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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
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by: Sergio - Sun, 24 Jul 2022 14:50 UTC

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 - Sun, 24 Jul 2022 15:35 UTC

Re: Natural numbers and vases III

<92503a09-7803-4aeb-405c-c5349bb1556b@att.net>

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From: james.g....@att.net (Jim Burns)
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Subject: Re: Natural numbers and vases III
Date: Sun, 24 Jul 2022 12:02:11 -0400
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by: Jim Burns - Sun, 24 Jul 2022 16:02 UTC

On 7/24/2022 10:13 AM, WM wrote:
> Jim Burns schrieb
> am Samstag, 23. Juli 2022 um 20:14:46 UTC+2:

>> Induction works for each k in each ⟨0⋯i,i+1⋯n⟩
>
> Yes.

>>> Induction does not work.
>>> I works for every finite set
>>> but not for the infinite set.

No,
there are infinitely-many k for which
induction works.

>> However,
>> no ⟨0⋯i,i+1⋯𝔊⟩ is possible which contains
>> each k for which induction works.
>
> Induction shows that
> each k for which induction works has finitely many
> predecessors and by far mr successors.

Induction works for each predecessor and for
each successor.

We know that it works for each successor
because
induction works for each k in each ⟨0⋯i,i+1⋯n⟩
and
each successor is in some ⟨0⋯i,i+1⋯n⟩

> Therefore a rough estimation shows that
> induction works for less than half of all
> natnumbers.

You're basing your estimation on a comparison
between predecessors upon which induction works
and successors upon which induction ALSO works.

Each k in ℕ = ⋃{⟨0⋯i,i+1⋯?⟩} is
in some ⟨0⋯i,i+1⋯n⟩
Each predecessor of each natural number.
Each successor of each natural number.

Each k in some ⟨0⋯i,i+1⋯n⟩ is
some k for which,
if P(0) and ∀i ∈ ℕ, P(i) -> ~P(i+1)
then P(k)

Each k in ℕ = ⋃{⟨0⋯i,i+1⋯?⟩} is
some k for which,
if P(0) and ∀i ∈ ℕ, P(i) -> ~P(i+1)
then P(k)

There is no k in ℕ = ⋃{⟨0⋯i,i+1⋯?⟩}
for which induction does NOT work.

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 - Sun, 24 Jul 2022 16:13 UTC

Re: Natural numbers and vases III

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by: zelos...@gmail.com - Mon, 25 Jul 2022 09:39 UTC

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 - Mon, 25 Jul 2022 12:55 UTC

William schrieb am Sonntag, 24. Juli 2022 um 16:41:04 UTC+2:
> On Sunday, July 24, 2022 at 11:08:40 AM UTC-3, WM wrote:
>
> > Removing all elements which have infinitely many successors leaves these successors.
> Piffle. "All elements which have infinitely many successors" is the set |N. Removing the set |N leaves nothing.

Let the first number remain. Remove all other numbers. That is not the set. It should be possible.

> So removing "all elements which have infinitely many successors" leaves nothing.

Why can the numbers not be removed individually? Why must the whole set be addressed? Why is it a difference to say |N \ |N = { } and |N \ the sequence of all n = ?

Note that I did not say the uinion of all natnumbers but the sequence. That's a difference.

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 - Mon, 25 Jul 2022 12:59 UTC

Fritz Feldhase schrieb am Sonntag, 24. Juli 2022 um 17:35:30 UTC+2:
> On Sunday, July 24, 2022 at 4:08:40 PM UTC+2, WM wrote:
>
> > Removing all elements which have infinitely many successors leaves these successors.
> "These" successors do not exist,

No?

> True: For all n e IN: there are infinitely many m e IN such that m > n.
>
> False: There are infinitely many m e IN such that for all n e IN: m > n.
>
> You committed a quantifier shift,

Here it is justified.

If there are infinitely many successors for all n, and if it is possible to use all n for enumerating purposes, then it is also possible to remove all these n. Why not?

