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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
     | | | | |  |||`- 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 RobinsonWM
     | | | | |  ||+* 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: timbandt...@gmail.com (Timothy Golden)
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 by: Timothy Golden - Wed, 18 May 2022 19:58 UTC

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?

>
> > 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: timbandt...@gmail.com (Timothy Golden)
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 by: Timothy Golden - Wed, 18 May 2022 20:14 UTC

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, ...

> >
> > > 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: franz.fr...@gmail.com (Fritz Feldhase)
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 by: Fritz Feldhase - Wed, 18 May 2022 20:35 UTC

On Wednesday, May 18, 2022 at 10:15:00 PM UTC+2, timba...@gmail.com 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...29, 333...30, 333...31, 333...32, 333...33 |split| 333...34, 333...35, 333...36, ...

You are sure that you are talking about _natural_ numbers (i.e. the elements in IN) AND the usual order defined on them?

Say, n < m :<-> Ek e IN \ {0}: n + k = m ,

where + is the usual addition defined on IN.

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 - Wed, 18 May 2022 20:48 UTC

Fritz Feldhase schrieb am Mittwoch, 18. Mai 2022 um 20:56:22 UTC+2:

> Hint: "Mathematical induction is a mathematical proof technique. It is essentially used to prove that a statement P(n) holds for every natural number n = 0, 1, 2, 3, ... ; that is, the overall statement is a sequence of infinitely many cases P(0), P(1), P(2), P(3), ... ." (Wikipedia)

This statement is wrong if there is omega existing. 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.

numbers [that] have ℵo successors before ω [...] belong to a finite set.
>
> Sure, each and every "number" n belongs to the finite set {n}.

If each and every number were incapable of spanning an actually infinite set, there could not exist an actually infinite set. But most natural numbers belong to a subset larger than every FISON.

> On the other hand, each and every number in IN has ℵo successors "before ω" (< ω), but IN is infinite.

Try to find your mistake. Hint: There are not two consecutive aleph_0-sets in IN.

Regards, WM

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Subject: Re: Mathematics, science and Abraham Robinson
From: askaske...@gmail.com (WM)
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 by: WM - Wed, 18 May 2022 20:55 UTC

Fritz Feldhase schrieb am Mittwoch, 18. Mai 2022 um 20:39:58 UTC+2:
> After all, your "business" is crankery and psychotic mumbo-jumbo.

You claim that every natural number has aleph_0 successors between itself and omega.
That implies that no natural number has less successors between itself and omega.
That implies that between all elements of the the set of all natural numbers and omega there are aleph_0 successors.
That implies that between the set of all natural numbers and omega there are aleph_0 successors.

The latter can only be denied by crankery and psychotic mumbo-jumbo. That is your business.

Regards, WM

Re: Mathematics, science and Abraham Robinson

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From: nom...@afraid.org (FromTheRafters)
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Subject: Re: Mathematics, science and Abraham Robinson
Date: Wed, 18 May 2022 13:57:38 -0700
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 by: FromTheRafters - Wed, 18 May 2022 20:57 UTC

Timothy Golden laid this down on his screen :
> 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?

Not even close. :) They are partitioning sets and those aren't even
sets.

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Subject: Re: Mathematics, science and Abraham Robinson
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 by: FromTheRafters - Wed, 18 May 2022 21:16 UTC

WM wrote :
> Fritz Feldhase schrieb am Mittwoch, 18. Mai 2022 um 20:39:58 UTC+2:
>> After all, your "business" is crankery and psychotic mumbo-jumbo.
>
> You claim that every natural number has aleph_0 successors between itself and
> omega.

You claim this. This means that you are accepting the successor
function (each and every element of N has one successor) and induction,
so don't try to disclaim it later.

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 - Wed, 18 May 2022 21:25 UTC

On Wednesday, May 18, 2022 at 4:35:09 PM UTC-4, Fritz Feldhase wrote:
> On Wednesday, May 18, 2022 at 10:15:00 PM UTC+2, timba...@gmail.com 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...29, 333...30, 333...31, 333...32, 333...33 |split| 333...34, 333...35, 333...36, ...
> You are sure that you are talking about _natural_ numbers (i.e. the elements in IN) AND the usual order defined on them?
>
> Say, n < m :<-> Ek e IN \ {0}: n + k = m ,
>
> where + is the usual addition defined on IN.

