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That wouldn't be good enough. -- Larry Wall in <199710131621.JAA14907@wall.org>

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
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Re: Natural numbers and vases III	WM
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Re: Natural numbers and vases III	WM
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Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
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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
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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
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Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	WM
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Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
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Re: Natural numbers and vases III	WM
Re: Natural numbers and vases III	William
Re: Natural numbers and vases III	WM
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Re: Natural numbers and vases III	William
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Re: Natural numbers and vases III	WM
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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

<956cc96d-6ea8-82e3-04a5-73b5df7b9bc1@att.net>

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From: james.g....@att.net (Jim Burns)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases III
Date: Fri, 26 Aug 2022 12:50:35 -0400
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by: Jim Burns - Fri, 26 Aug 2022 16:50 UTC

On 8/26/2022 11:18 AM, WM wrote:
> Jim Burns schrieb am Donnerstag,
> 25. August 2022 um 20:00:51 UTC+2:
>> On 8/25/2022 11:20 AM, FromTheRafters wrote:

>>> "Our intuition gained from finite sets
>>> breaks down when dealing with infinite sets."
>>> https://en.wikipedia.org/wiki/Cardinality#Infinite_sets
>>
>> One way to proceed despite broken down
>> intuitions is to depend only upon such
>> facts as hold in both the finite and the
>> infinite realms.
>
> Like exchanging X's and O's
> will never delete any X or O.
> Yes, only such deep principles may be applied.

Like
there can't be BOTH
an all-white row and an all-clack column,
but
there might be NEITHER
an all-white row nor an all-back column.

Because there might be NEITHER,
it is insufficient to know that
each natural number is followed by
an infinite end segment
to conclude that
one infinite end segment follows all
natural numbers.

Re: Natural numbers and vases III

<6d424b47-056a-4c94-86a1-9ff2f80bc093n@googlegroups.com>

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Subject: Re: Natural numbers and vases III
From: monteu...@t-online.de (WM)
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by: WM - Fri, 26 Aug 2022 17:06 UTC

Jim Burns schrieb am Donnerstag, 25. August 2022 um 20:26:04 UTC+2:
> On 8/25/2022 9:40 AM, WM wrote:
>
> > What makes the difference between
> > all elements and the set?
> I think that this gets answered very well
> with _pluralities_ which you and I have
> been giving the local name of "collections"

> With collections, we can do things
> like describe one of the natural numbers,
> so that, by reasoning from our description,
> we can reason about all infinitely-many
> of them.

But if there is nothing besides these infinitely many elements, then we cannot explain the different effects:
All infinitely many elements of the collection ℕ_def fail to empty the intersection:
∀k ∈ ℕ_def: ∩{E(1), E(2), ..., E(k)} = E(k) /\ |E(k)| = ℵ₀
All elements of the *set* however cause the empty set
∩{E(k) : k ∈ ℕ} = { }.
The set contains more.

Regards, WM

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: monteu...@t-online.de (WM)
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by: WM - Fri, 26 Aug 2022 17:18 UTC

Jim Burns schrieb am Freitag, 26. August 2022 um 00:32:17 UTC+2:
> On 8/25/2022 5:50 PM, Fritz Feldhase wrote:

> > Mückenheim just calls a set sequence
> > (S_n)_(n e IN) "inclusion monotonic"
> > iff
> > An e IN: S_n c S_(n+1) (*)
> > or
> > An e IN: S_(n+1) c S_n. (**)
> No.
> Mückenheim also requires a last set.

No!. There cannot be a last set because the dark sets have no (discernible) order. That is just the justification for dark sets.
>
> It's essential to his "proof" of dark numbers.
> Because "our" sets don't include a last one,

Your collections are potentially infinite. But complete sets must include all elements. All means a last one, if the elements are in linear order. In such an order "all" is proved by a "last". Therefore such an order cannot exist.
> the WM-necessarily-existing WM-last one
> must be dark.

and cannot be distinguished from aleph_0 other dark elements.

There is no smallest positive fraction. Either there is nothing next to zero, a gap, or there is something that cannot be found. The result is the same. All mathematics is based upon potential infinity. Whether or not dark numbers exists - only the definable numbers play a role in mathematics.

