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Re: Natural numbers and vases III

Subject	Author
Natural numbers and vases III	WM
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	sergi o
Re: Natural numbers and vases III	sergi o
Re: Natural numbers and vases III	FromTheRafters
Re: Natural numbers and vases III	sergi o
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	sergi o
Re: Natural numbers and vases III	sergi o
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	sergi o
Re: Natural numbers and vases III	Eram semper recta
Re: Natural numbers and vases III	Eram semper recta
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	sergi o
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	FromTheRafters
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	FromTheRafters
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	FromTheRafters
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	FromTheRafters
Re: Natural numbers and vases III	Surgio
Re: Natural numbers and vases III	Jim Burns
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	Sergio
Re: Natural numbers and vases III	Jim Burns
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	Sergio
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	Sergio
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	Surgio
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	Surgio
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	Sergio
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	Sergio
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	Sergio
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	Sergio
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	Sergio
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	Sergio
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	Sergio
Re: Natural numbers and vases III	Jim Africani
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	Sergio
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	FredJeffries
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	FredJeffries
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	FredJeffries
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Eram semper recta
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Eram semper recta
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Eram semper recta
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	Eram semper recta
Re: Natural numbers and vases III	Eram semper recta
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	Eram semper recta
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	Eram semper recta
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	Gus Gassmann
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	FromTheRafters
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	Fritz Feldhase
Re: Natural numbers and vases III	zelos...@gmail.com
Re: Natural numbers and vases III	Eram semper recta
Re: Natural numbers and vases III	Eduardo Faqtardo
Re: Natural numbers and vases III	Eram semper recta
Re: Natural numbers and vases III	Archimedes Plutonium
Re: Natural numbers and vases III	Archimedes Plutonium

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Re: Natural numbers and vases III

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From: inva...@invalid.com (Sergio)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases III
Date: Mon, 29 Aug 2022 16:08:11 -0500
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by: Sergio - Mon, 29 Aug 2022 21:08 UTC

On 8/29/2022 1:28 PM, Jim Burns wrote:
> On 8/29/2022 12:43 PM, Gus Gassmann wrote:
>> On Monday, 29 August 2022
>> at 12:32:41 UTC-3, Jim Burns wrote:
>
>>> WM seems to at least passively accept
>>> when I translate
>>> "stepwise ordered"
>>> as
>>> "for each split, a step exists with
>>> a last before and a first after".
>>>
>>> It is true of the standard order of ℕ that,
>>> for each split, a step exists with
>>> a last before and a first after.
>>> I would say that makes it a stepwise order.
>>
>> I see. And for the order type
>> {1, 3, 5, ..., 2, 4, 6, ...}
>> (or whatever bracketing is appropriate),
>> there is a split for which this is not the case?
>> But how is that different from a well-order?
>> (Perhaps I am not understanding what
>> you mean by "last before" and "first after".
>> "Last before" what? "First after" what?
>
> Sorry.
> Shorter is not always better.
>
> Last before the split,
> first after the split.
>
> I assume a total order.
>
> A split is a two-piece partition,
> both non-empty,
> with each of one before each of the other.
>
> A step from (last) before the split to
> (first) after the split is required to exist.
>
> If we add that there is a first and a last of
> the whole collection, we've described a finite set
> -- as defined by Paul Stäckel, well-ordered
> in both directions.
>
> I like how this approach appeals to our intuitions.
> If we can only advance step-by-step, for each
> split of the collection, there must be
> a step across that split, or "you can't get
> there from here". I think that it draws near
> to an explanation of what's so special
> about finite set. You _can_ get there from
> here, from anywhere in the set to anywhere else
> in the set.
>
> What I call a _counting-order_ is a stepwise
> order in which the required step is
> an arithmetic successor, i,i+1
>
> For the movement of an O, swapping with
> X's in WM's matrix, the required step p/q,n/1
> uses Cantor's formula n = (p+1-1)(p+q-2)/2+p
>
>> I see. And for the order type
>> {1, 3, 5, ..., 2, 4, 6, ...}
>> (or whatever bracketing is appropriate),
>> there is a split for which this is not the case?
>
> Yes, this is an essential point.
> For some orders, we can't always
> "get there from here".
> We can get from any odd number to any other
> (odd number) (allowing back steps), and
> we can get from any even number to any other
> even number. We can't get from odd numbers
> to even numbers or vice versa.
>
> We can prove we can't get there, even if
> we don't know what is on the other side of
> the split-without-a-step (dark numbers),
> because each step which starts before it
> does not end after it.
>
> My pride and joy:
> a solid proof,
> not so much that dark numbers don't exist,
> who could say that for something so vague,
> but that dark numbers don't appear in
> any of the "paradoxes" that WM presents.
>
> This, despite a near-complete absence of
> information about dark numbers.

