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Re: Natural numbers and vases

Subject	Author
Natural numbers and vases	William
Re: Natural numbers and vases	Ross A. Finlayson
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	William
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	Gus Gassmann
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	Gus Gassmann
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	Gus Gassmann
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	Gus Gassmann
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	FromTheRafters
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	Gus Gassmann
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	Gus Gassmann
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	Gus Gassmann
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	William
Re: Natural numbers and vases	Gus Gassmann
Re: Natural numbers and vases	Chris M. Thomasson
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	Gus Gassmann
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	Jim Burns
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	Jim Burns
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	Jim Burns
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	Jim Burns
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	Chet Hirasi
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	Gus Gassmann
Re: Natural numbers and vases	Chet Hirasi
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	FredJeffries
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	Chris M. Thomasson
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	Chris M. Thomasson
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	Chris M. Thomasson
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	Chris M. Thomasson
Re: Natural numbers and vases	Chris M. Thomasson
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	Chris M. Thomasson
Re: Natural numbers and vases	FromTheRafters
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	Gus Gassmann
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	Gus Gassmann
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	Gus Gassmann
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	William
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	zelos...@gmail.com
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	zelos...@gmail.com
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	zelos...@gmail.com
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	zelos...@gmail.com
Re: Natural numbers and vases	zelos...@gmail.com
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	zelos...@gmail.com
Re: Natural numbers and vases	zelos...@gmail.com
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	WM
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	Jim Burns
Re: Natural numbers and vases	sergio
Re: Natural numbers and vases	Ross A. Finlayson
Re: Natural numbers and vases	Fritz Feldhase
Re: Natural numbers and vases	Chris M. Thomasson
Re: Natural numbers and vases	Mostowski Collapse
Re: Natural numbers and vases	Socratis T.n.p.

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Re: Natural numbers and vases

<t7vj2h$s5s$1@gioia.aioe.org>

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From: inva...@invalid.com (sergi o)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases
Date: Fri, 10 Jun 2022 09:05:04 -0500
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by: sergi o - Fri, 10 Jun 2022 14:05 UTC

On 6/10/2022 8:49 AM, WM wrote:
> Fritz Feldhase schrieb am Donnerstag, 9. Juni 2022 um 22:41:01 UTC+2:
>
>> You can't decide TECHNICAL questions (in the context of set theory) by reference to "common sense",
>
> I know. No-one with common sense can accept set theory.

generalization red herring.

>
>> What's your point?
>
> The elements n of ℕ cannot exhaustively count ℕ as is shown by the attempted counting
> ∀n ∈ ℕ: |ℕ \ {1, 2, 3, ..., n}| = ℵo
Fail. that is not a "counting" equation

> How should they exhaustively count all fractions?

just use ℕ

>
> Regards, WM

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: wpihug...@gmail.com (William)
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by: William - Fri, 10 Jun 2022 14:30 UTC

On Friday, June 10, 2022 at 10:40:36 AM UTC-3, WM wrote:
> ∀n∈N says "for all n without exception".

For each element of |N without exception, not for the set "all n". Something that is true for each element of |N without exception (each element is finite) need not be true for the set |N (|N is infinite).

--
William Hughes

Re: Natural numbers and vases

<t7vpd8$50s$1@dont-email.me>

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From: nom...@afraid.org (FromTheRafters)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases
Date: Fri, 10 Jun 2022 11:53:06 -0400
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by: FromTheRafters - Fri, 10 Jun 2022 15:53 UTC