Regards, WM

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: monteu...@t-online.de (WM)
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by: WM - Fri, 26 Aug 2022 17:20 UTC

Fritz Feldhase schrieb am Freitag, 26. August 2022 um 00:34:00 UTC+2:
> On Friday, August 26, 2022 at 12:32:17 AM UTC+2, Jim Burns wrote:
> > On 8/25/2022 5:50 PM, Fritz Feldhase wrote:
> > > On Thursday, August 25, 2022
> > > at 11:41:57 PM UTC+2, Fritz Feldhase wrote:
> >
> > >> What's the matter with you man, these are
> > > just some trivial variants of terminology.
> > >
> > > To make a long story short:
> > >
> > > Mückenheim just calls a set sequence (S_n)_(n e IN) "inclusion monotonic" iff
> > >
> > > An e IN: S_n c S_(n+1) (*)
> > >
> > > or
> > >
> > > An e IN: S_(n+1) c S_n. (**)
> > >
> > No.
> Yes.

Yes, either it is ascending or descending.

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 - Fri, 26 Aug 2022 17:34 UTC

On Friday, 26 August 2022 at 14:06:07 UTC-3, WM wrote:
[...]
> But if there is nothing besides these infinitely many elements, then we cannot explain the different effects:
> All infinitely many elements of the collection ℕ_def fail to empty the intersection:
> ∀k ∈ ℕ_def: ∩{E(1), E(2), ..., E(k)} = E(k) /\ |E(k)| = ℵ₀
> All elements of the *set* however cause the empty set
> ∩{E(k) : k ∈ ℕ} = { }.
> The set contains more.

Muckenheim, you are delusional. Your quantifier swaps were cute at a time, but they have become, like you, tedious and boring. The set of natural numbers contains all the natural numbers, and nothing *BUT* the natural numbers.. That the set limit of the sequence [E(k)] as k -> oo is the empty set should not be a particularly big surprise, even to a moron like you.

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 - Fri, 26 Aug 2022 17:38 UTC

On Friday, 26 August 2022 at 14:20:15 UTC-3, WM wrote:
> Fritz Feldhase schrieb am Freitag, 26. August 2022 um 00:34:00 UTC+2:
> > On Friday, August 26, 2022 at 12:32:17 AM UTC+2, Jim Burns wrote:
> > > On 8/25/2022 5:50 PM, Fritz Feldhase wrote:
> > > > On Thursday, August 25, 2022
> > > > at 11:41:57 PM UTC+2, Fritz Feldhase wrote:
> > >
> > > >> What's the matter with you man, these are
> > > > just some trivial variants of terminology.
> > > >
> > > > To make a long story short:
> > > >
> > > > Mückenheim just calls a set sequence (S_n)_(n e IN) "inclusion monotonic" iff
> > > >
> > > > An e IN: S_n c S_(n+1) (*)
> > > >
> > > > or
> > > >
> > > > An e IN: S_(n+1) c S_n. (**)
> > > >
> > > No.
> > Yes.
> Yes, either it is ascending or descending.

OK. Is the sequence {1, 2, 3}, {2, 3}, {3}, {} "inclusion monotonic", or does it have to be an infinite sequence?

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Subject: Re: Natural numbers and vases III
From: monteu...@t-online.de (WM)
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by: WM - Fri, 26 Aug 2022 18:12 UTC

FromTheRafters schrieb am Freitag, 26. August 2022 um 03:32:23 UTC+2:

> I asked WM what he means by "complete" and Fritz hasn't answered me yet
> :(

"Cantor's work was well received by some of the prominent mathematicians of his day, such as Richard Dedekind. But his willingness to regard infinite sets as objects to be treated in much the same way as finite sets was bitterly attacked by others, particularly Kronecker. There was no objection to a 'potential infinity' in the form of an unending process, but an 'actual infinity' in the form of a completed infinite set was harder to accept." [H.B.. Enderton: "Elements of set theory", Academic Press, New York (1977) p. 14f]

