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

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* 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
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   |   |`* 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
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     | | | | | `* 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
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     | | | | |  ||| |`* 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 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
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     | | | | |  |`- 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
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     | | |   `- 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

<t65ntn$tu0$1@gioia.aioe.org>

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https://www.novabbs.com/tech/article-flat.php?id=100506&group=sci.math#100506

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From: inva...@invalid.com (sergi o)
Newsgroups: sci.math
Subject: Re: Mathematics, science and Abraham Robinson
Date: Thu, 19 May 2022 10:32:06 -0500
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 by: sergi o - Thu, 19 May 2022 15:32 UTC

On 5/19/2022 8:44 AM, WM wrote:
> 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?

your direction always is 'not math'.

>
>> 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.

there are no dark numbers.

>>
>> 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.

sorry, infinity is bigger than finite.

>
> Every finite number of +1 is contained in a potentially infinite set which is followed by an infinite set - if infinity is actual.

meaningless drivel.

>
> Regards, WM
>

Re: Mathematics, science and Abraham Robinson

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From: james.g....@att.net (Jim Burns)
Newsgroups: sci.math
Subject: Re: Mathematics, science and Abraham Robinson
Date: Thu, 19 May 2022 13:23:33 -0400
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 by: Jim Burns - Thu, 19 May 2022 17:23 UTC

On 5/19/2022 9:44 AM, WM wrote:
> 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.

Both dark numbers and omega are irrelevant to
natural-number induction.
It applies where it applies because,
for each BEFORE and AFTER =< where it applies,
some j ends BEFORE and j+1 begins AFTER.

What excludes dark numbers immediately before omega
is the definition of omega as the first infinite
ordinal.

>> Infinity enters the story because there are
>> infinitely-many numbers which are finitely-many '+1'
>> from 0.
>
> Impossible if actual infinity is meant,

You (WM) do not use "actual infinity" in the usual way.

How you use it is self-contradictory, so, yes,
meaning it that way produces impossibilities.
Of course, that's not what you think you're saying.

> that is *more* than any finite number of +1.

Each finite number is less than
as many as all finite numbers there are.

Re: Mathematics, science and Abraham Robinson

<t65vdf$kae$1@gioia.aioe.org>

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Newsgroups: sci.math
Subject: Re: Mathematics, science and Abraham Robinson
Date: Thu, 19 May 2022 12:39:59 -0500
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 by: sergi o - Thu, 19 May 2022 17:39 UTC

On 5/18/2022 3:14 PM, Timothy Golden wrote:
> On Wednesday, May 18, 2022 at 3:58:31 PM UTC-4, Timothy Golden wrote:
>> On Wednesday, May 18, 2022 at 3:28:27 PM UTC-4, Fritz Feldhase wrote:
>>> On Wednesday, May 18, 2022 at 9:07:55 PM UTC+2, sergi o wrote:
>>>> On 5/18/2022 1:25 PM, WM wrote:
>>>>>
>>>>> It is impossible to subdivide ℕ into two consecutive infinite [...] sets.
>>>
>>> That's indeed true! (Big surprise!)
>> Well, going off of my new interpretation I can split them at:
>> 333...33 versus 333...34.
>> How's that?
>
> More carefully:
> ..., 333...29, 333...30, 333...31, 333...32, 333...33 |split| 333...34, 333...35, 333...36, ...

whoa! you should have used |chop|

>
>>>
>>>> wrong, odd and even...
>>>
>>> Not "consecutive" (though infiite).
>>>
>>> What he's talking about here is a partition of IN into two _infinite_ sets {a_1, a_2, a_3 ...} and {b_1, b_2, b_3, ...} such that
>>>
>>> a_i < b_j, for all i.j e IN.
>>>
>>> (/Partition/ of IN into two sets {a_1, a_2, a_3 ...}, {b_1, b_2, b_3, ...} here means: {a_1, a_2, a_3 ...} =/= {}, {b_1, b_2, b_3, ...} =/= {}, {a_1, a_2, a_3 ...} n {b_1, b_2, b_3, ...} = {} and {a_1, a_2, a_3 ...} u {b_1, b_2, b_3, ...} = IN.)
>>>
>>> Need to see a proof? :-P
>>>
>>>>> ... that holds for every definable subdivision into n parts.
>>>>>
>>>> your "definable" is meaningless, with its beeps, flashes, raps, hoofs, giggles
>>> Indeed. :-)

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 - Thu, 19 May 2022 18:32 UTC

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

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 - Thu, 19 May 2022 19:10 UTC

timba...@gmail.com schrieb am Donnerstag, 19. Mai 2022 um 16:10:04 UTC+2:

> A long time ago I did ask for an instance of a dark number from WM.