WM submitted this idea :
> FromTheRafters schrieb am Freitag, 10. Juni 2022 um 15:55:25 UTC+2:
>> WM formulated on Friday :
>>> William schrieb am Donnerstag, 9. Juni 2022 um 20:05:56 UTC+2:
>>>> On Thursday, June 9, 2022 at 2:00:19 PM UTC-3, WM wrote:
>>>>
>>>>> If all are subtracted, then an infinite set is subtracted.
>>>> Correct. ∀n∈N:|N \ {1,2,3,...,n}| = ℵo is an infinite number of
>>>> statements. Each statement is about subtracting a finite set. You cannot
>>>> combine these statements to a statement about subtracting an infinite
>>>> set.
>>>
>>> Why not? ∀n∈N says "for all n without exception".
>>>
>>> Look ∀n∈N: N \ {1,2,3,...,n} means counting the natural numbers by
>>> individuals n.
>> Does it? It looks to me like the special symbol for the 'set' of
>> natural numbers is being compared (set difference) to the 'set' of
>> 'elements' in a FISON ending at n
>
> You overlooked the universal quantifier. All n are applied.

So what, still no order implied nor step-by-step sequence necessary.

Re: Natural numbers and vases

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From: james.g....@att.net (Jim Burns)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases
Date: Fri, 10 Jun 2022 15:29:18 -0400
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by: Jim Burns - Fri, 10 Jun 2022 19:29 UTC

On 6/10/2022 9:34 AM, WM wrote:
> Jim Burns schrieb
> am Donnerstag, 9. Juni 2022 um 19:44:16 UTC+2:
>> On 6/9/2022 1:01 PM, WM wrote:

>>> Not by doing it,
>>> and not in a way that could be checked.
>>
>> The bijection,
>> the collection of pairs ⟨m/n,k/1⟩,
>> is known to exist.
>
> It is believed so,

The important thing here is *HOW WE KNOW*

And *HOW WE KNOW* is that the truth of later
claims follows inevitably from the truth
of earlier claims.

If x exists and y exists,
then ordered pair ⟨x,y⟩ exists.

( ⟨x,y⟩ = ⟨u,v⟩ iff x = u and y = v

Do you know that _ordered pairs_ exist
as I describe them?

If,
for each existing x,
one and only one of P(x) and ~P(x) is true,

then
a collection B exists, such that
all and only existing x such that P(x)
are in B.

( For collections B and C, B = C iff,
( for each existing x, x in B iff x in C

Do you know that _collections_ exist
as I describe them?

Do you know that the matrix M of fractions m/n
and its first column C[1] of fractions k/1
exists as I have described them earlier?
(We can return to this point in more detail.)

Let P(x) be true iff
x = ⟨m/n,k/1⟩
m/n is in M
k/1 is in C[1]
k = (m+n-1)*(m+n-2)/2+m

Let B be the collection of
all and only existing x such that P(x)
B is the collection of pairs ⟨m/n,k/1⟩
B exists.

> but it cannot be constructed.
> On the contrary when constructing it,
> we see that construction fails.

| The bijection B,
| the collection of pairs ⟨m/n,k/1⟩,
| is known to exist.

That's not a claim that it can be constructed.
It's a claim that it exists.

>> We start without checking each one.
>> We advance without checking each one.
>> We know,
>> after advancing to finitely-many further claims,
>> that this last claim is true of the same
>> infinitely-many that the first claim was true of.
>> No checks.
>
> That's why hitherto no-one has recognized the failure.

We have the option of talking about _all and only_
individuals which can (in principle) be constructed.
I've done that for you. You're welcome.

I'm doing that when I describe m,n,k such that,
for each BEFORE and AFTER ⊂ ⟨1,...,m⟩
and each BEFORE and AFTER ⊂ ⟨1,...,m⟩
and each BEFORE and AFTER ⊂ ⟨1,...,m⟩,
some j ends BEFORE and j+1 begins AFTER.

Each m,n,k _like that_ can (in principle)
be constructed.

The collection of individuals which can
(in principle) be constructed cannot
(even in principle) be constructed.
We know it exists.