"I protest firstly against the use of an infinite magnitude as a completed one, which never has been allowed in mathematics. The infinite is only a mode of speaking, when we in principle talk about limits which are approached by certain ratios as closely as desired whereas others are allowed to grow without reservation." [C.F. Gauß, letter to H.C. Schumacher (12 Jul 1831)]

Complete: "In spite of significant difference between the notions of the potential and actual infinite, where the former is a variable finite magnitude, growing above all limits, the latter a constant quantity fixed in itself but beyond all finite magnitudes, it happens deplorably often that the one is confused with the other." [Cantor, p. 374]

Completed infinity, or actual infinity, is an infinity that one actually reaches; the process is already done. For instance, let's put braces around that sequence mentioned earlier: {1, 2, 3, 4, ...}. With this notation, we are indicating the set of all positive integers. This is just one object, a set. But that set has infinitely many members. By that I don't mean that it has a large finite number of members and it keeps getting more members. Rather, I mean that it already has infinitely many members.
We can also indicate the completed infinity geometrically. For instance, the diagram at right shows a one-to-one correspondence between points on an infinitely long line and points on a semicircle. There are no points for plus or minus infinity on the line, but it is natural to attach those 'numbers' to the endpoints of the semicircle.
Isn't that 'cheating', to simply add numbers in this fashion? Not really; it just depends on what we want to use those numbers for. For instance, f(x) = 1/(1 + x2) is a continuous function defined for all real numbers x, and it also tends to a limit of 0 when x 'goes to' plus or minus infinity (in the sense of potential infinity, described earlier). Consequently, if we add those two 'numbers' to the real line, to get the so-called 'extended real line', and we equip that set with the same topology as that of the closed semicircle (i.e., the semicircle including the endpoints), then the function f is continuous everywhere on the extended real line." [E. Schechter: "Potential versus completed infinity: Its history and controversy" (5 Dec 2009)]

Regards, WM

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: monteu...@t-online.de (WM)
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by: WM - Fri, 26 Aug 2022 18:20 UTC

zelos...@gmail.com schrieb am Freitag, 26. August 2022 um 06:56:21 UTC+2:
> torsdag 25 augusti 2022 kl. 15:40:21 UTC+2 skrev WM:

> > This which a serious mathematician cannot deny:
> > (1) The intersection of inclusion-monotonic infinite sets is infinite.
> Not necessarily.

Necessarily!
>
> >
> > (2) By exchanging X and O the matrix
> > XOO...
> > XOO...
> > XOO...
> > ...
> > cannot lose any O.
>
> Only if you think in a step wise process which mathematics IS NOT!

For every exchange defined by Cantor it is a stepwise process. All steps can be checked in principle. But even if it were not, no shufflig could eliminate an O.
Only the belief in miracles of matheology could remove all O's. But who believes in miracles in real mathematics?

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 - Fri, 26 Aug 2022 18:24 UTC

On Friday, August 26, 2022 at 11:57:21 AM UTC-3, WM wrote:
> William schrieb am Donnerstag, 25. August 2022 um 17:17:39 UTC+2:
> > On Thursday, August 25, 2022 at 10:37:04 AM UTC-3, WM wrote:
>
> > > each nonempty endsegment has at least one natnumber in common with all non-empty endsegments. Assume not.
> > This assumption is true (and does not lead to a contradiction).
> > >Assume for simplicity that every endsegment has only one element.
> > This assumption may be simple but it is false. Unsurprisingly it leads to a contradiction.
>
> It is not necessary for my proof.

Yes it is

> Each nonempty endsegment has at least one natnumber in common with all non-empty endsegments. Assume not. Then there are at least two natnumbers a and b in endsegments A and B, such that
> (1) a ∈ A
> (2) b ∈ B
> (3) a ∉ B
> (4) b ∉ A.

Nope, a exists but b does not exist. Assuming that b exists is equivalent to assuming A is finite.