dark numbers is oxymoron,
there is no information about dark numbers, except as officially specified in THE ANT LIST 6.0
but one characteristic that does stands out,
they (dark numbers) cannot be separated from each other, so it is a dark glob, which is not numbers....
some believe it is really WMs "dark synapses" which are not connected anymore to his internal grid.

>
>>> On the other hand,
>>> for each split of the numbers from 1 to
>>> Graham's number, there is a last i before
>>> and a first j after, with j = i+1.
>>
>> Ah, OK.
>> But again, how is that different from
>> a well-order (on |N)?
>
> You give a good example, {odds then evens}
> You can't get from the odds to the evens,
> step-by-step.
>
> But {odds then evens} is a well-order.
>
> _For the standard-ordered natural numbers_
> you can always get from here to there,
> step-by-step.
>

Re: Natural numbers and vases III

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Newsgroups: sci.math
Subject: Re: Natural numbers and vases III
Date: Mon, 29 Aug 2022 16:17:10 -0500
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by: Sergio - Mon, 29 Aug 2022 21:17 UTC

On 8/28/2022 8:55 AM, WM wrote:
> Jim Burns schrieb am Samstag, 27. August 2022 um 20:40:21 UTC+2:
>> On 8/27/2022 1:56 PM, Gus Gassmann wrote:
>>> On Saturday, 27 August 2022
>
>> The set of all FISONs contains only FISONs.
>> And so on.
>
> But it is not a set, only a potentially infinite collection.
>
> Regards, WM

wrong. this is a set: The set of all FISONs. Deal with it.

You can't continue to lie your way out of your deception.

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: horand.g...@gmail.com (Gus Gassmann)
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by: Gus Gassmann - Tue, 30 Aug 2022 01:37 UTC