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: askaske...@gmail.com (WM)
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by: WM - Sat, 11 Jun 2022 21:10 UTC

FromTheRafters schrieb am Freitag, 10. Juni 2022 um 17:53:23 UTC+2:
> WM submitted this idea :

> >>> Look ∀n∈N: N \ {1,2,3,...,n} means counting the natural numbers by
> >>> individuals n.
> >> Does it? It looks to me like the special symbol for the 'set' of
> >> natural numbers is being compared (set difference) to the 'set' of
> >> 'elements' in a FISON ending at n
> >
> > You overlooked the universal quantifier. All n are applied.
> So what, still no order implied nor step-by-step sequence necessary.

Ever heard of Peano? n ==> n+1

Ever heard of Cantor? Zwei „wohlgeordnete" Mengen (E) und (F) werden von derselben Anzahl genannt, wenn sie so gegenseitig eindeutig einander zugeordnet werden können,

Note that well-order is the basis for bijection.

Regards, WM

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: askaske...@gmail.com (WM)
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by: WM - Sat, 11 Jun 2022 21:14 UTC

William schrieb am Freitag, 10. Juni 2022 um 16:30:21 UTC+2:
> On Friday, June 10, 2022 at 10:40:36 AM UTC-3, WM wrote:
> > ∀n∈N says "for all n without exception".
> For each element of |N without exception, not for the set "all n".

Does ∀n∈N: f(n) = n define a bijection for the set N of all n?

Does ∀n∈N in this case cover all natnumbers without leaving infinitely many like in subtraction

∀n ∈ ℕ: |ℕ \ {1, 2, 3, ..., n}| = ℵo ?

Regards, WM

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: askaske...@gmail.com (WM)
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by: WM - Sat, 11 Jun 2022 21:26 UTC

Jim Burns schrieb am Freitag, 10. Juni 2022 um 21:29:29 UTC+2:
> On 6/10/2022 9:34 AM, WM wrote:
> > Jim Burns schrieb
> > am Donnerstag, 9. Juni 2022 um 19:44:16 UTC+2:
> >> On 6/9/2022 1:01 PM, WM wrote:
>
> >>> Not by doing it,
> >>> and not in a way that could be checked.
> >>
> >> The bijection,
> >> the collection of pairs ⟨m/n,k/1⟩,
> >> is known to exist.
> >
> > It is believed so,
> The important thing here is *HOW WE KNOW*

You know that ∀n∈N: f(n) = n defines a bijection for the set N of all n.

You know that ∀n∈N in this case covers all natnumbers without exception.

You know that, contrary to this, subtraction always leaves ℵo nat numbers:

∀n ∈ ℕ: |ℕ \ {1, 2, 3, ..., n}| = ℵo .

It does not cover all.

> If x exists and y exists,
> then ordered pair ⟨x,y⟩ exists.

But you assume that more exists than you can define. For instance you deny that ℵo numbers cannot be defined.

> The collection of individuals which can
> (in principle) be constructed cannot
> (even in principle) be constructed.
> We know it exists.

The collection of individuals which can (in principle) be constructed is potentially infinite.
If what you "know" to exist beyond that is either not existing or dark.

Regards, WM

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: wpihug...@gmail.com (William)
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by: William - Sat, 11 Jun 2022 21:31 UTC

On Saturday, June 11, 2022 at 6:14:31 PM UTC-3, WM wrote:
> William schrieb am Freitag, 10. Juni 2022 um 16:30:21 UTC+2:
> > On Friday, June 10, 2022 at 10:40:36 AM UTC-3, WM wrote:
> > > ∀n∈N says "for all n without exception".
> > For each element of |N without exception, not for the set "all n".
> Does ∀n∈N: f(n) = n define a bijection for the set N of all n?

Indeed,

For every element of "the set |N of all n": f(n)=n

defines a bijection.