--
William Hughes

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: monteu...@t-online.de (WM)
Injection-Date: Fri, 26 Aug 2022 18:25:09 +0000
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by: WM - Fri, 26 Aug 2022 18:25 UTC

Gus Gassmann schrieb am Freitag, 26. August 2022 um 19:38:43 UTC+2:
> On Friday, 26 August 2022 at 14:20:15 UTC-3, WM wrote:
> > Fritz Feldhase schrieb am Freitag, 26. August 2022 um 00:34:00 UTC+2:
> > > On Friday, August 26, 2022 at 12:32:17 AM UTC+2, Jim Burns wrote:
> > > > On 8/25/2022 5:50 PM, Fritz Feldhase wrote:
> > > > > On Thursday, August 25, 2022
> > > > > at 11:41:57 PM UTC+2, Fritz Feldhase wrote:
> > > >
> > > > >> What's the matter with you man, these are
> > > > > just some trivial variants of terminology.
> > > > >
> > > > > To make a long story short:
> > > > >
> > > > > Mückenheim just calls a set sequence (S_n)_(n e IN) "inclusion monotonic" iff
> > > > >
> > > > > An e IN: S_n c S_(n+1) (*)
> > > > >
> > > > > or
> > > > >
> > > > > An e IN: S_(n+1) c S_n. (**)
> > > > >
> > > > No.
> > > Yes.
> > Yes, either it is ascending or descending.
> OK. Is the sequence {1, 2, 3}, {2, 3}, {3}, {} "inclusion monotonic", or does it have to be an infinite sequence?

That depends on the definition. I would say yes.

Regards, WM

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: monteu...@t-online.de (WM)
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by: WM - Fri, 26 Aug 2022 18:29 UTC

William schrieb am Freitag, 26. August 2022 um 20:24:54 UTC+2:
> On Friday, August 26, 2022 at 11:57:21 AM UTC-3, WM wrote:
> > William schrieb am Donnerstag, 25. August 2022 um 17:17:39 UTC+2:
> > > On Thursday, August 25, 2022 at 10:37:04 AM UTC-3, WM wrote:
> >
> > > > each nonempty endsegment has at least one natnumber in common with all non-empty endsegments. Assume not.
> > > This assumption is true (and does not lead to a contradiction).
> > > >Assume for simplicity that every endsegment has only one element.
> > > This assumption may be simple but it is false. Unsurprisingly it leads to a contradiction.
> >
> > It is not necessary for my proof.
> Yes it is.

No, see below.

> > Each nonempty endsegment has at least one natnumber in common with all non-empty endsegments. Assume not. Then there are at least two natnumbers a and b in endsegments A and B, such that
> > (1) a ∈ A
> > (2) b ∈ B
> > (3) a ∉ B
> > (4) b ∉ A.
> Nope, a exists but b does not exist. Assuming that b exists is equivalent to assuming A is finite.

Finite or infinite: If no such b exists, then the intersection cannot be empty.

Regards, WM

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: monteu...@t-online.de (WM)
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by: WM - Fri, 26 Aug 2022 18:34 UTC

Gus Gassmann schrieb am Freitag, 26. August 2022 um 19:34:43 UTC+2:
> On Friday, 26 August 2022 at 14:06:07 UTC-3, WM wrote:
> [...]
> > But if there is nothing besides these infinitely many elements, then we cannot explain the different effects:
> > All infinitely many elements of the collection ℕ_def fail to empty the intersection:
> > ∀k ∈ ℕ_def: ∩{E(1), E(2), ..., E(k)} = E(k) /\ |E(k)| = ℵ₀
> > All elements of the *set* however cause the empty set
> > ∩{E(k) : k ∈ ℕ} = { }.
> > The set contains more.
> Your quantifier swaps

Your quantifiers are nonsense.
∀k ∈ ℕ_def: ∩{E(1), E(2), ..., E(k)} = E(k) /\ |E(k)| = ℵ₀
and
∩{E(k) : k ∈ ℕ} = { }
proves ℕ_def =/= ℕ.