Those are number whihc can on,y be defined collectively.

For every definable number we find ℵo successors
∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo
after n and before omega.

For all natural numbers, defined collectively, we find no successors
ℕ \ {1, 2, 3, ...} = { } .

This proves the existence numbers which can only be defined collectively.

Regards, WM

Re: Mathematics, science and Abraham Robinson

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Subject: Re: Mathematics, science and Abraham Robinson
Date: Thu, 19 May 2022 14:29:30 -0500
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 by: sergi o - Thu, 19 May 2022 19:29 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.
>
<snip crap>
>
>> You (WM) do not use "actual infinity" in the usual way.
>
<snip crap>
>
> Regards, WM

Re: Mathematics, science and Abraham Robinson

<t665rm$1n8v$2@gioia.aioe.org>

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Subject: Re: Mathematics, science and Abraham Robinson
Date: Thu, 19 May 2022 14:29:57 -0500
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 by: sergi o - Thu, 19 May 2022 19:29 UTC

On 5/19/2022 2:10 PM, WM wrote:
> timba...@gmail.com schrieb am Donnerstag, 19. Mai 2022 um 16:10:04 UTC+2:
>
>> A long time ago I did ask for an instance of a dark number from WM.
>
<snip crap>
>
> Regards, WM
>
>

Re: Mathematics, science and Abraham Robinson

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From: james.g....@att.net (Jim Burns)
Newsgroups: sci.math
Subject: Re: Mathematics, science and Abraham Robinson
Date: Thu, 19 May 2022 15:45:37 -0400
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 by: Jim Burns - Thu, 19 May 2022 19:45 UTC

On 5/19/2022 2:32 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.

Natural numbers are only definable ==
only finitely-many '+1' from 0 ==
only k such that, for each BEFORE and AFTER =< k,
some j ends BEFORE and j+1 begins AFTER.

This is why induction does not have any exceptions
where it is intended to be applied, in the naturals
== the definable naturals.

> That does not exclude aleph_0 dark numbers
> immediately before omega.

Something else excludes dark numbers immediately
before omega, the definition of omega.

>> You (WM) do not use "actual infinity" in the usual way.
>
> I use it in the correct way:

A WM-potentially-infinite collection can match
a proper subset. It lacks Bob-conservation.

A WM-actually-infinite collection _cannot_ match
a proper subset, but one of its subsets is
WM-potentially-infinite.

There are no WM-actually-infinite collections.

> 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]

A claim about _one of_ a collection which is true
no matter which one of the collection is referred to
can be used to _beat the bounds_ of the collection.

We know that how this method proceeds from one claim
to the next stays inside the bounds of the collection.

We can describe an infinite collection by describing
_one of_ the intended collection and then saying
that's true of each of them.

We can reason about an infinite collection by
reasoning from our description of _one of_ them.

The difference between potential and actual is
a pointless distinction as far as this method is
concerned. We _begin_ complete with respect to
whatever we're talking about, and we _stay_ complete
until we've finished (for now) reasoning.

> But you succeed always to distract from the clear
> evidence proving dark fractions:
....
> All definable fractions get indexed.
> Most fractions don 't get indexed.
> Dark fractions.

All definable fractions get indexed ==
all fractions get indexed.
No dark fractions.

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 - Fri, 20 May 2022 00:35 UTC

On Thursday, May 19, 2022 at 3:11:03 PM UTC-4, WM wrote:
> timba...@gmail.com schrieb am Donnerstag, 19. Mai 2022 um 16:10:04 UTC+2:
>
> > A long time ago I did ask for an instance of a dark number from WM.
> Those are number whihc can on,y be defined collectively.
>
> For every definable number we find ℵo successors
> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo
> after n and before omega.
>
> For all natural numbers, defined collectively, we find no successors
> ℕ \ {1, 2, 3, ...} = { } .
>
> This proves the existence numbers which can only be defined collectively.
>
> Regards, WM

'collectively' is a terminology that seems uncontroversial and acceptable. To what degree we work collectively here on usenet I wouldn't care to say, yet somehow it does hold true. Clearly definitions only work within their collective. If I reject natural numbers then I am outside of the collective that works with natural numbers and so your condition is somewhat a tautological situation.

You've chosen to delete the relevant input of mine here. I suggest that I have instantiated a dark number and you choose to ignore it. This is not a collective situation. Once again I will put the data back in:
333...33, 333...34, 333...35
are three adjacent infinite values aren't they? Is it a collective of three? I suppose. There are issues. Add two of them and land at 666...67 but add all three and land at 1000...02 then add in the first again to get 4333...35? Some of the addition seems to work uncontroversially, but this last bit seems like trouble. Anyway the ability to present a digital representation of your dark numbers does not mean that we should be able to do too much with them. It seems pretty clear that 666...6 is twice as large as 333...3 doesn't it?