--
William Hughes

Re: Natural numbers and vases

<t833bs$17vd$1@gioia.aioe.org>

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Subject: Re: Natural numbers and vases
Date: Sat, 11 Jun 2022 17:01:31 -0500
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by: sergi o - Sat, 11 Jun 2022 22:01 UTC

On 6/11/2022 4:26 PM, WM wrote:
> Jim Burns schrieb am Freitag, 10. Juni 2022 um 21:29:29 UTC+2:
>> On 6/10/2022 9:34 AM, WM wrote:
>>> Jim Burns schrieb
>>> am Donnerstag, 9. Juni 2022 um 19:44:16 UTC+2:
>>>> On 6/9/2022 1:01 PM, WM wrote:
>>
>>>>> Not by doing it,
>>>>> and not in a way that could be checked.
>>>>
>>>> The bijection,
>>>> the collection of pairs ⟨m/n,k/1⟩,
>>>> is known to exist.
>>>
>>> It is believed so,
>> The important thing here is *HOW WE KNOW*
>
> You know that ∀n∈N: f(n) = n defines a bijection for the set N of all n.

no. functions are not one to one mapping.

>
> You know that ∀n∈N in this case covers all natnumbers without exception.

no. it only states n is an element of N

>
> You know that, contrary to this, subtraction always leaves ℵo nat numbers:
>
> ∀n ∈ ℕ: |ℕ \ {1, 2, 3, ..., n}| = ℵo .

No. that is not subtraction, it is removal of elements.

>
> It does not cover all.

the equation says you STOPPED at n, this part right here see ? => "..., n}"

>
>> If x exists and y exists,
>> then ordered pair ⟨x,y⟩ exists.
>
> But you assume that more exists than you can define.

he made no such statement.

> For instance you deny that ℵo numbers cannot be defined.

Any natural number can be defined, so he is right.

>
>> The collection of individuals which can
>> (in principle) be constructed cannot
>> (even in principle) be constructed.
>> We know it exists.
>
> The collection of individuals which can (in principle) be constructed is potentially infinite.

Potentially infinite = Finite.

> If what you "know" to exist beyond that is either not existing or dark.

no, it is simply an Endsegment.

>
> Regards, WM

Re: Natural numbers and vases

<t837th$jri$1@gioia.aioe.org>

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Subject: Re: Natural numbers and vases
Date: Sat, 11 Jun 2022 18:19:12 -0500
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by: sergi o - Sat, 11 Jun 2022 23:19 UTC

On 6/11/2022 4:10 PM, WM wrote:
> FromTheRafters schrieb am Freitag, 10. Juni 2022 um 17:53:23 UTC+2:
>> WM submitted this idea :
>
>>>>> Look ∀n∈N: N \ {1,2,3,...,n} means counting the natural numbers by
>>>>> individuals n.
>>>> Does it? It looks to me like the special symbol for the 'set' of
>>>> natural numbers is being compared (set difference) to the 'set' of
>>>> 'elements' in a FISON ending at n
>>>
>>> You overlooked the universal quantifier. All n are applied.
>> So what, still no order implied nor step-by-step sequence necessary.
>
> Ever heard of Peano? n ==> n+1
>
> Ever heard of Cantor? Zwei „wohlgeordnete" Mengen (E) und (F) werden von derselben Anzahl genannt, wenn sie so gegenseitig eindeutig einander zugeordnet werden können,
>
> Note that well-order is the basis for bijection.
>
> Regards, WM

WM => "Look ∀n∈N: N \ {1,2,3,...,n} means counting the natural numbers by
individuals n."

your interpretation of this equation is wrong. there is no counting. There are no individuals.

It describes an Endsegment(n+1). It is the set of natural numbers with a FISON (elements) excluded.

If you mean something else, be clear, and write the correct equations for it.