Regards, WM

Re: Natural numbers and vases III

<teb3o0$3ul$1@gioia.aioe.org>

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by: Sergio - Fri, 26 Aug 2022 18:34 UTC

On 8/26/2022 11:20 AM, Fritz Feldhase wrote:
> On Friday, August 26, 2022 at 5:57:29 PM UTC+2, Sergio wrote:
>> On 8/26/2022 10:22 AM, WM wrote:
>>> Fritz Feldhase schrieb am Freitag, 26. August 2022 um 00:10:24 UTC+2:
>>>> On Thursday, August 25, 2022 at 11:49:42 PM UTC+2, Sergio wrote:
>>>>
>>>>> OK, are these WN unique words and WM ideas ?
>>>>>
>>>>> FISON
>>>>
>>>> Nope. IIRC "Virgil" coined that term. It 's just short for "finite initial segment of natural numbers" or "finite initial segment of IN".
>>>
>>> Yes, it is Virgil's word.
>>>
>> some other troll on the internet.
>
> Fuck you, you silly idiot.

I was not on sci.mat that far back, 2015 ?
Or he got on my plonk list, is he still posting here ?

Virgil <vir...@ligriv.com>Virgil <VIR...@VIRGIL.com> ?

"It vanished with the smoke of whatever WM was smoking when he dreamed up
Union {} = {1,2,3 ... }"
--
Virgil
"Mit der Dummheit kampfen Gotter selbst vergebens." (Schiller)

yea, the guy looks sane

and WM was/is still bullshitting same old stuff

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 - Fri, 26 Aug 2022 18:39 UTC

On Friday, August 26, 2022 at 3:29:56 PM UTC-3, WM wrote:
> William schrieb am Freitag, 26. August 2022 um 20:24:54 UTC+2:
> > On Friday, August 26, 2022 at 11:57:21 AM UTC-3, WM wrote:
> > > William schrieb am Donnerstag, 25. August 2022 um 17:17:39 UTC+2:
> > > > On Thursday, August 25, 2022 at 10:37:04 AM UTC-3, WM wrote:
> > >
> > > > > each nonempty endsegment has at least one natnumber in common with all non-empty endsegments. Assume not.
> > > > This assumption is true (and does not lead to a contradiction).
> > > > >Assume for simplicity that every endsegment has only one element.
> > > > This assumption may be simple but it is false. Unsurprisingly it leads to a contradiction.
> > >
> > > It is not necessary for my proof.
> > Yes it is.
>
> No, see below.
> > > Each nonempty endsegment has at least one natnumber in common with all non-empty endsegments. Assume not. Then there are at least two natnumbers a and b in endsegments A and B, such that
> > > (1) a ∈ A
> > > (2) b ∈ B
> > > (3) a ∉ B
> > > (4) b ∉ A.
> > Nope, a exists but b does not exist. Assuming that b exists is equivalent to assuming A is finite.
> Finite or infinite: If no such b exists, then the intersection cannot be empty.

Nope. As long as for any A and B, a exists then the intersection is empty.

--
William Hughes

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: monteu...@t-online.de (WM)
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by: WM - Fri, 26 Aug 2022 18:41 UTC

Eram semper recta schrieb am Freitag, 26. August 2022 um 18:35:33 UTC+2:
> On Friday, 26 August 2022 at 00:56:21 UTC-4, zelos idiot wrote:
> > torsdag 25 augusti 2022 kl. 15:40:21 UTC+2 skrev WM:

> > > This which a serious mathematician cannot deny:
> > > (1) The intersection of inclusion-monotonic infinite sets is infinite.
> > Not necessarily.
> >
> > >
> > > (2) By exchanging X and O the matrix
> > > XOO...
> > > XOO...
> > > XOO...
> > > ...
> > > cannot lose any O.

> >
> > Only if you think in a step wise process which mathematics IS NOT!
> Newsflash: Everything in mathematics is a stepwise process. Set theory is not mathematics.

Every sequence can be checked, in principle, to every term. Only set theorists deny it because it would rob them of their delusions.

Regards, WM

Re: Natural numbers and vases III

<4ad814a0-7ec0-3b5b-f6c6-deca2a202cff@att.net>

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From: james.g....@att.net (Jim Burns)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases III
Date: Fri, 26 Aug 2022 14:43:14 -0400
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by: Jim Burns - Fri, 26 Aug 2022 18:43 UTC

On 8/26/2022 1:18 PM, WM wrote:

> All means a last one,
> if the elements are in linear order.