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 - Mon, 25 Jul 2022 13:04 UTC

Jim Burns schrieb am Sonntag, 24. Juli 2022 um 18:02:22 UTC+2:
> On 7/24/2022 10:13 AM, WM wrote:
> > Jim Burns schrieb
> > am Samstag, 23. Juli 2022 um 20:14:46 UTC+2:
>
> >> Induction works for each k in each ⟨0⋯i,i+1⋯n⟩
> >
> > Yes.
> >>> Induction does not work.
> >>> I works for every finite set
> >>> but not for the infinite set.
> No,
> there are infinitely-many k for which
> induction works.

Every k is the end of a FISON. All FISONs are finite and therefore they are a finite set:
by pigeon-holeprinciple.
>
> >> However,
> >> no ⟨0⋯i,i+1⋯𝔊⟩ is possible which contains
> >> each k for which induction works.
> >
> > Induction shows that
> > each k for which induction works has finitely many
> > predecessors and by far mr successors.
> Induction works for each predecessor and for
> each successor.

No, then the successors would be used up.

> > Therefore a rough estimation shows that
> > induction works for less than half of all
> > natnumbers.
> You're basing your estimation on a comparison
> between predecessors upon which induction works
> and successors upon which induction ALSO works.

No. ℵo successors remain provably beyond those numbers which induction can be applied to.

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 - Mon, 25 Jul 2022 13:07 UTC

Fritz Feldhase schrieb am Sonntag, 24. Juli 2022 um 18:13:13 UTC+2:
> On Sunday, July 24, 2022 at 4:13:17 PM UTC+2, WM wrote:
> >
> > a rough estimation shows that induction works for less than half of all natnumbers.
> Mathematics: induction works for _all_ natnumbers.
>
> Hint: A proof by induction typically ends with a/the statement An e IN: Phi[n].

Then you should be able to exhaust the successors of the numbers which induction can be applied to. This however is impossible.

> This means that Phi[n] holds for ALL natural numbers n. That is what induction proves.

That is what thoughtless or stupid matheologians usually claim.

Regards, WM

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
Date: Mon, 25 Jul 2022 08:54:05 -0500
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by: Sergio - Mon, 25 Jul 2022 13:54 UTC

On 7/25/2022 7:59 AM, WM wrote:
> Fritz Feldhase schrieb am Sonntag, 24. Juli 2022 um 17:35:30 UTC+2:
>> On Sunday, July 24, 2022 at 4:08:40 PM UTC+2, WM wrote:
>>
>>> Removing all elements which have infinitely many successors leaves these successors.
>> "These" successors do not exist,
>
> No?
>
>> True: For all n e IN: there are infinitely many m e IN such that m > n.
>>
>> False: There are infinitely many m e IN such that for all n e IN: m > n.
>>
>> You committed a quantifier shift,
>
> Here it is justified.
>
> If there are infinitely many successors for all n, and if it is possible to use all n for enumerating purposes, then it is also possible to remove all these n. Why not?
>
> Regards, WM
>
>

did you remove the "Last One" ?

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, 25 Jul 2022 09:33:15 -0500
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by: Sergio - Mon, 25 Jul 2022 14:33 UTC

On 7/25/2022 7:55 AM, WM wrote:
> William schrieb am Sonntag, 24. Juli 2022 um 16:41:04 UTC+2:
>> On Sunday, July 24, 2022 at 11:08:40 AM UTC-3, WM wrote:
>>
>>> Removing all elements which have infinitely many successors leaves these successors.
>> Piffle. "All elements which have infinitely many successors" is the set |N. Removing the set |N leaves nothing.
>
> Let the first number remain. Remove all other numbers.

So you changed from the set of natural numbers to the set {1} right ?

> That is not the set.

what set are you referring too?

>> So removing "all elements which have infinitely many successors" leaves nothing.
>
> Why can the numbers not be removed individually? Why must the whole set be addressed? Why is it a difference to say |N \ |N = { } and |N \ the sequence of all n = ?

one uses deletion of an infinite set, the other uses deletion of single elements. Can't you see that ?