To be honest with you I suspect that this form will eventually be treated as invalid. This form is already in use though it is not appreciated.
I am referring to the decimal value as when we write:
1/3 = 0.333...
and we observe that the decimal point of this representation may be treated as an augmentation to the natural value. It seems that to make it stick to these infinite forms we should explain its position from the left (high) side of the natural value. This would be to maintain consistency with these existing forms. Please note that the digits themselves haven't a care how small or large they may be. They are each connected to their adjacents in the same way regardless of their size. In this way, and as it is written above the form has been in use for some time already. All that is needed is to appreciate that the decimal point is an addition to the structure of the natural value. It is certainly easily removed. It's replacement is readily coded. In this case we'll specify the form
x e
where x is the natural value and e is the digital position of the decimal point with zero being at the far left of the value. This is of course to be deleted from the format now and we are left to deal with the rational value in natural terms. No dirty reradixers need apply themselves here. Everything is modulo 10, sir. Perfect values...

Of course the farce is with you and ends when you realize that the decimal format is in fact a gray format; in finite terms, where these ellipses will never require usage, we can play out epsilon/delta theory on any value merely by chasing digits. At least this is how I see it. As we pull the taffy the other way now it does seem to be working out in WM's favor. I guess we have a bifurcation, but I insist that it comes from real analysis. If you allow the ellipsis form 0.333... then you have chosen one side of the bifurcation. If you reject the ellipsis then the continuum goes gray. It's dark numbers, or it's gray numbers.

Oddly the notion of recovering the continuum from the naturals is here as well. As odd, under my interpretation the continuum always takes natural values too... augmented with a secondary unity aka the decimal point.

Could it be of interest that some quite complicated values like
141421356237309504880168872420969807856967187537694807317667973799073247846210703885038753432764157273501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147010955997160597027453459686201472851741864088919860955232923048430871432145083976260362799525140799...

whose square is 2000... are available to work with... if you believe in infinite fairly tales. Remarkably though this is where that epsilon/delta business crept in in the first place. Why it would apply to some values but not to others is problematic.

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Subject: Re: Mathematics, science and Abraham Robinson
From: timbandt...@gmail.com (Timothy Golden)
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 by: Timothy Golden - Wed, 18 May 2022 21:36 UTC

On Wednesday, May 18, 2022 at 4:35:09 PM UTC-4, Fritz Feldhase wrote:
> On Wednesday, May 18, 2022 at 10:15:00 PM UTC+2, timba...@gmail.com 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...29, 333...30, 333...31, 333...32, 333...33 |split| 333...34, 333...35, 333...36, ...
> You are sure that you are talking about _natural_ numbers (i.e. the elements in IN) AND the usual order defined on them?
>
> Say, n < m :<-> Ek e IN \ {0}: n + k = m ,
>
> where + is the usual addition defined on IN.

Sorry I didn't answer your question directly: yes: addition works:
333...29 + 5 = 333...34
333...29 + 333...34 = 666...63

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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 - Wed, 18 May 2022 22:17 UTC

On Wednesday, May 18, 2022 at 11:36:34 PM UTC+2, timba...@gmail.com wrote:

> Sorry I didn't answer your question directly: yes: addition works:

> 333...29 + 5 = 333...34
> 333...29 + 333...34 = 666...63

You know there's a smallest natural number, usually called 0 (or 1).

Which of your symbols donotes 0 (or 1)?

And if we are already at that, if X denotes the natural number n which symbol denotes the natural number n+1?

You see there's a system of names which is called "unary"?

In this system 1 is denoted by the symbol "|" and if n is denoted by the symbol X, then n+1 is denoted by the symbol X appended by the symbol "|".

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 - Wed, 18 May 2022 22:54 UTC

On Wednesday, May 18, 2022 at 6:17:15 PM UTC-4, Fritz Feldhase wrote:
> On Wednesday, May 18, 2022 at 11:36:34 PM UTC+2, timba...@gmail.com wrote:
>
> > Sorry I didn't answer your question directly: yes: addition works:
>
> > 333...29 + 5 = 333...34
> > 333...29 + 333...34 = 666...63
> You know there's a smallest natural number, usually called 0 (or 1).
>
> Which of your symbols donotes 0 (or 1)?
>
> And if we are already at that, if X denotes the natural number n which symbol denotes the natural number n+1?
>
> You see there's a system of names which is called "unary"?
>
> In this system 1 is denoted by the symbol "|" and if n is denoted by the symbol X, then n+1 is denoted by the symbol X appended by the symbol "|".