For all FISONs 𝐹ᵢ 𝐹ⱼ 𝐹ₖ

𝐹ⱼ ⊆ 𝐹ₖ or 𝐹ₖ ⊆ 𝐹ⱼ
[1]

𝐹ⱼ ⊆ 𝐹ₖ and 𝐹ₖ ⊆ 𝐹ⱼ imply 𝐹ⱼ = 𝐹ₖ

𝐹ᵢ ⊆ 𝐹ⱼ and 𝐹ⱼ ⊆ 𝐹ₖ imply 𝐹ᵢ ⊆ 𝐹ₖ

All the FISONs are in linear order with '⊆'
That's what "linear order" means.

Each FISON 𝐹ₖ ends somewhere, call it k.

𝐹ₖ∪⟨k+1⟩ is also a FISON, and
𝐹ₖ∪⟨k+1⟩ ⊈ 𝐹ₖ
𝐹ₖ is not last with '⊆'

The FISONs are linearly ordered,
but there is no last FISON.

> In such an order "all" is proved
> by a "last".

We begin with
claims we know are true of all.

We prove more
claims we know are true of all
by stepping only to new
claims we know are true of all
which we know must be true of all because
of the other claims we already know.

----
[1]
𝐹ⱼ ⊆ 𝐹ₖ or 𝐹ₖ ⊆ 𝐹ⱼ

j ∈ 𝐹ₖ or j ∉ 𝐹ₖ

If j ∈ 𝐹ₖ then 𝐹ⱼ ⊆ 𝐹ₖ

If j ∉ 𝐹ₖ then a split of 𝐹ⱼ exists
BEFORE = { i₁ : ∀i₂=<i₁, i₂ ∈ 𝐹ₖ }
AFTER = { i₁ : ∃i₂=<i₁, i₂ ∉ 𝐹ₖ }

A step i₃,i₃+1 exists
i₃ is last in BEFORE
i₃+1 is first in AFTER

i₃ ∈ 𝐹ₖ
i₃+1 ∉ 𝐹ₖ

k is the only element m ∈ 𝐹ₖ
for which m+1 ∉ 𝐹ₖ
Thus,
k = i₃ ∈ 𝐹ⱼ
𝐹ₖ ⊆ 𝐹ⱼ

Therefore,
𝐹ⱼ ⊆ 𝐹ₖ or 𝐹ₖ ⊆ 𝐹ⱼ

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: monteu...@t-online.de (WM)
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by: WM - Fri, 26 Aug 2022 18:47 UTC

Jim Burns schrieb am Freitag, 26. August 2022 um 18:50:45 UTC+2:

> it is insufficient to know that
> each natural number is followed by
> an infinite end segment
> to conclude that
> one

not one and the same but an

> infinite end segment
> follows all
> natural numbers.

Then by
∀k ∈ ℕ: E(k+1) = E(k) \ {k}
there must be a first natural number which is not followed by an infinite endsegment. Alas, it cannot be found.

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 - Fri, 26 Aug 2022 18:51 UTC

On Friday, August 26, 2022 at 3:47:09 PM UTC-3, WM wrote:
> Jim Burns schrieb am Freitag, 26. August 2022 um 18:50:45 UTC+2:
>
> > it is insufficient to know that
> > each natural number is followed by
> > an infinite end segment
> > to conclude that
> > one
> not one and the same but an
> > infinite end segment
> > follows all
> > natural numbers.
> Then by
> ∀k ∈ ℕ: E(k+1) = E(k) \ {k}
> there must be a first natural number which is not followed by an infinite endsegment. Alas, it cannot be found.
>
> Regards, WM

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Subject: Re: Natural numbers and vases III
From: monteu...@t-online.de (WM)
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by: WM - Fri, 26 Aug 2022 18:51 UTC

Jim Burns schrieb am Freitag, 26. August 2022 um 18:43:06 UTC+2:
> On 8/26/2022 10:57 AM, WM wrote:
>
> > It is not necessary for my proof.
> > Each nonempty endsegment has at least one
> > natnumber in common with all non-empty
> > endsegments.
> > Assume not.
> Assume not.
> Assume no natural number is in common
> with all non-empty end segments.
> A is a non-empty end segment.
> There is some a ∈ A
> and some non-empty B, a ∉ B
>
> But B is also non-empty.
> There is some b ∈ B
> and some non-empty C, b ∉ C
> > Then there are at least two natnumbers
> > a and b in endsegments A and B, such that
> > (1) a ∈ A
> > (2) b ∈ B
> > (3) a ∉ B
> > (4) b ∉ A.
> How do you know A = C?