And of course these values gain their legitimacy from the old well accepted 1/3=0.333...3.
Well, close enough...
is good enough?

The question of legitimacy of these values I do want to discuss. That these happen to come very close to some of the conditions of dark numbers does seem interesting. Regardless of their darkness they are infinite values. Their legitimacy imo was built when the mathematician accepted 0.3333... as a valid representation. Falsify one and you falsify the other. I actually do accept this falsification but on your side of things it seems as though you'd want to let these large values in. The only difference between the decimal value and the natural value is a little dot in the notation. That little dot is a piece of data that has been augmented to the natural value. Of course I already said all this. It's not so carefully stated here.

Re: Mathematics, science and Abraham Robinson

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Subject: Re: Mathematics, science and Abraham Robinson
From: davidlpe...@gmail.com (David Petry)
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 by: David Petry - Fri, 20 May 2022 02:23 UTC

On Wednesday, May 18, 2022 at 9:14:16 AM UTC-7, FredJeffries wrote:

> <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>

I got the Robinson quotes from a Wikipedia article on "actual infinity". That article didn't include the second paragraph.

What Fred Jeffries wrote in a previous article is almost exactly what he should be saying in response to (ii)

[from Fred Jeffries' previous article]
> Even though you can't do something we can still obtain the result that we would get if we could do it?!

> What kind of silliness is that? How do we KNOW what we would get if we can't do it? Not only can't do it, but have no idea what it would mean to do it?

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 - Fri, 20 May 2022 10:03 UTC

Jim Burns schrieb am Donnerstag, 19. Mai 2022 um 21:45:49 UTC+2:
> On 5/19/2022 2:32 PM, WM wrote:

> > All definable fractions get indexed.
> > Most fractions don 't get indexed.
> > Dark fractions.
> All definable fractions get indexed ==
> all fractions get indexed.
> No dark fractions.

Wo you are not willing to analyse the facts.

Enumerate all positive fractions.

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, ...
....

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 although all definable fractions get indexes. It remains constant in all infinitely many cases because every applied index is taken from an indexed fraction.

What is difficult to understand here?

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 - Fri, 20 May 2022 10:08 UTC

timba...@gmail.com schrieb am Freitag, 20. Mai 2022 um 02:35:39 UTC+2:

> You've chosen to delete the relevant input of mine here. I suggest that I have instantiated a dark number and you choose to ignore it. This is not a collective situation. Once again I will put the data back in:
> 333...33, 333...34, 333...35
> are three adjacent infinite values aren't they?

No. The last digits cannot be known because they would put three dark numbers in an order. Dark numbers however cannot be ordered. This is proved best by dark fractions.

Try to enumerate all positive fractions.

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, ...
....

We check the number of available 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 according to Cantor 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 although all definable fractions get indexes. The number of indexes remains constant in all cases because the newly applied indexes are taken from already indexed fractions.

Regards, WM

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 - Fri, 20 May 2022 12:01 UTC

On Friday, May 20, 2022 at 6:08:15 AM UTC-4, WM wrote:
> timba...@gmail.com schrieb am Freitag, 20. Mai 2022 um 02:35:39 UTC+2:
>
> > You've chosen to delete the relevant input of mine here. I suggest that I have instantiated a dark number and you choose to ignore it. This is not a collective situation. Once again I will put the data back in:
> > 333...33, 333...34, 333...35
> > are three adjacent infinite values aren't they?
> No. The last digits cannot be known because they would put three dark numbers in an order. Dark numbers however cannot be ordered. This is proved best by dark fractions.

Well, perhaps you are wrong.
Here we have some infinite numbers whose last digits are specified. I know because I specified them.
I can at least increment the damn thing and I can do it many times.
I do find it interesting that you lean on the rationals. But they are not the most organized bunch.
By remaining in the modulo 10 system we can keep things simple.
It seems as though you are allowing in this format.
I guess you are saying that these are infinite numbers and that they are adjacent, but that they still are not the dark numbers that you've postulated..
I suppose that is ground I hadn't considered within the structural possibilities. I thought there was a sort of one way or the other sort of positioning that would result. You seem to have found a third rail, so to speak. I'll try to mind the gap.

>
> Try to enumerate all positive fractions.
> 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, ...
> ...
> We check the number of available 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 according to Cantor 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 although all definable fractions get indexes. The number of indexes remains constant in all cases because the newly applied indexes are taken from already indexed fractions.
>
> Regards, WM

So the language of 'already indexed fractions' is ambiguous because there is tremendous redundancy. For instance in your own list 2/2 is an already indexed fraction. It's sort of like having to bring up the concept of equivalence, but it is at such a contorted level that it shouldn't have to be so twisted. The rational number somehow is in the basis of your argumentation here, as it is in mine. I find this aspect peculiar and as well have provided for the falsification of it as a fundamental value. The redundancies that your system include are a fine instance of its failings.