> Regards, WM

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, 25 Jul 2022 09:36:22 -0500
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by: Sergio - Mon, 25 Jul 2022 14:36 UTC

On 7/25/2022 8:04 AM, WM wrote:
> Jim Burns schrieb am Sonntag, 24. Juli 2022 um 18:02:22 UTC+2:
>> On 7/24/2022 10:13 AM, WM wrote:
>>> Jim Burns schrieb
>>> am Samstag, 23. Juli 2022 um 20:14:46 UTC+2:
>>
>>>> Induction works for each k in each ⟨0⋯i,i+1⋯n⟩
>>>
>>> Yes.
>>>>> Induction does not work.
>>>>> I works for every finite set
>>>>> but not for the infinite set.
>> No,
>> there are infinitely-many k for which
>> induction works.
>
> Every k is the end of a FISON. All FISONs are finite and therefore they are a finite set:
> by pigeon-holeprinciple.

no, we are not stuffing birds into holes here.

>>
>>>> However,
>>>> no ⟨0⋯i,i+1⋯𝔊⟩ is possible which contains
>>>> each k for which induction works.
>>>
>>> Induction shows that
>>> each k for which induction works has finitely many
>>> predecessors and by far mr successors.
>> Induction works for each predecessor and for
>> each successor.
>
> No, then the successors would be used up.

Never use a Used Ant.

>
>>> Therefore a rough estimation shows that
>>> induction works for less than half of all
>>> natnumbers.
>> You're basing your estimation on a comparison
>> between predecessors upon which induction works
>> and successors upon which induction ALSO works.
>
> No. ℵo successors remain provably beyond those numbers which induction can be applied to.

Wrong. Show a specific case where this happens. Use Math and you will amaze everyone.

>
> Regards, WM

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, 25 Jul 2022 09:37:52 -0500
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by: Sergio - Mon, 25 Jul 2022 14:37 UTC

On 7/25/2022 8:07 AM, WM wrote:
> Fritz Feldhase schrieb am Sonntag, 24. Juli 2022 um 18:13:13 UTC+2:
>> On Sunday, July 24, 2022 at 4:13:17 PM UTC+2, WM wrote:
>>>
>>> a rough estimation shows that induction works for less than half of all natnumbers.
>> Mathematics: induction works for _all_ natnumbers.
>>
>> Hint: A proof by induction typically ends with a/the statement An e IN: Phi[n].
>
> Then you should be able to exhaust the successors of the numbers which induction can be applied to. This however is impossible.
>
>> This means that Phi[n] holds for ALL natural numbers n. That is what induction proves.
>
> That is what thoughtless or stupid matheologians usually claim.
>
> Regards, WM

you attack because Mathematical induction proves there are no dark numbers.

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 - Mon, 25 Jul 2022 14:53 UTC

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 - Mon, 25 Jul 2022 14:53 UTC

On Monday, 25 July 2022 at 10:00:01 UTC-3, WM wrote:
> Fritz Feldhase schrieb am Sonntag, 24. Juli 2022 um 17:35:30 UTC+2:
> > On Sunday, July 24, 2022 at 4:08:40 PM UTC+2, WM wrote:
> >
> > > Removing all elements which have infinitely many successors leaves these successors.
> > "These" successors do not exist,
> No?
> > True: For all n e IN: there are infinitely many m e IN such that m > n.
> >
> > False: There are infinitely many m e IN such that for all n e IN: m > n.
> >
> > You committed a quantifier shift,
> Here it is justified.

Bullshit, hog piss, and assorted other manure. You have shit for brains, and you crap your shit all over this thread (and others).

> If there are infinitely many successors for all n, and if it is possible to use all n for enumerating purposes, then it is also possible to remove all these n. Why not?