Meaningless question. There must be two non-empty sets with condition (1-4).. If you dislike A and B then call them without loss of generality X and Y..

Regards, WM

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Subject: Re: Natural numbers and vases III
From: wpihug...@gmail.com (William)
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by: William - Fri, 26 Aug 2022 18:55 UTC

On Friday, August 26, 2022 at 3:47:09 PM UTC-3, WM wrote:
>... by
> ∀k ∈ ℕ: E(k+1) = E(k) \ {k}
> there must be a first natural number which is not followed by an infinite endsegment.

Nope, does not follow and is not true.

--
William Hughes

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 - Fri, 26 Aug 2022 18:58 UTC

On Friday, 26 August 2022 at 15:34:17 UTC-3, WM wrote:
> Gus Gassmann schrieb am Freitag, 26. August 2022 um 19:34:43 UTC+2:
> > On Friday, 26 August 2022 at 14:06:07 UTC-3, WM wrote:
> > [...]
> > > But if there is nothing besides these infinitely many elements, then we cannot explain the different effects:
> > > All infinitely many elements of the collection ℕ_def fail to empty the intersection:
> > > ∀k ∈ ℕ_def: ∩{E(1), E(2), ..., E(k)} = E(k) /\ |E(k)| = ℵ₀
> > > All elements of the *set* however cause the empty set
> > > ∩{E(k) : k ∈ ℕ} = { }.
> > > The set contains more.
> > Your quantifier swaps
>
> Your quantifiers are nonsense.

They're not my quantifiers, stupid. But it is clear that you aree waging war with logic. Logic will win.

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Subject: Re: Natural numbers and vases III
From: monteu...@t-online.de (WM)
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by: WM - Fri, 26 Aug 2022 19:02 UTC

William schrieb am Freitag, 26. August 2022 um 20:55:21 UTC+2:
> On Friday, August 26, 2022 at 3:47:09 PM UTC-3, WM wrote:
> >... by
> > ∀k ∈ ℕ: E(k+1) = E(k) \ {k}
> > there must be a first natural number which is not followed by an infinite endsegment.
> Nope, does not follow and is not true.
>
It follows because above you see the universal quantifier.

Regards, WM

Re: Natural numbers and vases III

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by: FromTheRafters - Fri, 26 Aug 2022 19:11 UTC

Sergio explained :
> On 8/26/2022 11:20 AM, Fritz Feldhase wrote:
>> On Friday, August 26, 2022 at 5:57:29 PM UTC+2, Sergio wrote:
>>> On 8/26/2022 10:22 AM, WM wrote:
>>>> Fritz Feldhase schrieb am Freitag, 26. August 2022 um 00:10:24 UTC+2:
>>>>> On Thursday, August 25, 2022 at 11:49:42 PM UTC+2, Sergio wrote:
>>>>>
>>>>>> OK, are these WN unique words and WM ideas ?
>>>>>>
>>>>>> FISON
>>>>>
>>>>> Nope. IIRC "Virgil" coined that term. It 's just short for "finite
>>>>> initial segment of natural numbers" or "finite initial segment of IN".
>>>>
>>>> Yes, it is Virgil's word.
>>>>
>>> some other troll on the internet.
>>
>> Fuck you, you silly idiot.
>
> I was not on sci.mat that far back, 2015 ?
> Or he got on my plonk list, is he still posting here ?
>
> Virgil <vir...@ligriv.com>Virgil <VIR...@VIRGIL.com> ?
>
>
>
> "It vanished with the smoke of whatever WM was smoking when he dreamed up
> Union {} = {1,2,3 ... }"

https://proofwiki.org/wiki/Definition:Initial_Segment_of_Natural_Numbers

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Subject: Re: Natural numbers and vases III
From: wpihug...@gmail.com (William)
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by: William - Fri, 26 Aug 2022 19:16 UTC