On Monday, 29 August 2022 at 15:28:24 UTC-3, Jim Burns wrote:
> On 8/29/2022 12:43 PM, Gus Gassmann wrote:
> > On Monday, 29 August 2022
> > at 12:32:41 UTC-3, Jim Burns wrote:
>
> >> WM seems to at least passively accept
> >> when I translate
> >> "stepwise ordered"
> >> as
> >> "for each split, a step exists with
> >> a last before and a first after".
> >>
> >> It is true of the standard order of ℕ that,
> >> for each split, a step exists with
> >> a last before and a first after.
> >> I would say that makes it a stepwise order.
> >
> > I see. And for the order type
> > {1, 3, 5, ..., 2, 4, 6, ...}
> > (or whatever bracketing is appropriate),
> > there is a split for which this is not the case?
> > But how is that different from a well-order?
> > (Perhaps I am not understanding what
> > you mean by "last before" and "first after".
> > "Last before" what? "First after" what?
> Sorry.
> Shorter is not always better.
>
> Last before the split,
> first after the split.
>
> I assume a total order.
>
> A split is a two-piece partition,
> both non-empty,
> with each of one before each of the other.
>
> A step from (last) before the split to
> (first) after the split is required to exist.
>
> If we add that there is a first and a last of
> the whole collection, we've described a finite set
> -- as defined by Paul Stäckel, well-ordered
> in both directions.
>
> I like how this approach appeals to our intuitions.
> If we can only advance step-by-step, for each
> split of the collection, there must be
> a step across that split, or "you can't get
> there from here". I think that it draws near
> to an explanation of what's so special
> about finite set. You _can_ get there from
> here, from anywhere in the set to anywhere else
> in the set.
>
> What I call a _counting-order_ is a stepwise
> order in which the required step is
> an arithmetic successor, i,i+1
>
> For the movement of an O, swapping with
> X's in WM's matrix, the required step p/q,n/1
> uses Cantor's formula n = (p+1-1)(p+q-2)/2+p
> > I see. And for the order type
> > {1, 3, 5, ..., 2, 4, 6, ...}
> > (or whatever bracketing is appropriate),
> > there is a split for which this is not the case?
> Yes, this is an essential point.
> For some orders, we can't always
> "get there from here".
> We can get from any odd number to any other
> (odd number) (allowing back steps), and
> we can get from any even number to any other
> even number. We can't get from odd numbers
> to even numbers or vice versa.
>
> We can prove we can't get there, even if
> we don't know what is on the other side of
> the split-without-a-step (dark numbers),
> because each step which starts before it
> does not end after it.
>
> My pride and joy:
> a solid proof,
> not so much that dark numbers don't exist,
> who could say that for something so vague,
> but that dark numbers don't appear in
> any of the "paradoxes" that WM presents.

Months ago I called them "non-instantiated" numbers, which is much more apt and jibes with computing applications. But of course non-instantiated natural numbers are still natural numbers and, as soon as WM figured this out (yes, WM actually figured something out), he stopped using the term.

> This, despite a near-complete absence of
> information about dark numbers.
> >> On the other hand,
> >> for each split of the numbers from 1 to
> >> Graham's number, there is a last i before
> >> and a first j after, with j = i+1.
> >
> > Ah, OK.
> > But again, how is that different from
> > a well-order (on |N)?
> You give a good example, {odds then evens}
> You can't get from the odds to the evens,
> step-by-step.
>
> But {odds then evens} is a well-order.

Ah right! Should have quite while I was ahead. (Good point - wasn't thinking!)

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
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by: Sergio - Tue, 30 Aug 2022 02:17 UTC

On 8/29/2022 4:23 AM, WM wrote:
> Fritz Feldhase schrieb am Sonntag, 28. August 2022 um 18:44:44 UTC+2:
>> On Sunday, August 28, 2022 at 4:24:11 PM UTC+2, WM wrote:
>>> Jim Burns schrieb am Samstag, 27. August 2022 um 20:14:07 UTC+2:
>>>>
>>>> Elements [for] which [there is an endsegment they are not-in]
>>>> are not-in [the] intersection [of all endsegments]
>>
>> Right. All elements in IN are
>>
>>> such elements
>>>> a ∈ A and a ∉ B
>> Especially, n ∈ E(n) and n ∉ E(n+1) for all n e IN.
>
> All endsegments which are there fail to satisfy
> (1) a ∈ A
> (2) b ∈ B
> (3) a ∉ B
> (4) b ∉ A.
> All which fail have an element in common.

So the intersection of all endsegments is empty, as there are no common elements.

>
> Regards, WM

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: zelos.ma...@gmail.com (zelos...@gmail.com)
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by: zelos...@gmail.com - Tue, 30 Aug 2022 04:55 UTC