I don't get your nonsense. Let n = 333...34. then n+1 is 333...35. zero is still zero and one is still one. Already at what? we're discussing WM's dark numbers, roughly. As you dodge the usage of the ellipsis and play dumb, the rational value 1/3 = 0.333... is a similar instance. I've explained my analysis carefully enough. You talk past it as if it hasn't occurred; as if what you see is nonsense. It's for you to falsify, sir. Just talk straight, please.

Re: Mathematics, science and Abraham Robinson

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 by: sergi o - Thu, 19 May 2022 00:03 UTC

On 5/18/2022 3:55 PM, WM wrote:
> Fritz Feldhase schrieb am Mittwoch, 18. Mai 2022 um 20:39:58 UTC+2:
>> After all, your "business" is crankery and psychotic mumbo-jumbo.
>
> You claim that every natural number has aleph_0 successors between itself and omega.

ok

> That implies that no natural number has less successors between itself and omega.

so you state 17 has as many successors as 100^100^100 ?

> That implies that between all elements of the the set of all natural numbers and omega there are aleph_0 successors.

Wrong. try re-wording that.

> That implies that between the set of all natural numbers and omega there are aleph_0 successors.

Wrong. The set is not a natural number, and neither is omega.

>
> The latter can only be denied by crankery and psychotic mumbo-jumbo. That is your business.

you are the one with the wrong implied mumbo-jumbo, but you are crank, and that is your business here.

>
> Regards, WM
>

Re: Mathematics, science and Abraham Robinson

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 by: sergi o - Thu, 19 May 2022 00:09 UTC

On 5/18/2022 3:48 PM, WM wrote:
> Fritz Feldhase schrieb am Mittwoch, 18. Mai 2022 um 20:56:22 UTC+2:
>
>> Hint: "Mathematical induction is a mathematical proof technique. It is essentially used to prove that a statement P(n) holds for every natural number n = 0, 1, 2, 3, ... ; that is, the overall statement is a sequence of infinitely many cases P(0), P(1), P(2), P(3), ... ." (Wikipedia)
>
> This statement is wrong if there is omega existing.

wrong, omega is not needed.

> 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.

"vanish" is not a mathematical term. Omega is diversion.

>
> numbers [that] have ℵo successors before ω [...] belong to a finite set.

wrong, no set is specified.

>>
>> Sure, each and every "number" n belongs to the finite set {n}.
>
> If each and every number were incapable of spanning an actually infinite set,

So, how does the element 17 span an infinite set ?

> there could not exist an actually infinite set.

hokie-pokie

> But most natural numbers belong to a subset larger than every FISON.

baloney, prove it. you are spouting nonsense.
use equations

>
>> On the other hand, each and every number in IN has ℵo successors "before ω" (< ω), but IN is infinite.
>
> Try to find your mistake. Hint: There are not two consecutive aleph_0-sets in IN.

another red herring.

>
> 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 - Thu, 19 May 2022 00:22 UTC

On Thursday, May 19, 2022 at 12:54:22 AM UTC+2, timba...@gmail.com wrote:

> the rational value 1/3 = 0.333... is a similar instance.

Fuck off, you silly troll.

Hint: We were talking about NATURAL NUMBERS! GOT THAT, you silly asshole full of shit?!

*PLONK*

Re: Mathematics, science and Abraham Robinson

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 by: Jim Burns - Thu, 19 May 2022 00:30 UTC

On 5/18/2022 4:48 PM, WM wrote:
> Fritz Feldhase schrieb
> am Mittwoch, 18. Mai 2022 um 20:56:22 UTC+2:

>> Hint:
>> "Mathematical induction is a mathematical proof technique.
>> It is essentially used to prove that a statement P(n)
>> holds for every natural number n = 0, 1, 2, 3, ... ;
>> that is, the overall statement is a sequence of
>> infinitely many cases P(0), P(1), P(2), P(3), ... ."
>> (Wikipedia)
>
> This statement is wrong if there is omega existing.

No.

> 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.
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.

If the truth-value of P(i) changes, in going from P(0)
to ~P(k), then, somewhere, the truth-value must change
in a single step, in going from P(j) to ~P(j+1).
[1]

If the predicate P(i) does not permit the truth-value
to change in a single step, then the truth-value
cannot change in finitely-many steps, either.

