
6/1:21/1 6/2:27/1 6/3:34/1 6/4:42/1 ...
7/1:28/1 7/2:35/1 7/3:43/1 7/4:52/1 ...
8/1:36/1 8/2:44/1 8/3:53/1 8/4:63/1 ...
9/1:45/1 9/2:54/1 9/3:64/1 9/4:75/1 ...
.... ... ... ...

> I am not willing to discuss with stubborn stupidity.
> Come back when you will have recovered.
> EOD.

Re: Mathematics, science and Abraham Robinson

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Subject: Re: Mathematics, science and Abraham Robinson
Date: Mon, 23 May 2022 14:18:48 -0500
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 by: sergi o - Mon, 23 May 2022 19:18 UTC

On 5/19/2022 1:32 PM, WM wrote:
> Jim Burns schrieb am Donnerstag, 19. Mai 2022 um 19:23:45 UTC+2:
>> On 5/19/2022 9:44 AM, WM wrote:
>
>>> That does not exclude aleph_0 dark numbers
>>> immediately before omega.
>> Both dark numbers and omega are irrelevant to
>> natural-number induction.
>
> Of course. Induction covers only the collection of definable natural numbers.
> That does not exclude aleph_0 dark numbers immediately before omega.
>
>> You (WM) do not use "actual infinity" in the usual way.
>
> I use it in the correct way: A quantity larger than all finite quantities. "In spite of significant difference between the notions of the potential and actual infinite, where the former is a variable finite magnitude, growing above all limits, the latter a constant quantity fixed in itself but beyond all finite magnitudes, it happens deplorably often that the one is confused with the other." [Cantor, p. 374]
>
> But you succeed always to distract from the clear evidence proving dark fractions:
>
> We check the number of indexes by bijecting them with the fractions of the first column (we could use every other column or line as well). When applying the indexes for indexing fractions such that m/n gets the index
> k = (m + n - 1)(m + n - 2)/2 + m
> with the resulting sequence
> 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, ...
> then the integer fractions have to supply these indexes. They are stripped off these indexes. The number of not indexed fractions remains constant.
>
> All definable fractions get indexed. Most fractions don 't get indexed. Dark fractions.
>
> Regards, WM

I am not willing to discuss with stubborn stupidity.

Re: Mathematics, science and Abraham Robinson

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Subject: Re: Mathematics, science and Abraham Robinson
Date: Mon, 23 May 2022 14:19:08 -0500
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 by: sergi o - Mon, 23 May 2022 19:19 UTC

On 5/23/2022 7:13 AM, WM wrote:
> Fritz Feldhase schrieb am Sonntag, 22. Mai 2022 um 17:12:21 UTC+2:
>> On Sunday, May 22, 2022 at 3:32:46 PM UTC+2, WM wrote:
> That means, there is a [distribution] such that the X of the first column
>>>
>>> XOOOO...
>>> XOOOO...
>>> XOOOO...
>>> XOOOO...
>>> XOOOO...
>>> ...
>>>
>>> [...] cover all matrix positions.
>>
>> Indeed!
>>
>> There's a simple distribution of the X to achieve this goal:
>>
>>> 1/1, 1/2, 1/4, 1/7, ...
>>> 1/3, 1/5, 1/8, ...
>>> 1/6, 1/9, ...
>>> 1/10, ...
>>> ...
>>
> Please explain why the sequence -1 + 1 - 1 + 1 - 1 + 1 -+ has sum oo or write down the complete matrix in order to prove your claim.
>
> Regards, WM

I am not willing to discuss with stubborn stupidity.

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