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: talwus...@gmail.com (talwusapa)
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by: talwusapa - Sat, 11 Jun 2022 23:44 UTC

One of the reason why i kick people with my rain boot

Damn... nasty... I’m out of here

Re: Natural numbers and vases

<t83are$gh8$1@dont-email.me>

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From: nom...@afraid.org (FromTheRafters)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases
Date: Sat, 11 Jun 2022 20:09:11 -0400
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by: FromTheRafters - Sun, 12 Jun 2022 00:09 UTC

WM was thinking very hard :
> FromTheRafters schrieb am Freitag, 10. Juni 2022 um 17:53:23 UTC+2:
>> WM submitted this idea :
>
>>>>> Look ∀n∈N: N \ {1,2,3,...,n} means counting the natural numbers by
>>>>> individuals n.
>>>> Does it? It looks to me like the special symbol for the 'set' of
>>>> natural numbers is being compared (set difference) to the 'set' of
>>>> 'elements' in a FISON ending at n
>>>
>>> You overlooked the universal quantifier. All n are applied.
>> So what, still no order implied nor step-by-step sequence necessary.
>
> Ever heard of Peano? n ==> n+1

Sure, you can generate the naturals' sequence in their natural
sequential order, but the set's order type isn't affected by this, and
bijections between it and other sets don't depend upon the sequence or
your imagined step-by-step treatments.

> Ever heard of Cantor? Zwei „wohlgeordnete" Mengen (E) und (F) werden von
> derselben Anzahl genannt, wenn sie so gegenseitig eindeutig einander
> zugeordnet werden können,
>
> Note that well-order is the basis for bijection.

Is there a bijection between sets {elephant, donkey, walrus} and
{monkey, garden hoe, geode}?

Are these 'well ordered' sets?

Re: Natural numbers and vases

<t83b23$hsv$1@dont-email.me>

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From: nom...@afraid.org (FromTheRafters)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases
Date: Sat, 11 Jun 2022 20:12:44 -0400
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by: FromTheRafters - Sun, 12 Jun 2022 00:12 UTC

William explained :
> On Saturday, June 11, 2022 at 6:14:31 PM UTC-3, WM wrote:
>> William schrieb am Freitag, 10. Juni 2022 um 16:30:21 UTC+2:
>>> On Friday, June 10, 2022 at 10:40:36 AM UTC-3, WM wrote:
>>>> ∀n∈N says "for all n without exception".
>>> For each element of |N without exception, not for the set "all n".
>> Does ∀n∈N: f(n) = n define a bijection for the set N of all n?
>
> Indeed,
>
> For every element of "the set |N of all n": f(n)=n
>
> defines a bijection.

Narrowly avoiding an identity crisis.

Re: Natural numbers and vases

<37e3f6f5-de74-d1ce-ac22-a877b2f2ff6b@att.net>

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From: james.g....@att.net (Jim Burns)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases
Date: Sat, 11 Jun 2022 20:19:17 -0400
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by: Jim Burns - Sun, 12 Jun 2022 00:19 UTC

On 6/11/2022 5:26 PM, WM wrote:
> Jim Burns schrieb
> am Freitag, 10. Juni 2022 um 21:29:29 UTC+2:

>> The collection of individuals which can
>> (in principle) be constructed cannot
>> (even in principle) be constructed.
>> We know it exists.
>
> The collection of individuals which can
> (in principle) be constructed is
> potentially infinite.

_Each one_ in the collection of individuals which
can (in principle) be constructed
can (in principle) be constructed.
(Not a typo.)

Knowing that an individual can (in principle)
be constructed is quite a lot to know about
that individual.

Since we know what it means _here_ to be able to be
constructed, we know that a collection exists of
things _on the way to_ that individual such that,
for each split BEFORE and AFTER of things on the way,
some j ends BEFORE and j+1 begins AFTER.

That barely begins what we know, what we might know
at some point. There are steps we can take to
further claims such that, if we know the earlier
claims we also will know the further claims.
And we do know the earlier claims, as securely as
we know that an individual which can be constructed
can be constructed.

> If what you "know" to exist beyond that is
> either not existing or dark.

I have told you how we know what we know.

You appear to be unable to take that information
in. It's as though all explanations are printed
in a shade of ink to which you are color-blind.