This is natural-number, finitely-many-+1 induction.
As I've stated it here, induction says
| If k exists such that P(0) and ~P(k),
| then j exists such that P(j) and ~P(j+1)

That is equivalent to the usual formulation
| If P(0) and, for all j, P(j) -> P(j+1),
| then, for all k, P(k)

Infinity enters the story because there are
infinitely-many numbers which are finitely-many '+1'
from 0.

[1]
If the truth-value of P(i) changes, in going from P(0)
to ~P(k), then, somewhere, the truth-value must change
in a single step, in going from P(j) to ~P(j+1).

Assume that P(0) and ~P(k), and
k is finitely-many '+1" from 0.

Define _finitely-many_ to mean that,
for each split BEFORE and AFTER of ⟨0,...,k⟩
some j ends BEFORE and j+1 begins AFTER.

Define AFTERnot to be the collection of all i in ⟨0,...,k⟩
such that ~P(i) or i > i' such that ~P(i').

Define BEFOREnot to be the collection of all i in ⟨0,...,k⟩
such that i is not in AFTERnot.

P(0), so BEFOREnot is not empty.
~P(k), so AFTERnot is not empty.

BEFOREnot ∪ AFTERnot = ⟨0,...,k⟩
Each of BEFOREnot is before each of AFTERnot.

Therefore, BEFOREnot and AFTERnot is a split of ⟨0,...,k⟩

_Because k is finite_
some j ends BEFOREnot and j+1 begins AFTERnot.

Given the definitions of BEFOREnot and AFTERnot,

any i in ⟨0,...,k⟩ for which ~P(i) is in AFTERnot
j is in BEFOREnot
P(j)

j+1 is first in AFTERnot.
Either ~P(j+1)
or, for some i' in BEFOREnot, ~P(i')
However,
there is no i' in BEFOREnot, ~P(i')
~P(j+1)

Therefore,
P(j) and ~P(j+1)

Re: Mathematics, science and Abraham Robinson

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 by: Fritz Feldhase - Thu, 19 May 2022 00:50 UTC

On Wednesday, May 18, 2022 at 10:48:58 PM UTC+2, WM wrote:
> Fritz Feldhase schrieb am Mittwoch, 18. Mai 2022 um 20:56:22 UTC+2:
> >
> > Hint: "Mathematical induction is a mathematical proof technique. It is essentially used to prove that a statement P(n) holds for every natural number n = 0, 1, 2, 3, ..."
> >
> wrong [...]. Between every number subject to induction and omega there are [...] natural numbers which are not subject to induction

Fascinating, you psychotic asshole full of shit.

So in your psychotic kingdom

P(0) & An(n e IN & P(n) -> P(n+1))

does NOT imply

An e IN: P(n) ?

Hint: An e IN: P(n) means that THERE IS NO natural number n such that P(n) does NOT HOLD.

Fuck off, you silly crank!

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 - Thu, 19 May 2022 00:54 UTC

On Thursday, May 19, 2022 at 2:30:59 AM UTC+2, Jim Burns wrote:

> You are facing in the wrong direction.

Concerning Mückenheim that's _always_ the case.

Re: Mathematics, science and Abraham Robinson

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Subject: Re: Mathematics, science and Abraham Robinson
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 by: sergi o - Thu, 19 May 2022 01:46 UTC

On 5/18/2022 5:54 PM, Timothy Golden wrote:
> On Wednesday, May 18, 2022 at 6:17:15 PM UTC-4, Fritz Feldhase wrote:
>> On Wednesday, May 18, 2022 at 11:36:34 PM UTC+2, timba...@gmail.com wrote:
>>
>>> Sorry I didn't answer your question directly: yes: addition works:
>>
>>> 333...29 + 5 = 333...34
>>> 333...29 + 333...34 = 666...63
>> You know there's a smallest natural number, usually called 0 (or 1).
>>
>> Which of your symbols donotes 0 (or 1)?
>>
>> And if we are already at that, if X denotes the natural number n which symbol denotes the natural number n+1?
>>
>> You see there's a system of names which is called "unary"?
>>
>> In this system 1 is denoted by the symbol "|" and if n is denoted by the symbol X, then n+1 is denoted by the symbol X appended by the symbol "|".
>
> I don't get your nonsense. Let n = 333...34. then n+1 is 333...35. zero is still zero and one is still one. Already at what? we're discussing WM's dark numbers, roughly. As you dodge the usage of the ellipsis and play dumb, the rational value 1/3 = 0.333... is a similar instance. I've explained my analysis carefully enough. You talk past it as if it hasn't occurred; as if what you see is nonsense. It's for you to falsify, sir. Just talk straight, please.
>
>

?