As you've mentioned 'collective' thinking what happens when you claim your collective to be larger than it actually is? Would you say that you have marginalized your own movement? It's like you've built in twitter bots in your list. Literally built them into your organization. If this happens to be true of twitter then their stock will fail. Markets do crash. It certainly does feel like a dark age is coming. How appropriate that the dark number could be a part of the crash. In time, at least. Fake America Fake Again.

Getting to human philosophy, which is something only the orbs can transcend; and certainly they will understand; we see the accumulation of power or control and its preservation as working elements in the academic system of mathematics. That we've all been clucking like chickens for generations... under threat of failure... it is as if anarchy could meet mathematics. I'm afraid we'll have to leave it to the orbs. We'll be following them shortly. Particularly that which the modern mathematician refuses to dismantle even though its structural content is proven to be such that can be dismantled is a failing that places him in the past. It is not so much a fraud as it is that we are at the early beginnings still. We all start as a blank slate. As to what accumulated along the way: the same old slate year after year for how many generations? Out of space yet? Just write over those older portions in a new color. We do face this sort of accumulation. It is overwhelming. I am overwhelmed by it. Further ,I find its accuracy to be dubious. That simplicity can still rule: this is just the sort of awareness that the structural argument rests upon. How odd that your natural value which I've looked askance at all these years rears its head down there. As the natural value develops the continuum in its own terms all that is needed is the secondary unity that the decimal point brings. All this business of endless digits is optional, especially when epsilon/delta shows the way.

Re: Mathematics, science and Abraham Robinson

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 by: sergi o - Fri, 20 May 2022 13:26 UTC

On 5/20/2022 5:03 AM, WM wrote:
> Jim Burns schrieb am Donnerstag, 19. Mai 2022 um 21:45:49 UTC+2:
>> On 5/19/2022 2:32 PM, WM wrote:
>
>>> All definable fractions get indexed.
>>> Most fractions don 't get indexed.
>>> Dark fractions.
>> All definable fractions get indexed ==
>> all fractions get indexed.
>> No dark fractions.
>
> Wo you are not willing to analyse the facts.

they are not facts at all, its your misleading bullshit

>
> Enumerate all positive fractions.
>
> 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, ...
> ...
>
> We check the number of indexes

Wrong.

by bijecting them with the fractions of the first column (we could use every other column or line as well

Wrong.

). When applying the indexes for indexing fractions such that m/n gets the index

wrong.

.................................
We re-index the matrix of rationals using k

> 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, ...

done.
.........................................
> then the integer fractions have to supply these indexes.

wrong.

>They are stripped off these indexes.

Wrong

>The number of not indexed fractions remains constant

Wrong.

>although all definable fractions get indexes.

wrong.

> It remains constant in all infinitely

Wrong.

> many cases because every applied index is taken from an indexed fraction.

wrong.

>
> What is difficult to understand here?

*why do you make so many intentional mistakes to justify your Darkies* ?.

>
> Regards, WM
>

Re: Mathematics, science and Abraham Robinson

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 by: Jim Burns - Fri, 20 May 2022 18:22 UTC

On 5/20/2022 6:03 AM, WM wrote:
> Jim Burns schrieb
> am Donnerstag, 19. Mai 2022 um 21:45:49 UTC+2:
>> On 5/19/2022 2:32 PM, WM wrote:

>>> All definable fractions get indexed.
>>> Most fractions don 't get indexed.
>>> Dark fractions.
>>
>> All definable fractions get indexed ==
>> all fractions get indexed.
>> No dark fractions.
>
> Wo you are not willing to analyse the facts.

"Calculemus." -- Gottfried Leibniz

> Enumerate all positive fractions.

These fractions:
m/n such thatm and n are _definable_ ==
only finite-many '+1' from 0 ==
for each BEFORE and AFTER =< m
and each BEFORE and AFTER =< n,
some j ends BEFORE and j+1 begins AFTER.

> 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, ...
> ...
>
> 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

Calculemus.

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

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

....etc, etc, etc...

....and so,
for each BEFORE and AFTER =< (m+n-1)
and each BEFORE and AFTER =< (m+n-2),
some j ends BEFORE and j+1 begins AFTER.