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: askaske...@gmail.com (WM)
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by: WM - Sun, 12 Jun 2022 12:36 UTC

William schrieb am Samstag, 11. Juni 2022 um 23:31:36 UTC+2:
> On Saturday, June 11, 2022 at 6:14:31 PM UTC-3, WM wrote:
> > William schrieb am Freitag, 10. Juni 2022 um 16:30:21 UTC+2:
> > > On Friday, June 10, 2022 at 10:40:36 AM UTC-3, WM wrote:
> > > > ∀n∈N says "for all n without exception".
> > > For each element of |N without exception, not for the set "all n".
> > Does ∀n∈N: f(n) = n define a bijection for the set N of all n?
> Indeed,
>
> For every element of "the set |N of all n": f(n)=n
>
> defines a bijection.

Here the universal quantifier works perfectly such that no n is missing. Why does it not work perfectly here

∀n ∈ ℕ: |ℕ \ {1, 2, 3, ..., n}| = ℵo ?

Note that subtracting finite initial segments from ℕ can be interptreted as a bijection n <--> {1, 2, 3, ..., n}

Regards, WM

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: askaske...@gmail.com (WM)
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by: WM - Sun, 12 Jun 2022 12:43 UTC

FromTheRafters schrieb am Sonntag, 12. Juni 2022 um 02:09:28 UTC+2:
> WM was thinking very hard :
> > FromTheRafters schrieb am Freitag, 10. Juni 2022 um 17:53:23 UTC+2:
> >> WM submitted this idea :
> >
> >>>>> Look ∀n∈N: N \ {1,2,3,...,n} means counting the natural numbers by
> >>>>> individuals n.
> >>>> Does it? It looks to me like the special symbol for the 'set' of
> >>>> natural numbers is being compared (set difference) to the 'set' of
> >>>> 'elements' in a FISON ending at n
> >>>
> >>> You overlooked the universal quantifier. All n are applied.
> >> So what, still no order implied nor step-by-step sequence necessary.
> >
> > Ever heard of Peano? n ==> n+1
> Sure, you can generate the naturals' sequence in their natural
> sequential order, but the set's order type isn't affected by this, and
> bijections between it and other sets don't depend upon the sequence or
> your imagined step-by-step treatments.
> > Ever heard of Cantor? Zwei „wohlgeordnete" Mengen (E) und (F) werden von
> > derselben Anzahl genannt, wenn sie so gegenseitig eindeutig einander
> > zugeordnet werden können,
> >
> > Note that well-order is the basis for bijection.
> Is there a bijection between sets {elephant, donkey, walrus} and
> {monkey, garden hoe, geode}?
>
> Are these 'well ordered' sets?

Every finite set can be well-ordered. Every extensionally given set is well-ordered by the order of writing.

Regards, WM

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: wpihug...@gmail.com (William)
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by: William - Sun, 12 Jun 2022 14:26 UTC

On Sunday, June 12, 2022 at 9:36:17 AM UTC-3, WM wrote:
> William schrieb am Samstag, 11. Juni 2022 um 23:31:36 UTC+2:
> > On Saturday, June 11, 2022 at 6:14:31 PM UTC-3, WM wrote:
> > > William schrieb am Freitag, 10. Juni 2022 um 16:30:21 UTC+2:
> > > > On Friday, June 10, 2022 at 10:40:36 AM UTC-3, WM wrote:
> > > > > ∀n∈N says "for all n without exception".
> > > > For each element of |N without exception, not for the set "all n".
> > > Does ∀n∈N: f(n) = n define a bijection for the set N of all n?
> > Indeed,
> >
> > For every element of "the set |N of all n": f(n)=n
> >
> > defines a bijection.
> Here the universal quantifier works perfectly such that no n is missing.

Indeed, no n, element of the set "all n", is missing.

Look! Over there! A pink elephant!

The set is not missing.