Re: Mathematics, science and Abraham Robinson

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Subject: Re: Mathematics, science and Abraham Robinson
From: zelos.ma...@gmail.com (zelos...@gmail.com)
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 by: zelos...@gmail.com - Thu, 19 May 2022 06:21 UTC

onsdag 18 maj 2022 kl. 14:42:22 UTC+2 skrev WM:
> sobriquet schrieb am Dienstag, 17. Mai 2022 um 19:52:54 UTC+2:
>
> > I see no conceptual difference between the claim that there is no biggest natural number (or biggest prime number) and the claim that there is no smallest number, since we can always add 1 to a number or divide a number by 2.
> >
> > Infinitely subdividing an area of two is equivalent to the claim that there is no biggest finite number of subdivisions beyond which the area we're subdividing would suddenly no longer sum up to two.
> Both supply potential infinity, i.e., numbers growing larger than any given number and (positive) numbers shrinking smaller than any given positive number. But that is not the infinite of set theory, namely actual infinity.
>
> All numbers you get by induction have ℵo successors before ω:
> ∀n ∈ ℕ_ind: |ℕ \ {1, 2, 3, ..., n}| = ℵo .
> They cannot be exhausted, because they remain always there.
>
> But the set of all natural numbers exhausts also these successors:
> |ℕ \ {1, 2, 3, ...}| = 0
> or
> {0, 1, 2, 3, ..., ω} \ ℕ = {0, ω}.
>
> Regards, WM
this is meaningless for what you want to do

Re: Mathematics, science and Abraham Robinson

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 by: zelos...@gmail.com - Thu, 19 May 2022 06:22 UTC

onsdag 18 maj 2022 kl. 20:25:52 UTC+2 skrev WM:
> sobriquet schrieb am Mittwoch, 18. Mai 2022 um 18:25:04 UTC+2:
> > On Wednesday, May 18, 2022 at 2:42:22 PM UTC+2, WM wrote:
> > > sobriquet schrieb am Dienstag, 17. Mai 2022 um 19:52:54 UTC+2:
> > >
> > > > I see no conceptual difference between the claim that there is no biggest natural number (or biggest prime number) and the claim that there is no smallest number, since we can always add 1 to a number or divide a number by 2.
> > > >
> > > > Infinitely subdividing an area of two is equivalent to the claim that there is no biggest finite number of subdivisions beyond which the area we're subdividing would suddenly no longer sum up to two.
> > > Both supply potential infinity, i.e., numbers growing larger than any given number and (positive) numbers shrinking smaller than any given positive number. But that is not the infinite of set theory, namely actual infinity.
> > >
> > > All numbers you get by induction have ℵo successors before ω:
> > > ∀n ∈ ℕ_ind: |ℕ \ {1, 2, 3, ..., n}| = ℵo .
> > > They cannot be exhausted, because they remain always there.
>
> > But then what does it mean when we prove something with mathematical induction?
> We prove it for all definable natural numbers.

your "definable" is MEANINGLESS

> > I would assume that if we proof a property P(n) for natural numbers n with induction, that
> > means that for all natural numbers n, P(n) holds.
> You know that the set of numbers which this is proved for is always finite?
> > But since there is an infinite set of natural numbers, the property has been proven for an actual
> > infinite number of cases.
> By induction we can prove that the numbers reached by induction have ℵo successors before ω.Therefore they belong to a finite set. It is impossible to subdivide ℕ into two consecutive infinite aleph_0-sets.The successor numbers cannot be identified by induction. They can only be used collectively.

False, they belong to N which is not finite.