Of course it is possible to remove all n, but not sequentially, at least not without invoking a limit. You have shown yourself incapable of comprehending this simple step for two decades now. What a mental giant you are!

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 - Mon, 25 Jul 2022 15:19 UTC

On Monday, July 25, 2022 at 3:00:01 PM UTC+2, WM wrote:

> If there are infinitely many successors [...] it is also possible to remove all these [successors].

Of course it is possible, with the appropriate notions. That's why I devised the "\\" operation.

This way we can state

IN \\ {{s(n)} : n e IN} = IN \\ {{2}, {3}, {4}, ...} = {1}

This can be written as

IN \ {2} \ {3} \ {4} \ ... = {1}.

Why are you so bad at math, Mückenheim?

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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 - Mon, 25 Jul 2022 15:22 UTC

On Monday, July 25, 2022 at 3:07:21 PM UTC+2, WM wrote:
> Fritz Feldhase schrieb am Sonntag, 24. Juli 2022 um 18:13:13 UTC+2:
> > On Sunday, July 24, 2022 at 4:13:17 PM UTC+2, WM wrote:
> > > >
> > > a rough estimation shows that induction works for less than half of all natnumbers. #

Psychotic mumbo-jumbo.

Hint:

> > Mathematics: induction works for _all_ natnumbers.
> >
> > Note: A proof by induction typically ends with a/the statement An e IN: Phi[n].
> >
> > This means that Phi[n] holds for ALL natural numbers n. That is what induction proves.
> >
> That is what thoughtless or stupid matheologians usually claim.

No, that's just mathematics.

Hint: "Mathematical induction is a mathematical proof technique. It is essentially used to prove that a statement P(n) holds for every natural number n = 0, 1, 2, 3, ..." (Wikipedia)

Learn same logic, learn some math, Mückenheim.

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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 - Mon, 25 Jul 2022 15:23 UTC

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 - Mon, 25 Jul 2022 15:59 UTC

måndag 25 juli 2022 kl. 15:07:21 UTC+2 skrev WM:
> Fritz Feldhase schrieb am Sonntag, 24. Juli 2022 um 18:13:13 UTC+2:
> > On Sunday, July 24, 2022 at 4:13:17 PM UTC+2, WM wrote:
> > >
> > > a rough estimation shows that induction works for less than half of all natnumbers.
> > Mathematics: induction works for _all_ natnumbers.
> >
> > Hint: A proof by induction typically ends with a/the statement An e IN: Phi[n].
> Then you should be able to exhaust the successors of the numbers which induction can be applied to. This however is impossible.
> > This means that Phi[n] holds for ALL natural numbers n. That is what induction proves.
> That is what thoughtless or stupid matheologians usually claim.
>
> Regards, WM
no, it is what MATHEMATICIANS think, know and claim. You are an IDIOT!

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Subject: Re: Natural numbers and vases III
From: askaske...@gmail.com (WM)
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by: WM - Mon, 25 Jul 2022 17:05 UTC

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 - Mon, 25 Jul 2022 17:09 UTC

Gus Gassmann schrieb am Montag, 25. Juli 2022 um 16:53:41 UTC+2:
> On Monday, 25 July 2022 at 10:00:01 UTC-3, WM wrote:
> > Fritz Feldhase schrieb am Sonntag, 24. Juli 2022 um 17:35:30 UTC+2:
> > > On Sunday, July 24, 2022 at 4:08:40 PM UTC+2, WM wrote:
> > >
> > > > Removing all elements which have infinitely many successors leaves these successors.
> > > "These" successors do not exist,
> > No?
> > > True: For all n e IN: there are infinitely many m e IN such that m > n.
> > >
> > > False: There are infinitely many m e IN such that for all n e IN: m > n.
> > >
> > > You committed a quantifier shift,
> > Here it is justified.
> Bullshit, hog piss,

If every endsegment contains a number, then, by inclusion monotony, all endsegments contain the same number. This is unavoidable.