--
William Hughes

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
Date: Sun, 12 Jun 2022 09:29:39 -0500
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by: sergi o - Sun, 12 Jun 2022 14:29 UTC

On 6/12/2022 7:36 AM, WM wrote:
> William schrieb am Samstag, 11. Juni 2022 um 23:31:36 UTC+2:
>> On Saturday, June 11, 2022 at 6:14:31 PM UTC-3, WM wrote:
>>> William schrieb am Freitag, 10. Juni 2022 um 16:30:21 UTC+2:
>>>> On Friday, June 10, 2022 at 10:40:36 AM UTC-3, WM wrote:
>>>>> ∀n∈N says "for all n without exception".
>>>> For each element of |N without exception, not for the set "all n".
>>> Does ∀n∈N: f(n) = n define a bijection for the set N of all n?
>> Indeed,
>>
>> For every element of "the set |N of all n": f(n)=n
>>
>> defines a bijection.
>
> Here the universal quantifier works perfectly such that no n is missing. Why does it not work perfectly here
>
> ∀n ∈ ℕ: |ℕ \ {1, 2, 3, ..., n}| = ℵo ?

that is called an Endsegment(n+1),
all Endsegments are infinite.

why do you say it is not working ?

> Regards, WM

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: franz.fr...@gmail.com (Fritz Feldhase)
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by: Fritz Feldhase - Sun, 12 Jun 2022 14:32 UTC

On Sunday, June 12, 2022 at 2:43:30 PM UTC+2, WM wrote:
> FromTheRafters schrieb am Sonntag, 12. Juni 2022 um 02:09:28 UTC+2:
> > WM was thinking very hard:
> > >
> > > Note that well-order is the basis for bijection.
> > >
> > Is there a bijection between sets {elephant, donkey, walrus} and
> > {monkey, garden hoe, geode}?
> >
> > Are these 'well ordered' sets?
> >
> Every extensionally given set is well-ordered by the order of writing.

No, it isn't, you silly crank, since

{elephant, donkey, walrus} = {donkey, elephant, walrus} = {walrus, donkey, elephant} = ...

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: franz.fr...@gmail.com (Fritz Feldhase)
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by: Fritz Feldhase - Sun, 12 Jun 2022 15:14 UTC

On Saturday, June 11, 2022 at 11:10:38 PM UTC+2, WM wrote:

> well-order is the basis for bijection.

Huh?!

Re: Natural numbers and vases

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by: sergi o - Sun, 12 Jun 2022 15:34 UTC

On 6/12/2022 7:43 AM, WM wrote:
> FromTheRafters schrieb am Sonntag, 12. Juni 2022 um 02:09:28 UTC+2:
>> WM was thinking very hard :
>>> FromTheRafters schrieb am Freitag, 10. Juni 2022 um 17:53:23 UTC+2:
>>>> WM submitted this idea :
>>>
>>>>>>> Look ∀n∈N: N \ {1,2,3,...,n} means counting the natural numbers by
>>>>>>> individuals n.
>>>>>> Does it? It looks to me like the special symbol for the 'set' of
>>>>>> natural numbers is being compared (set difference) to the 'set' of
>>>>>> 'elements' in a FISON ending at n
>>>>>
>>>>> You overlooked the universal quantifier. All n are applied.
>>>> So what, still no order implied nor step-by-step sequence necessary.
>>>
>>> Ever heard of Peano? n ==> n+1
>> Sure, you can generate the naturals' sequence in their natural
>> sequential order, but the set's order type isn't affected by this, and
>> bijections between it and other sets don't depend upon the sequence or
>> your imagined step-by-step treatments.
>>> Ever heard of Cantor? Zwei „wohlgeordnete" Mengen (E) und (F) werden von
>>> derselben Anzahl genannt, wenn sie so gegenseitig eindeutig einander
>>> zugeordnet werden können,
>>>
>>> Note that well-order is the basis for bijection.
>> Is there a bijection between sets {elephant, donkey, walrus} and
>> {monkey, garden hoe, geode}?
>>
>> Are these 'well ordered' sets?
>
> Every finite set can be well-ordered.

no. "Every finite totally ordered set is well ordered."
"A totally ordered set (A,<=) is said to be well ordered (or have a well-founded order) iff every nonempty subset of A has a least element."