måndag 29 augusti 2022 kl. 15:27:52 UTC+2 skrev WM:
> Fritz Feldhase schrieb am Montag, 29. August 2022 um 11:56:37 UTC+2:
> > On Monday, August 29, 2022 at 11:29:04 AM UTC+2, WM wrote:
> > > zelos...@gmail.com schrieb am Montag, 29. August 2022 um 07:09:03 UTC+2:
> > > > fredag 26 augusti 2022 kl. 18:35:33 UTC+2 skrev Eram semper recta:
> > > > >
> > > > > Newsflash: Everything in mathematics is a stepwise process. Set theory is not mathematics.
> > Complete nonsense.
> > > All sequences are stepwise processes
> > > and are stepwise analyzable as long as the terms can be individually defined.
> > So what? Who cares?
> Everyone who wants to refute your above "complete nonsense" should care.
> >
> > You know, we usually don't check infinitely many mathematical facts, when doing proofs. Actually, that's why we USE proofs.
> But you are not aware of the fact that dark numbers exist and that your proofs are invalid for dark numbers.
> >
> > We can prove that for all natural numbers n: n + n = 2*n.
> And we can check that for each definable number.
> >
> > We DON'T prove this by explicitly checking that
> >
> > 1 + 1 = 2 = 2*1, 2 + 2 = 4 = 2*2, 3 + 3 = 6 = 2*3, ad infinitum
> No-one does. But we know that for every definable number this can be done.. For undefinable numbers it cannot be done.
> >
> > Seems that you are confusing mathematics with using a pocket calculator..
> The pocket calculator contains more mathematics than all ZFC-matheology.
>
> Regards, WM

You cannot prove that they exist. All your "definable" applies to all natural numbers, making itso your dark shit does not exist.

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: zelos.ma...@gmail.com (zelos...@gmail.com)
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by: zelos...@gmail.com - Tue, 30 Aug 2022 04:55 UTC

måndag 29 augusti 2022 kl. 15:32:58 UTC+2 skrev WM:
> zelos...@gmail.com schrieb am Montag, 29. August 2022 um 12:31:44 UTC+2:
> > måndag 29 augusti 2022 kl. 11:26:27 UTC+2 skrev WM:
>
> > > But that is wrong without an empty endsegment, contradicted by
> > > (1) a ∈ A
> > > (2) b ∈ B
> > > (3) a ∉ B
> > > (4) b ∉ A.
> > > All which fail and are not empty have an element in common.
> > you still do not understand how intersections work
> If they work, they are working, they are processes. Otherwise they could only *be*.
>
> Regards, WM
There are no processes, how hardi s this to understand?

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: askaske...@gmail.com (WM)
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by: WM - Tue, 30 Aug 2022 13:42 UTC

Fritz Feldhase schrieb am Montag, 29. August 2022 um 16:07:22 UTC+2:
> On Monday, August 29, 2022 at 3:27:52 PM UTC+2, WM wrote:
>
> > you are not aware of the fact that dark numbers exist and that your proofs are invalid for dark numbers.
> Yeah, I have to admit that I'm indeed not aware of these "facts".

But you need them to understand set theory correctly.
If no dark numbers would blow up the set ℕ to accomplish
∩{E(k) : k ∈ ℕ} = { }
then only the elements of
∀k ∈ ℕ: ∩{E(1), E(2), ..., E(k)} = E(k) /\ |E(k)| = ℵ₀
would exist and therefore
∩{E(k) : k ∈ ℕ} =/= { } .

Regards, WM

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: askaske...@gmail.com (WM)
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by: WM - Tue, 30 Aug 2022 13:46 UTC

Fritz Feldhase schrieb am Montag, 29. August 2022 um 16:48:12 UTC+2:

> In other words, __the successor of the smallest endsegent in {E(n) | n ∈ ℕ_def}__ does not belong to ∩{E(n) | n ∈ ℕ_def}.

There is no smallest endsegment in {E(n) | n ∈ ℕ_def}.
>
> Hint: If there were no smallest endsegment in {E(n) | n ∈ ℕ_def}, then {E(n) | n ∈ ℕ_def} would be infinite, and hence we would have ∩{E(n) | n ∈ ℕ_def} = { }.

No, that is not caused by infinity. It is caused by an empty endsegment. Du musst noch viele lernen.