For each BEFORE and AFTER =< (m+n-1)*(m+n-2),
some j ends BEFORE and j+1 begins AFTER.
because
for each BEFORE and AFTER =< (m+n-1)
and each BEFORE and AFTER =< (m+n-2),
some j ends BEFORE and j+1 begins AFTER.
and,
otherwise, there are contradictions.

| Assume OTHERWISE[1].
| Assume,
| for each BEFORE and AFTER =< (m+n-1)
| and each BEFORE and AFTER =< (m+n-2),
| some j ends BEFORE and j+1 begins AFTER,
| but
| NOT, for each BEFORE and AFTER =< (m+n-1)*(m+n-2),
| some j ends BEFORE and j+1 begins AFTER.
| | There is a _first_ p₁+1 =< (m+n-2) such that
| NOT, for each BEFORE and AFTER =< (m+n-1)*(p₁+1),
| some j ends BEFORE and j+1 begins AFTER,
| AND it is
| TRUE, for each BEFORE and AFTER =< (m+n-1)*p₁,
| some j ends BEFORE and j+1 begins AFTER.
| | By definition of '*',
| (m+n-1)*(p₁+1) = (m+n-1)*p₁+(m+n-1)
| | However,
| for each BEFORE and AFTER =< (m+n-1)*p₁+(m+n-1),
| some j ends BEFORE and j+1 begins AFTER,
| because
| for each BEFORE and AFTER =< (m+n-1)*p₁,
| for each BEFORE and AFTER =< (m+n-1),
| some j ends BEFORE and j+1 begins AFTER,
| and,
| otherwise, there are contradictions.
| || Assume OTHERWISE[2].
|| Assume,
|| for each BEFORE and AFTER =< (m+n-1)*p₁
|| and each BEFORE and AFTER =< (m+n-1),
|| some j ends BEFORE and j+1 begins AFTER,
|| but
|| NOT, for each BEFORE and AFTER =< (m+n-1)*p₁+(m+n-1),
|| some j ends BEFORE and j+1 begins AFTER.
||
|| There is a _first_ s₁+1 =< (m+n-1) such that
|| NOT, for each BEFORE and AFTER =< (m+n-1)*p₁+(s₁+1),
|| some j ends BEFORE and j+1 begins AFTER,
|| AND it is
|| TRUE, for each BEFORE and AFTER =< (m+n-1)*p₁+s₁,
|| some j ends BEFORE and j+1 begins AFTER.
||
|| By definition of '+',
|| (m+n-1)*p₁+(s₁+1) = ((m+n-1)*p₁+s₁)+1
||
|| However,
|| for each BEFORE and AFTER =< ((m+n-1)*p₁+s₁)+1
|| some j ends BEFORE and j+1 begins AFTER,
|| because
|| for each BEFOREless and AFTERless =< (m+n-1)*p₁+s₁,
|| some j ends BEFOREless and j+1 begins AFTERless
||
|| For each BEFORE and AFTER =< ((m+n-1)*p₁+s₁)+1
||
|| either (i)
|| BEFORE = BEFOREless
|| AFTER = AFTERless∪{((m+n-1)*p₁+s₁)+1}
|| and the j which ends BEFOREless ends BEFORE
|| and the j+1 which begins AFTERless begins AFTER
||
|| or (ii)
|| BEFORE = {all =< (m+n-1)*p₁+s₁}
|| AFTER = {((m+n-1)*p₁+s₁)+1}
|| and the j which ends BEFORE = (m+n-1)*p₁+s₁
|| and the j+1 which begins AFTER = ((m+n-1)*p₁+s₁)+1
||
|| CONTRADICTION[2]:
|| NOT, for each BEFORE and AFTER =< (m+n-1)*p₁+(s₁+1),
|| some j ends BEFORE and j+1 begins AFTER,
|| AND it is
|| TRUE, for each BEFORE and AFTER =< (m+n-1)*p₁+(s₁+1),
|| some j ends BEFORE and j+1 begins AFTER.
| | Therefore,
| for each BEFORE and AFTER =< (m+n-1)*p₁+(m+n-1),
| some j ends BEFORE and j+1 begins AFTER.
| and,
| for each BEFORE and AFTER =< (m+n-1)*(p₁+1),
| some j ends BEFORE and j+1 begins AFTER.
| | CONTRADICTION[1]:
| NOT, for each BEFORE and AFTER =< (m+n-1)*(p₁+1),
| some j ends BEFORE and j+1 begins AFTER,
| AND it is
| TRUE, for each BEFORE and AFTER =< (m+n-1)*(p₁+1),
| some j ends BEFORE and j+1 begins AFTER.

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

....etc, etc, etc...

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

Therefore,
k/1 is one of "these fractions",
k is _definable_
k is only finite-many '+1' from 0

> 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.

....which we just now saw that they do.

> They are stripped off these indexes.
> The number of not indexed fractions remains constant

For each indexed fraction,
the number of _indexed_ fractions after it
remains constant and infinite.