On Friday, August 26, 2022 at 4:03:03 PM UTC-3, WM wrote:
> William schrieb am Freitag, 26. August 2022 um 20:55:21 UTC+2:
> > On Friday, August 26, 2022 at 3:47:09 PM UTC-3, WM wrote:
> > >... by
> > > ∀k ∈ ℕ: E(k+1) = E(k) \ {k}
> > > there must be a first natural number which is not followed by an infinite endsegment.
> > Nope, does not follow and is not true.
> >
> It follows because above you see the universal quantifier.
>
Nope, There is no last |N so the fact that there is a universal quantifier means
∀k ∈ ℕ: | E(k)|= aleph_0

--
William Hughes

Re: Natural numbers and vases III

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From: FTR...@nomail.afraid.org (FromTheRafters)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases III
Date: Fri, 26 Aug 2022 15:30:54 -0400
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by: FromTheRafters - Fri, 26 Aug 2022 19:30 UTC

It happens that WM formulated :
> FromTheRafters schrieb am Freitag, 26. August 2022 um 03:32:23 UTC+2:
>
>> I asked WM what he means by "complete" and Fritz hasn't answered me yet
>> :(
>
> "Cantor's work was well received by some of the prominent mathematicians of
> his day, such as Richard Dedekind. But his willingness to regard infinite
> sets as objects to be treated in much the same way as finite sets was
> bitterly attacked by others, particularly Kronecker. There was no objection
> to a 'potential infinity' in the form of an unending process, but an 'actual
> infinity' in the form of a completed infinite set was harder to accept."
> [H.B. Enderton: "Elements of set theory", Academic Press, New York (1977) p.
> 14f]
>
> "I protest firstly against the use of an infinite magnitude as a completed
> one, which never has been allowed in mathematics. The infinite is only a mode
> of speaking, when we in principle talk about limits which are approached by
> certain ratios as closely as desired whereas others are allowed to grow
> without reservation." [C.F. Gauß, letter to H.C. Schumacher (12 Jul 1831)]
>
> Complete: "In spite of significant difference between the notions of the
> potential and actual infinite, where the former is a variable finite
> magnitude, growing above all limits, the latter a constant quantity fixed in
> itself but beyond all finite magnitudes, it happens deplorably often that the
> one is confused with the other." [Cantor, p. 374]
>
> Completed infinity, or actual infinity, is an infinity that one actually
> reaches; the process is already done. For instance, let's put braces around
> that sequence mentioned earlier: {1, 2, 3, 4, ...}. With this notation, we
> are indicating the set of all positive integers. This is just one object, a
> set. But that set has infinitely many members. By that I don't mean that it
> has a large finite number of members and it keeps getting more members.
> Rather, I mean that it already has infinitely many members. We can also
> indicate the completed infinity geometrically. For instance, the diagram at
> right shows a one-to-one correspondence between points on an infinitely long
> line and points on a semicircle. There are no points for plus or minus
> infinity on the line, but it is natural to attach those 'numbers' to the
> endpoints of the semicircle. Isn't that 'cheating', to simply add numbers in
> this fashion? Not really; it just depends on what we want to use those
> numbers for. For instance, f(x) = 1/(1 + x2) is a continuous function defined
> for all real numbers x, and it also tends to a limit of 0 when x 'goes to'
> plus or minus infinity (in the sense of potential infinity, described
> earlier). Consequently, if we add those two 'numbers' to the real line, to
> get the so-called 'extended real line', and we equip that set with the same
> topology as that of the closed semicircle (i.e., the semicircle including the
> endpoints), then the function f is continuous everywhere on the extended real
> line." [E. Schechter: "Potential versus completed infinity: Its history and
> controversy" (5 Dec 2009)]
>
> Regards, WM

Okay, so it is about your reluctance to accept an infinite set? If
that's the case, how can you be using ZFC to prove anything about the
natural numbers?

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

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