Regards, WM

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: franz.fr...@gmail.com (Fritz Feldhase)
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by: Fritz Feldhase - Tue, 30 Aug 2022 14:02 UTC

On Tuesday, August 30, 2022 at 3:46:11 PM UTC+2, WM wrote:
>
> It is caused by an empty endsegment.

An empty endsegment?

Huh?!

No entsegment is empty ***BY DEFINITION***, you silly idiot.

Holy shit! You are dumb like shit!!!

Re: Natural numbers and vases III

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by: WM - Tue, 30 Aug 2022 14:03 UTC

Jim Burns schrieb am Montag, 29. August 2022 um 16:48:13 UTC+2:

> Very closely related to the set of all FISONs,
> there is the set of all FISON-ends.

ℕ_def
>
> However,
> unlike a FISON, the set of all FISON-ends
> does not end anywhere.

With n also n^n^n^n is an element.
>
> The set of all FISON-ends
> (the set of all FISONs)
> does not represent any one process.

The collection of all FISON-ends can be checked and can be used to index the steps of a process.

Regards, WM

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Subject: Re: Natural numbers and vases III
From: askaske...@gmail.com (WM)
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by: WM - Tue, 30 Aug 2022 14:06 UTC

Gus Gassmann schrieb am Montag, 29. August 2022 um 16:49:38 UTC+2:
> On Monday, 29 August 2022 at 10:30:55 UTC-3, WM wrote:
> [...]
> > ℕ is a stepwise ordered set.
> ℕ is a set that can be ordered, but there is nothing stepwise about the order.

ℕ _def is used to order sets.

> Hint: How do you establish that Graham's number is larger than 1? Certainly not by establishing that 2 > 1, 3 > 1, 4 > 1, ..., (Graham's number - 1) > 1, Graham's number > 1.

But every step from n to n+1 can be checked in principle.

Regards, WM

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: askaske...@gmail.com (WM)
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by: WM - Tue, 30 Aug 2022 14:12 UTC

William schrieb am Montag, 29. August 2022 um 17:14:43 UTC+2:
> On Monday, August 29, 2022 at 6:26:27 AM UTC-3, WM wrote:
>
> > (1) a ∈ A
> > (2) b ∈ B
> > (3) a ∉ B
> > (4) b ∉ A.
> > All which fail and are not empty have an element in common.

> But not the same element.

Show and endsegment that has an element not on common with all its predecessors.

> Given and two endsegments A and B there is a common element b(A,B). However, there is no one element b which works for any A and B

It cannot be found because the set of definable endsegments is potentially infinite. The smallest element can be larger than any hotherto known number.. But its existence can be provedt.

All infinite endsegments have infinitely many elements in common. Otherwise there would be a first one with less numbers in common with its predecessors.

Regards, WM

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: askaske...@gmail.com (WM)
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by: WM - Tue, 30 Aug 2022 14:17 UTC

William schrieb am Montag, 29. August 2022 um 17:10:18 UTC+2:
> On Monday, August 29, 2022 at 6:20:57 AM UTC-3, WM wrote:
> > William schrieb am Sonntag, 28. August 2022 um 18:27:31 UTC+2:
> > > On Sunday, August 28, 2022 at 10:52:02 AM UTC-3, WM wrote:
> > > > William schrieb am Samstag, 27. August 2022 um 17:19:36 UTC+2:
> > > > > On Saturday, August 27, 2022 at 10:44:04 AM UTC-3, WM wrote:
> > > > > > ... you must find an endsegment which contains some elements not in a predecessor.
> > > > >
> > > > > Nope. No such endsegment is needed in order to get an empty intersection
> > > > It is needed if all endsegments are nonempty. The only alternative would be an empty endsegment.
> > > Piffle. There is also the alternative that there is no last endsegment.
> > All endsegments which are there fail to satisfy
> > (1) a ∈ A
> > (2) b ∈ B
> > (3) a ∉ B
> > (4) b ∉ A.
> Piffle. No pair of endsengmens satisfies 2 and 4. So what?