For k = (m + n - 1)(m + n - 2)/2 + m
there are no _not-indexed_ fractions.

Infinity is not a reallyreallyreallyreallyreallyreally
large number. It is a different kind of thing.

> although all definable fractions get indexes.

All definable fractions == all fractions.
All fractions get indexes.

> It remains constant in all infinitely many cases
> because every applied index is taken from
> an indexed fraction.
>
> What is difficult to understand here?

My best guess is that you don't understand that
infinity is not a reallyreallyreallyreallyreallyreally
large number. It is a different kind of thing.

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 - Sat, 21 May 2022 15:20 UTC

timba...@gmail.com schrieb am Freitag, 20. Mai 2022 um 14:02:10 UTC+2:
> On Friday, May 20, 2022 at 6:08:15 AM UTC-4, WM wrote:

> Here we have some infinite numbers whose last digits are specified.

But the number of their digits is not specified.

> > Try to enumerate all positive fractions.
> > 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, ...
> > ...
> > We check the number of available 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 according to Cantor 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 although all definable fractions get indexes. The number of indexes remains constant in all cases because the newly applied indexes are taken from already indexed fractions.
> >
> So the language of 'already indexed fractions' is ambiguous because there is tremendous redundancy. For instance in your own list 2/2 is an already indexed fraction.

No, the fractions 1/1 and 2/2 have the same value but are different fractions.

Regards, WM

Re: Mathematics, science and Abraham Robinson

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 by: WM - Sat, 21 May 2022 15:30 UTC

Jim Burns schrieb am Freitag, 20. Mai 2022 um 20:22:18 UTC+2:
> On 5/20/2022 6:03 AM, WM wrote:

> For k = (m + n - 1)(m + n - 2)/2 + m
> there are no _not-indexed_ fractions.

But these k don't cover all |N because they have aleph_0 successors in |N.
>
> Infinity is not a reallyreallyreallyreallyreallyreally
> large number. It is a different kind of thing.

That is why your k are irrelevant as a vanishing minority.

> > every applied index is taken from
> > an indexed fraction.
> >
> > What is difficult to understand here?
> My best guess is that you don't understand that
> infinity is not a reallyreallyreallyreallyreallyreally
> large number. It is a different kind of thing.

When we take an element and throw it away, then we will not collect a set. Do you believe that this will change in infinity?

Regards, WM

Re: Mathematics, science and Abraham Robinson

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 by: sergi o - Sat, 21 May 2022 16:02 UTC

On 5/21/2022 10:20 AM, WM wrote:
> timba...@gmail.com schrieb am Freitag, 20. Mai 2022 um 14:02:10 UTC+2:
>> On Friday, May 20, 2022 at 6:08:15 AM UTC-4, WM wrote:
>
>> Here we have some infinite numbers whose last digits are specified.
>
> But the number of their digits is not specified.
>
>>> Try to enumerate all positive fractions.
>>> 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, ...
>>> ...
>>> We check the number of available 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 according to Cantor 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 although all definable fractions get indexes. The number of indexes remains constant in all cases because the newly applied indexes are taken from already indexed fractions.
>>>
>> So the language of 'already indexed fractions' is ambiguous because there is tremendous redundancy. For instance in your own list 2/2 is an already indexed fraction.

there is infinite redundancy

>
> No, the fractions 1/1 and 2/2 have the same value but are different fractions.
>
> Regards, WM
>

WM remains confused, from above;

integer fractions
indexed fractions
definable fractions
already indexed fractions

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 - Sat, 21 May 2022 16:12 UTC

On Friday, May 20, 2022 at 2:22:18 PM UTC-4, Jim Burns wrote:
> On 5/20/2022 6:03 AM, WM wrote:
> > Jim Burns schrieb
> > am Donnerstag, 19. Mai 2022 um 21:45:49 UTC+2:
> >> On 5/19/2022 2:32 PM, WM wrote:
>
> >>> All definable fractions get indexed.
> >>> Most fractions don 't get indexed.
> >>> Dark fractions.
> >>
> >> All definable fractions get indexed ==
> >> all fractions get indexed.
> >> No dark fractions.
> >
> > Wo you are not willing to analyse the facts.
> "Calculemus." -- Gottfried Leibniz
>
> > Enumerate all positive fractions.
>
> These fractions:
> m/n such thatm and n are _definable_ ==
> only finite-many '+1' from 0 ==
> for each BEFORE and AFTER =< m
> and each BEFORE and AFTER =< n,
> some j ends BEFORE and j+1 begins AFTER.
> > 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, ...
> > ...
> >
> > 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
> Calculemus.
>
> For k = (m + n - 1)(m + n - 2)/2 + m
> for each BEFORE and AFTER =< k,
> some j ends BEFORE and j+1 begins AFTER,
> because...
>

big snip of lots of BEFORE and AFTER

> For each indexed fraction,
> the number of _indexed_ fractions after it
> remains constant and infinite.
>
> For k = (m + n - 1)(m + n - 2)/2 + m
> there are no _not-indexed_ fractions.
>
> Infinity is not a reallyreallyreallyreallyreallyreally
> large number. It is a different kind of thing.
> > although all definable fractions get indexes.
> All definable fractions == all fractions.
> All fractions get indexes.