> > If there are infinitely many successors for all n, and if it is possible to use all n for enumerating purposes, then it is also possible to remove all these n. Why not?
> Of course it is possible to remove all n, but not sequentially,

For all definable numbers it is possible sequentially.

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 - Mon, 25 Jul 2022 17:12 UTC

Fritz Feldhase schrieb am Montag, 25. Juli 2022 um 17:19:10 UTC+2:
> On Monday, July 25, 2022 at 3:00:01 PM UTC+2, WM wrote:
>
> > If there are infinitely many successors [...] it is also possible to remove all these [successors].
>
> Of course it is possible, with the appropriate notions.

For all definable numers it is possible using \.

> That's why I devised the "\\" operation.
>
> This way we can state
>
> IN \\ {{s(n)} : n e IN} = IN \\ {{2}, {3}, {4}, ...} = {1}
>
> This can be written as
>
> IN \ {2} \ {3} \ {4} \ ... = {1}.
>
For every {n} infinitely many successors provably are not subtracted.

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 - Mon, 25 Jul 2022 17:28 UTC

Re: Natural numbers and vases III

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by: Sergio - Mon, 25 Jul 2022 19:11 UTC

On 7/25/2022 12:28 PM, William wrote:
> On Monday, July 25, 2022 at 2:06:00 PM UTC-3, WM wrote:
>> William schrieb am Montag, 25. Juli 2022 um 16:53:30 UTC+2:
>>> On Monday, July 25, 2022 at 9:55:31 AM UTC-3, WM wrote:
>>>> ... removing "all elements which have infinitely many successors" leaves nothing.
>>>> Why can the numbers not be removed individually?
>>> A stepwise process, "removed individually", will not reach a final state.
>> But you can define it by induction.
>
> Nope. You can define a final state but not by induction. And the stepwise process will not reach a final state
>
>

shuffling along the infinite highway to no where, cause you cant ever get there, cause it is like "infinite" man, really.

Re: Natural numbers and vases III

<tbmq2i$j59$1@gioia.aioe.org>

copy mid

https://www.novabbs.com/tech/article-flat.php?id=107354&group=sci.math#107354

copy link Newsgroups: sci.math

Path: i2pn2.org!i2pn.org!aioe.org!jq9Zon5wYWPEc6MdU7JpBw.user.46.165.242.75.POSTED!not-for-mail
From: inva...@invalid.com (Sergio)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases III
Date: Mon, 25 Jul 2022 14:14:25 -0500
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by: Sergio - Mon, 25 Jul 2022 19:14 UTC

On 7/25/2022 12:09 PM, WM wrote:
> Gus Gassmann schrieb am Montag, 25. Juli 2022 um 16:53:41 UTC+2:
>> On Monday, 25 July 2022 at 10:00:01 UTC-3, WM wrote:
>>> Fritz Feldhase schrieb am Sonntag, 24. Juli 2022 um 17:35:30 UTC+2:
>>>> On Sunday, July 24, 2022 at 4:08:40 PM UTC+2, WM wrote:
>>>>
>>>>> Removing all elements which have infinitely many successors leaves these successors.
>>>> "These" successors do not exist,
>>> No?
>>>> True: For all n e IN: there are infinitely many m e IN such that m > n.
>>>>
>>>> False: There are infinitely many m e IN such that for all n e IN: m > n.
>>>>
>>>> You committed a quantifier shift,
>>> Here it is justified.
>> Bullshit, hog piss,
>
> If every endsegment contains a number,

that is LAMO, each endsegment contains an infinity of numbers.

> then, by inclusion monotony, all endsegments contain the same number.

Wrong, and Wrong.

This is unavoidable.

no, it is not possible.

>
>>> If there are infinitely many successors for all n, and if it is possible to use all n for enumerating purposes, then it is also possible to remove all these n. Why not?
>> Of course it is possible to remove all n, but not sequentially,
>
> For all definable numbers it is possible sequentially.

QuACK!

>
> Regards, WM

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
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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	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
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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	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	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
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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
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