Any endsegment contains only a subset of what its predecessors contained. This holds for all non-empty endsegments.

Regards, WM

Re: Natural numbers and vases III

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Date: Tue, 30 Aug 2022 07:21:32 -0700 (PDT)
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Subject: Re: Natural numbers and vases III
From: horand.g...@gmail.com (Gus Gassmann)
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by: Gus Gassmann - Tue, 30 Aug 2022 14:21 UTC

On Tuesday, 30 August 2022 at 10:42:23 UTC-3, WM wrote:
> Fritz Feldhase schrieb am Montag, 29. August 2022 um 16:07:22 UTC+2:
> > On Monday, August 29, 2022 at 3:27:52 PM UTC+2, WM wrote:
> >
> > > you are not aware of the fact that dark numbers exist and that your proofs are invalid for dark numbers.
> > Yeah, I have to admit that I'm indeed not aware of these "facts".
> But you need them to understand set theory correctly.

Oh bullshit. All you need is that any time t, the set (!) of "instantiated" natural numbers (that is, natural numbers that someone, somewhere has written down or talked about explicitly), ℕ_inst (t), is a finite subset of ℕ. All natural numbers have been defined (by Peano, for instance), and even your assholiness can't change that.

Then
∩{E(k) : k ∈ ℕ_inst (t)} = E(max(ℕ_inst (t))) --- the max exists because the set is finite ---
and
∩{E(k) : k ∈ ℕ} = { }.

There is no contradiction, no "potentially infinite set" (which is an oxymoron, moron), and of course due to the triviality of all of it, only infinitesimal glory.

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: horand.g...@gmail.com (Gus Gassmann)
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by: Gus Gassmann - Tue, 30 Aug 2022 14:23 UTC

On Tuesday, 30 August 2022 at 10:46:11 UTC-3, WM wrote:
> [...] Du musst noch viele lernen.

It is just a little inappropriate to use colloquial German in an English-speaking newsgroup, but at least do a proper spellcheck, asshole

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: askaske...@gmail.com (WM)
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by: WM - Tue, 30 Aug 2022 14:24 UTC

Jim Burns schrieb am Montag, 29. August 2022 um 17:45:40 UTC+2:

> Intersections *are*
>
> The intersection of a collection of sets
> *is* what contains each element which is
> in each set in the collection.

There *are* infinitely many natnumbers common to all *infinite* endsegments. Otherwise there would be a first endsegment having less than infinitely many natnumbers in common with some predecessor. Can you follow that logic?

Regards, WM

Re: Natural numbers and vases III

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From: inva...@invalid.com (Sergio)
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Subject: Re: Natural numbers and vases III
Date: Tue, 30 Aug 2022 09:24:36 -0500
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by: Sergio - Tue, 30 Aug 2022 14:24 UTC

On 8/30/2022 9:17 AM, WM wrote:
> William schrieb am Montag, 29. August 2022 um 17:10:18 UTC+2:
>> On Monday, August 29, 2022 at 6:20:57 AM UTC-3, WM wrote:
>>> William schrieb am Sonntag, 28. August 2022 um 18:27:31 UTC+2:
>>>> On Sunday, August 28, 2022 at 10:52:02 AM UTC-3, WM wrote:
>>>>> William schrieb am Samstag, 27. August 2022 um 17:19:36 UTC+2:
>>>>>> On Saturday, August 27, 2022 at 10:44:04 AM UTC-3, WM wrote:
>>>>>>> ... you must find an endsegment which contains some elements not in a predecessor.
>>>>>>
>>>>>> Nope. No such endsegment is needed in order to get an empty intersection
>>>>> It is needed if all endsegments are nonempty. The only alternative would be an empty endsegment.
>>>> Piffle. There is also the alternative that there is no last endsegment.
>>> All endsegments which are there fail to satisfy
>>> (1) a ∈ A
>>> (2) b ∈ B
>>> (3) a ∉ B
>>> (4) b ∉ A.
>> Piffle. No pair of endsengmens satisfies 2 and 4. So what?
>
> Any endsegment contains only a subset of what its predecessors contained.