In this parlor of the infinite the elaborate redundancies that are being worked on get wiped away by some large value under whose rational system all values up though n can be enlisted via multiples of 1/n!.

Then too, this is overgrown since the halves are already covered by the quarters and so the critical value is much smaller than the factorial.
You all would have a better sense of what this critical value is, and it is larger by far than n, but in its simple index all of those priors will be present.
Could this value somehow be of interest? How does it go?
1, 2, 6, 12, 60, 60, 420, 840, 2520, 2520, 27720, 27720,
360360, 360360, 360360, 720720, 12252240 (at 17 now),
12252240, 232792560 (19), 232792560, 232792560,
232792560, 5354228880 (23), ...
It's not pretty. It's more obtuse, but it is another way.

I'm still waiting for more to weigh in on the legitimacy of the large value:
333...35
as a working value. Sure it's large; infinite in nature, but doesn't it help knowing what the last few digits are?
Of course it does. They are critical to the mechanics of computation. Without them the notion of a successor is not going to happen.
I think it is absurd, yet it is as well workable.
It seems according to WM that by instantiating this instance I have sunk his own theory of dark numbers.
I hesitate to make this claim too strongly, and there are bizarre instances of things to do with these large values.
Still there are at least eight of them. That this is a result of the numerical base that we work in means that if you needed more you could make some more as well by cracking that open.

Is it interesting that the carry operation becomes significant in this analysis? Here we see the action of the digits making consequences on the workable values. This is good, but in this depth the question of whether our digital values have operations embedded within them has to be refreshed. It looks as if this carry operator arguable does exist, but this does not legitimate any claim that division exists. Operators and values seem to be distinct concepts by the works of humans. Linguistically this is close by to Chomsky's analysis of language possibly. These are nouns and verbs of the simplest form. As to what it means to have a 'he' who cannot be named specifically... this is problematic. This is the puzzle of instantiation which often enough evaporates the works of mathematicians who are stuck in the abstracted state. That a working instance could evaporate their work is entirely feasible.

Could it be said that these infinite values refuse to multiply? Would that be a problem?
Or is it even fair to vary the head and the tail uniquely so that
567333...36
is as well a valid infinite form? Leaning back on those pesky rationals they do happily instance things such as
1.121212...
and so I guess reusing their logic without any vindictive prejudice we should then allow in
1121212...1216
as a valid infinite value, no?
So then there is no shortage even in base ten.
This would be the permissive strategy I suppose.
Every digit is specified and counts. Therefor these are the best defined numbers in existence.

Sincerely I would think you will have to falsify these or accept them. Each choice is consequential.
Bifurcation.

For me the logic will extend back onto the rationals as they are taught with their digital ellipses.
This is a structural argument. The decimal value is the natural value augmented with a little dot.

> > It remains constant in all infinitely many cases
> > because every applied index is taken from
> > an indexed fraction.
> >
> > What is difficult to understand here?
> My best guess is that you don't understand that
> infinity is not a reallyreallyreallyreallyreallyreally
> large number. It is a different kind of thing.

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 - Sat, 21 May 2022 16:25 UTC

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.")

Yeah, exactly!

I'd like to call this fact the Hauptsatz of Mückenmath!

Re: Mathematics, science and Abraham Robinson

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Subject: Re: Mathematics, science and Abraham Robinson
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 by: Jim Burns - Sat, 21 May 2022 17:06 UTC

On 5/21/2022 11:30 AM, WM wrote:
> Jim Burns schrieb
> am Freitag, 20. Mai 2022 um 20:22:18 UTC+2:

>> For k = (m + n - 1)(m + n - 2)/2 + m
>> there are no _not-indexed_ fractions.
>
> But these k don't cover all |N because
> they have aleph_0 successors in |N.

Describe these fractions m/n.

Describe these k/1 as some, but not all, of the m/n.

From the descriptions, we know that

for each k, there is m/n such that
k = (m+n-1)*(m+n-2)/2+m

and
for each m/n, there is k such that
k = (m+n-1)*(m+n-2)/2+m

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 ...
.... ... ... ...

>> Infinity is not a reallyreallyreallyreallyreallyreally
>> large number. It is a different kind of thing.
>
> 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.