> Every extensionally given set is well-ordered by the order of writing.

no. You need to study up!

>
> Regards, WM

Re: Natural numbers and vases

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by: Ben Bacarisse - Sun, 12 Jun 2022 15:43 UTC

sergi o <invalad@invalid.com> writes:

> On 6/12/2022 7:43 AM, WM wrote:

>> Every finite set can be well-ordered.
>
> no. "Every finite totally ordered set is well ordered."
> "A totally ordered set (A,<=) is said to be well ordered (or have a
> well-founded order) iff every nonempty subset of A has a least
> element."

I'm not sure what your objection here is. That a finite set A "can be
well-ordered" just means that there exists some relation R on A such
that R is a well-founded order.

--
Ben.

Re: Natural numbers and vases

<t8541i$e9r$1@gioia.aioe.org>

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Newsgroups: sci.math
Subject: Re: Natural numbers and vases
Date: Sun, 12 Jun 2022 11:25:21 -0500
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by: sergi o - Sun, 12 Jun 2022 16:25 UTC

On 6/12/2022 10:43 AM, Ben Bacarisse wrote:
> sergi o <invalad@invalid.com> writes:
>
>> On 6/12/2022 7:43 AM, WM wrote:
>
>>> Every finite set can be well-ordered.
>>
>> no. "Every finite totally ordered set is well ordered."
>> "A totally ordered set (A,<=) is said to be well ordered (or have a
>> well-founded order) iff every nonempty subset of A has a least
>> element."
>
> I'm not sure what your objection here is. That a finite set A "can be
> well-ordered" just means that there exists some relation R on A such
> that R is a well-founded order.
>

he is too general

{stamps, letters, envelopes} is a finite set...

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: askaske...@gmail.com (WM)
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by: WM - Sun, 12 Jun 2022 16:55 UTC

Fritz Feldhase schrieb am Sonntag, 12. Juni 2022 um 17:14:42 UTC+2:
> On Saturday, June 11, 2022 at 11:10:38 PM UTC+2, WM wrote:
>
> > well-order is the basis for bijection.
> Huh?!

"Zwei wohlgeordnete Mengen M und N heissen von gleichem Typus oder auch von gleicher Anzahl, wenn sie sich gegenseitig eindeutig und vollständig unter beidseitiger Wahrung der Rangfolge ihrer Elemente auf einander beziehen, abbilden lassen;" [G. Cantor, letter to R. Lipschitz (19 Nov 1883)]

Regards, WM

Re: Natural numbers and vases

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Subject: Re: Natural numbers and vases
From: askaske...@gmail.com (WM)
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by: WM - Sun, 12 Jun 2022 16:58 UTC

Fritz Feldhase schrieb am Sonntag, 12. Juni 2022 um 16:32:30 UTC+2:
> On Sunday, June 12, 2022 at 2:43:30 PM UTC+2, WM wrote:
> > FromTheRafters schrieb am Sonntag, 12. Juni 2022 um 02:09:28 UTC+2:
> > > WM was thinking very hard:
> > > >
> > > > Note that well-order is the basis for bijection.
> > > >
> > > Is there a bijection between sets {elephant, donkey, walrus} and
> > > {monkey, garden hoe, geode}?
> > >
> > > Are these 'well ordered' sets?
> > >
> > Every extensionally given set is well-ordered by the order of writing.
> No, it isn't
>
> {elephant, donkey, walrus} = {donkey, elephant, walrus} = {walrus, donkey, elephant} = ...

This yields three well-orderings. One is sufficient.

Regards, WM

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