Wrong, E(1) contains no elements of its predecessor. (Sloppy Math)

>This holds for all non-empty endsegments.

Wrong, as all endsegments are infinite by definition AND E(1)

>
> Regards, WM
>

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: horand.g...@gmail.com (Gus Gassmann)
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by: Gus Gassmann - Tue, 30 Aug 2022 14:24 UTC

On Tuesday, 30 August 2022 at 11:06:08 UTC-3, WM wrote:
> Gus Gassmann schrieb am Montag, 29. August 2022 um 16:49:38 UTC+2:
> > On Monday, 29 August 2022 at 10:30:55 UTC-3, WM wrote:
> > [...]
> > > ℕ is a stepwise ordered set.
> > ℕ is a set that can be ordered, but there is nothing stepwise about the order.
> ℕ _def is used to order sets.
> > Hint: How do you establish that Graham's number is larger than 1? Certainly not by establishing that 2 > 1, 3 > 1, 4 > 1, ..., (Graham's number - 1) > 1, Graham's number > 1.
> But every step from n to n+1 can be checked in principle.

The order still is not a "stepwise order". How do YOU know that Graham's number is greater than 1? Did you go through the steps?

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: horand.g...@gmail.com (Gus Gassmann)
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by: Gus Gassmann - Tue, 30 Aug 2022 14:26 UTC

On Tuesday, 30 August 2022 at 11:12:20 UTC-3, WM wrote:
> All infinite endsegments have infinitely many elements in common. Otherwise there would be a first one with less numbers in common with its predecessors.

Quantifier salad! Don't you have *ANYTHING* else any longer?

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From: inva...@invalid.com (Sergio)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases III
Date: Tue, 30 Aug 2022 09:27:16 -0500
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by: Sergio - Tue, 30 Aug 2022 14:27 UTC

On 8/30/2022 9:12 AM, WM wrote:
> William schrieb am Montag, 29. August 2022 um 17:14:43 UTC+2:
>> On Monday, August 29, 2022 at 6:26:27 AM UTC-3, WM wrote:
>>
>>> (1) a ∈ A
>>> (2) b ∈ B
>>> (3) a ∉ B
>>> (4) b ∉ A.
>>> All which fail and are not empty have an element in common.
>
>> But not the same element.
>
> Show and endsegment that has an element not on common with all its predecessors.

E(1)

>
>> Given and two endsegments A and B there is a common element b(A,B). However, there is no one element b which works for any A and B
>
> It cannot be found because the set of definable endsegments is potentially infinite.

Wrong. "definable" and "potentially infinite" are two of your Fake Math keywords.

>
> Regards, WM

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: wpihug...@gmail.com (William)
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by: William - Tue, 30 Aug 2022 14:29 UTC

On Tuesday, August 30, 2022 at 11:12:20 AM UTC-3, WM wrote:
> William schrieb am Montag, 29. August 2022 um 17:14:43 UTC+2:
> > On Monday, August 29, 2022 at 6:26:27 AM UTC-3, WM wrote:
> >
> > > (1) a ∈ A
> > > (2) b ∈ B
> > > (3) a ∉ B
> > > (4) b ∉ A.
> > > All which fail and are not empty have an element in common.
>
> > But not the same element.
> Show and endsegment that has an element not on common with all its predecessors.

There is no such endsegment. So what? There is no endsegment that has an element in common with all its successors.