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. Sheep.
Pebbles. Sand grains. Stars.

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

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 - Sat, 21 May 2022 17:15 UTC

On Saturday, May 21, 2022 at 11:20:09 AM UTC-4, WM wrote:
> timba...@gmail.com schrieb am Freitag, 20. Mai 2022 um 14:02:10 UTC+2:
> > On Friday, May 20, 2022 at 6:08:15 AM UTC-4, WM wrote:
>
> > Here we have some infinite numbers whose last digits are specified.
> But the number of their digits is not specified.
> > > Try to enumerate all positive fractions.
> > > 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, ...
> > > ...
> > > We check the number of available 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 according to Cantor 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 although all definable fractions get indexes. The number of indexes remains constant in all cases because the newly applied indexes are taken from already indexed fractions.
> > >
> > So the language of 'already indexed fractions' is ambiguous because there is tremendous redundancy. For instance in your own list 2/2 is an already indexed fraction.
> No, the fractions 1/1 and 2/2 have the same value but are different fractions.
>
> Regards, WM

That's two very weak points, Wolfgang.
Could it be that infinite values are capable of having specific successors and predecessors while their specific ordering is not actually possible?
In other words is it naive to claim that
111...1113 < 222...2
In effect each unique head has a working tail but the uniqueness of the head places it in its own isolated space? This is sort of a dimensional argument, actually. Stating that
111...13 + 1 = 111...14
is not controversial is it? As you claim to be able to discriminate equivalent values perhaps you would care to ponder:
111...13 =?= 111...13
and I see no controversy here. This is a unique representation. It is as well readily evaluable I think as to whether
10111...13 =?= 111...13
where clearly the answer is no, but now as to which value is the lesser: this is why I argue that the heads of these numbers are isolated. We are somewhat establishing some form of dimensionality. Still their tails are operable. I guess it wouldn't be long to start cutting up such values and calling them multidimensional, which will then get us into another ellipses at our whim, just as the first occurred.

Certainly the form is demanding redundancy of digits in the middle otherwise no compaction can occur, which the ellipses develop. In effect all these infinite forms suffer this informational ambiguity... unless you try to let in the irrational instances... something I think I'll try not to consider here.
I think philosophically it would be great to develop a system that denies many usages of the ellipses and treats them as an abusive form that no compiler or procedure can actually handle; under this guise every usage of them is a bug within mathematical thinking. They are used in so many places that I wouldn't care to claim this universally. I do think that this level of discussion is possible within this topic. It is especially thanks to your usage of the rational values that these sorts of things come into being. Again, back in the establishment of the real number it is epsilon/delta that relaxes things. This is a procedure which occurs at the tail of numbers, but not at the tail of natural numbers. No: perfection is intact on these infinite forms. Please tell me a digit that is not specified, and to confront your first fib what infinite value should have finitely many digits? None of them, sir. You have to be engaged in this format.

Re: Mathematics, science and Abraham Robinson

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Subject: Re: Mathematics, science and Abraham Robinson
Date: Sat, 21 May 2022 13:56:53 -0500
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 by: sergi o - Sat, 21 May 2022 18:56 UTC

On 5/21/2022 10:30 AM, WM wrote:
> Jim Burns schrieb am Freitag, 20. Mai 2022 um 20:22:18 UTC+2:
>> On 5/20/2022 6:03 AM, WM wrote:
>
>> For k = (m + n - 1)(m + n - 2)/2 + m
>> there are no _not-indexed_ fractions.
>
> But these k don't cover all |N because they have aleph_0 successors in |N.

wrong!

For k = (m + n - 1)(m + n - 2)/2 + m holds for all m,n and k in |N

"because they have aleph_0 successors in |N" shows *you do not understand equations*!

>>
>> Infinity is not a reallyreallyreallyreallyreallyreally
>> large number. It is a different kind of thing.
>
> That is why your k are irrelevant as a vanishing minority.

you have lost the argument, and are irrelevant.

>
>>> every applied index is taken from
>>> an indexed fraction.
>>>
>>> What is difficult to understand here?
>> My best guess is that you don't understand that
>> infinity is not a reallyreallyreallyreallyreallyreally
>> large number. It is a different kind of thing.
>
> When we take an element and throw it away, then we will not collect a set.

when you take a sheep out of a set of sheeps, you have a different set of sheeps.

you will not collect the set of sheeps, until you pay for them.

> Do you believe that this will change in infinity?

you have no concept of infinity

>
> Regards, WM

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 - Sat, 21 May 2022 21:17 UTC

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.

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.

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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Subject: Re: Mathematics, science and Abraham Robinson
From: askaske...@gmail.com (WM)
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 by: WM - Sun, 22 May 2022 13:32 UTC

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

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