>
> The set of all natural numbers contains also these successors:
> ℕ \ {1, 2, 3, ...} = { }
> >
> > So if we prove that we can subdivide a given finite area into an arbitrary natural number of
> > parts decreasing in size, and the parts sum up to the original area we started
> > out with, that means we know that the actual infinite sum of subdivided areas decreasing
> > in size indefinitely yields the original area.
> Yes that holds for every definable subdivision into n parts.
>
> 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 - Thu, 19 May 2022 13:32 UTC

On Wednesday, May 18, 2022 at 8:22:28 PM UTC-4, Fritz Feldhase wrote:
> On Thursday, May 19, 2022 at 12:54:22 AM UTC+2, timba...@gmail.com wrote:
>
> > the rational value 1/3 = 0.333... is a similar instance.
> Fuck off, you silly troll.
>
> Hint: We were talking about NATURAL NUMBERS! GOT THAT, you silly asshole full of shit?!
>
> *PLONK*

Wow, that brings passive aggressive to a whole new level!

The values under discussion here are infinite forms right?

Is 333...3 an infinite form? Yes. Are ellipses in use in mathematics? Yes.
Are they used in number theory? Yes.
It happens that the series of digits does not really care what size they are. They act based on their adjacency.
That a decimal point might or might not be added to the structure: it is just this fact that it is an addition to an existing structure and that it has its own distinct character that allows me to provide this working interpretation.
It is readily stated that
333...33 + 1 = 333...34
so I've satisfied your requirement. Now you've completely blown off the argument why exactly?
It seems you can't handle it.
It's pretty clear though what else collapses as you formalize your rejection rather than simply send emotional rejection my way.
Such an emotional man must surely be sensitive enough to this argumentation.
Your rejection is well received, but as to the technical falsification: this is where I care.

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 13:44 UTC

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.
>
> 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.

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 - Thu, 19 May 2022 14:09 UTC

On Wednesday, May 18, 2022 at 6:17:15 PM UTC-4, Fritz Feldhase wrote:
> On Wednesday, May 18, 2022 at 11:36:34 PM UTC+2, timba...@gmail.com wrote:
>
> > Sorry I didn't answer your question directly: yes: addition works:
>
> > 333...29 + 5 = 333...34
> > 333...29 + 333...34 = 666...63
> You know there's a smallest natural number, usually called 0 (or 1).
>
> Which of your symbols donotes 0 (or 1)?

I haven't done anything to the notation. I have no idea where this question comes from.

>
> And if we are already at that, if X denotes the natural number n which symbol denotes the natural number n+1?
>
> You see there's a system of names which is called "unary"?
>
> In this system 1 is denoted by the symbol "|" and if n is denoted by the symbol X, then n+1 is denoted by the symbol X appended by the symbol "|".

I really don't see where you are going with this. I guess you are saying this is the use of IN in the prior text?

A long time ago I did ask for an instance of a dark number from WM. But W was was busy mucking with something.
Now through a long devolution I see it. But my own theory actually rejects the use of the ellipses and lands us on a continuum that always obeys epsilon/delta theory. This is upheld by physical correspondence. That this theory of the continuum now comes back and interacts at the natural value is of interest I believe, for it is as well a part of my theory that we fundamentally represent the continuum with the natural value. It is just that it has to be augmented with a new interpretation of unity.

Simply, and from a fairly bland starting point: observe that the rational value 3/5 is actually two values and an operator. In this way we have just falsified the rational value as fundamental in half a sentence. Furthermore that operator (division) is actually an inverse operator and so is even less fundamental than it already wasn't. Furthermore the notion of closure is nonexistent so long as those two values are treated as naturals. In effect we see a heap of lies that pass as the birthing of the continuum for the status quo.

Now we extend a bit further, realizing that another form of representation exists which does not embed any operator: the decimal value. So we go ahead and actually compute the value 3/5 and we get 0.6! What we see is a new format of numerical representation. Most view this form as if the dot acts as a separator of digits, as if the connection of the digits is somehow broken there, but it is not at all broken there. The chain of digits has not a care as to where that little dot goes. Instead what we see is a mark of unity; a secondary unity that does not even need to challenge the first form of unity which is that of the natural number, and that is all that is left when we remove the little dot known to most in this day as the decimal point.