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Subject: Re: Natural numbers and vases III
Date: Tue, 30 Aug 2022 09:30:39 -0500
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by: Sergio - Tue, 30 Aug 2022 14:30 UTC

On 8/30/2022 8:46 AM, WM wrote:
> Fritz Feldhase schrieb am Montag, 29. August 2022 um 16:48:12 UTC+2:
>
>> In other words, __the successor of the smallest endsegent in {E(n) | n ∈ ℕ_def}__ does not belong to ∩{E(n) | n ∈ ℕ_def}.
>
> There is no smallest endsegment in {E(n) | n ∈ ℕ_def}.
>>
>> Hint: If there were no smallest endsegment in {E(n) | n ∈ ℕ_def}, then {E(n) | n ∈ ℕ_def} would be infinite, and hence we would have ∩{E(n) | n ∈ ℕ_def} = { }.
>
> No, that is not caused by infinity. It is caused by an empty endsegment. Du musst noch viele lernen.
>
> Regards, WM

No, Your Math is caused by an empty headsegment.

so, where is your proof ? got EQUATIONS ? google for it

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: askaske...@gmail.com (WM)
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by: WM - Tue, 30 Aug 2022 14:38 UTC

Eram semper recta schrieb am Montag, 29. August 2022 um 18:09:28 UTC+2:
> On Monday, 29 August 2022 at 10:02:08 UTC-4, WM wrote:

> > > You still haven't told me what you think about the proof ∞-∞=0:
> > >
> > oo for me means a direction, not a quantity.
> Ah, I see you've revised your thoughts.... I think you've taken a turn for the worse here. 😊😊
>
> ∞ is not seen as a direction by the mainstream. It is seen as a cardinal number.

Das Zeichen ∞, welches ich in Nr. 2 dieses Aufsatzes gebraucht habe, ersetze ich von nun an durch ω, weil das Zeichen ∞ schon vielfach zur Bezeichnung von unbestimmten [d. h. potentiellen] Unendlichkeiten verwandt wird. [Cantor]

Later, when he found the first inconsistencies, he replaced it by ℵo.
Regards, WM

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Subject: Re: Natural numbers and vases III
From: franz.fr...@gmail.com (Fritz Feldhase)
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by: Fritz Feldhase - Tue, 30 Aug 2022 14:39 UTC

On Tuesday, August 30, 2022 at 4:03:43 PM UTC+2, WM wrote:
> Jim Burns schrieb am Montag, 29. August 2022 um 16:48:13 UTC+2:
> >
> > there is the set of all FISON-ends

{max({1, ..., n}) : n e IN} = IN

> ℕ_def

=> IN_def = IN once more, you silly asshole full of shit!

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Newsgroups: sci.math
Subject: Re: Natural numbers and vases III
Date: Tue, 30 Aug 2022 10:42:57 -0400
Organization: Peripheral Visions
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by: FromTheRafters - Tue, 30 Aug 2022 14:42 UTC

After serious thinking WM wrote :
> William schrieb am Montag, 29. August 2022 um 17:10:18 UTC+2:
>> On Monday, August 29, 2022 at 6:20:57 AM UTC-3, WM wrote:
>>> William schrieb am Sonntag, 28. August 2022 um 18:27:31 UTC+2:
>>>> On Sunday, August 28, 2022 at 10:52:02 AM UTC-3, WM wrote:
>>>>> William schrieb am Samstag, 27. August 2022 um 17:19:36 UTC+2:
>>>>>> On Saturday, August 27, 2022 at 10:44:04 AM UTC-3, WM wrote:
>>>>>>> ... you must find an endsegment which contains some elements not in a
>>>>>>> predecessor.
>>>>>>
>>>>>> Nope. No such endsegment is needed in order to get an empty intersection
>>>>> It is needed if all endsegments are nonempty. The only alternative would
>>>>> be an empty endsegment.
>>>> Piffle. There is also the alternative that there is no last endsegment.
>>> All endsegments which are there fail to satisfy
>>> (1) a ∈ A
>>> (2) b ∈ B
>>> (3) a ∉ B
>>> (4) b ∉ A.
>> Piffle. No pair of endsengmens satisfies 2 and 4. So what?
>
> Any endsegment contains only a subset of what its predecessors contained.
> This holds for all non-empty endsegments.

Does that make it smaller?

tech / sci.math / Re: Natural numbers and vases III

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