It's a can of worms that takes many tendrils of interpretation, but the fact that they can even hold up into here; into WM's theory is really quite a strange event imo. The issues of precision that put up a direct wall between the engineers and physicists on the one side and the mathematicians on the other has to be a false wall. By this reasoning the ellipsis form will not be accepted. Accepting a bifurcation here though, and by this allowing in the well accepted
1/3 = 0.333...
it is not much of a stretch to admit that
1/3 = 0.333...33
is it? This is interesting in that numbers are somewhat two-sided entities. But really the magic is done when we pull the decimal point. This is the nature of structured thought which the computer programmer understands well. He who obeys a strict compiler knows that such things must be done. And of course as to what compiler the mathematicians have been obeying... well, it does not look so good and strict through this lens does it?
No. Mathematics is a soft version of compilation and ultimately every check has been done by habituated humans who are passing along their rules under threat of failure; not just making monkeys of yourselves but guaranteeing that your inner monkey is locked down to boot. This is how programmable humans are, and the lack of attention to this detail within mathematics is a horrible abuse, as if to say that philosophy needn't have any part of this cult. Nor need the physicist nor the engineer usher his complaint about the perfection presumed of the rational value upon his continuum... until the hardware computer arises. Then too, jotting down the computations on a piece of paper we see the same thing: the numbers are all natural valued. They may contain a little dot somewhere, and you'd better keep track of it, but that is all.

So we can defend WM's theory now by instantiating constant valued infinite instances. This is something he himself has failed to do.
All of you who are caught in n and never care to instantiate a damn thing are fraught with a sense of superiority that fails you.
This is a big win in the short term here for me. Yet in the long term I do believe that it is the collapse of much of existing mathematical theory.
Whether your sort can think this big; well, no, I suppose not. You all have proven it time and time again. The practice of applying simple logic and falsifying somebody elses application of simple logic simply has not occurred. Slur away, please.

Re: Mathematics, science and Abraham Robinson

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From: nom...@afraid.org (FromTheRafters)
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Subject: Re: Mathematics, science and Abraham Robinson
Date: Thu, 19 May 2022 07:34:39 -0700
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 by: FromTheRafters - Thu, 19 May 2022 14:34 UTC

Timothy Golden laid this down on his screen :
> On Wednesday, May 18, 2022 at 8:22:28 PM UTC-4, Fritz Feldhase wrote:
>> On Thursday, May 19, 2022 at 12:54:22 AM UTC+2, timba...@gmail.com wrote:
>>
>>> the rational value 1/3 = 0.333... is a similar instance.
>> Fuck off, you silly troll.
>>
>> Hint: We were talking about NATURAL NUMBERS! GOT THAT, you silly asshole
>> full of shit?!
>>
>> *PLONK*
>
> Wow, that brings passive aggressive to a whole new level!
>
> The values under discussion here are infinite forms right?
>
> Is 333...3 an infinite form? Yes. Are ellipses in use in mathematics? Yes.
> Are they used in number theory? Yes.

Something similar is used to denote large primes, and it is not an
infinite form used this way.

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: Thu, 19 May 2022 10:15:10 -0500
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 by: sergi o - Thu, 19 May 2022 15:15 UTC

On 5/19/2022 8:32 AM, Timothy Golden wrote:
> On Wednesday, May 18, 2022 at 8:22:28 PM UTC-4, Fritz Feldhase wrote:
>> On Thursday, May 19, 2022 at 12:54:22 AM UTC+2, timba...@gmail.com wrote:
>>
>>> the rational value 1/3 = 0.333... is a similar instance.
>> Fuck off, you silly troll.
>>
>> Hint: We were talking about NATURAL NUMBERS! GOT THAT, you silly asshole full of shit?!
>>
>> *PLONK*
>
> Wow, that brings passive aggressive to a whole new level!
>
> The values under discussion here are infinite forms right?
>
> Is 333...3 an infinite form? Yes. Are ellipses in use in mathematics? Yes.
> Are they used in number theory? Yes.
> It happens that the series of digits does not really care what size they are. They act based on their adjacency.
> That a decimal point might or might not be added to the structure: it is just this fact that it is an addition to an existing structure and that it has its own distinct character that allows me to provide this working interpretation.
> It is readily stated that
> 333...33 + 1 = 333...34
> so I've satisfied your requirement. Now you've completely blown off the argument why exactly?
> It seems you can't handle it.
> It's pretty clear though what else collapses as you formalize your rejection rather than simply send emotional rejection my way.
> Such an emotional man must surely be sensitive enough to this argumentation.
> Your rejection is well received, but as to the technical falsification: this is where I care.

he called you a silly troll, which I disagree with.

But I have to ask, what type of program are you using to generate such large volumes of non-applicable fluff ?

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