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tech / sci.math / Re: |N is a Peano set

Re: |N is a Peano set

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Subject: Re: |N is a Peano set
From: wolfgang...@hs-augsburg.de (WM)
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by: WM - Sat, 5 Jun 2021 11:05 UTC

William schrieb am Freitag, 4. Juni 2021 um 23:15:05 UTC+2:
> On Friday, June 4, 2021 at 3:05:14 PM UTC-3, WM wrote:
>
> Playing with the ambiguity of the natural language word "all" . This does take me back

We can use strict mathematics.
Achilles passes more than any finite number of frames before frame omega.
You and others cannot name more than a FISON of them, which, how large ever, is finite and has an infinite complement in |N. That proves clearly and without any intuition, that aleph_0 frames cannot be named. I call that dark.. But any other name would not change the facts.

Similar with descending sequences: Between omega and the first accessible natnumbers there are aleph_0 natnumbers.

> > omega is next to all natnumbers n. (1)
> Using all natnumbers to mean the set of natural numbers

including the dark natnumbers.

> > omega - n = omega for all natnumbers n. (2)
> Using all natnumbers to mean each element of the set of natural numbers.

which is visible.

Note that the set cannot extend farther than its elements.

> Note that the set (0,1] is "next to" 0 even though no element of the set is "next to" 0.

Note that this is impossible. Dark elements are required, and the proof of dark elements is easy because between every accessible unit fraction and zero there remain aleph_0 dark unit fractions. You do not deny that. Alas in confusing logic you claim counterfactually that all these remaining elements could be named.

Regards, WM

Re: |N is a Peano set

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From: imva...@invalid.com (Sergio)
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Subject: Re: |N is a Peano set
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by: Sergio - Sat, 5 Jun 2021 13:00 UTC

On 6/5/2021 6:05 AM, WM wrote:
> William schrieb am Freitag, 4. Juni 2021 um 23:15:05 UTC+2:
>> On Friday, June 4, 2021 at 3:05:14 PM UTC-3, WM wrote:
>>
>> Playing with the ambiguity of the natural language word "all" . This does take me back
>
> We can use strict mathematics.
> Achilles passes more than any finite number of frames before frame omega.
> You and others cannot name more than a FISON of them, which, how large ever, is finite and has an infinite complement in |N. That proves clearly and without any intuition, that aleph_0 frames cannot be named. I call that dark. But any other name would not change the facts.
>
> Similar with descending sequences: Between omega and the first accessible natnumbers there are aleph_0 natnumbers.

Draw a line on paper with a pencil.

Put a ruler next to that line.

Between any two points on the ruler, there is an infinite number of points.

As you drew that line, your pencil passes over infinite number of
infinities of points, but only used a finite amount of graphite.

no dark numbers were ever needed.

The pencil won the race, and it "jumped" to multiple omegas as it passed
each number on the ruler.

You are using LAMO Math that relies on a fixed set of words you know,
and that is not math.

>
>>> omega is next to all natnumbers n. (1)
>> Using all natnumbers to mean the set of natural numbers
>
> including the dark natnumbers.
>
>>> omega - n = omega for all natnumbers n. (2)
>> Using all natnumbers to mean each element of the set of natural numbers.
>
> which is visible.
>
> Note that the set cannot extend farther than its elements.
>
>> Note that the set (0,1] is "next to" 0 even though no element of the set is "next to" 0.
>
> Note that this is impossible. Dark elements are required, and the proof of dark elements is easy because between every accessible unit fraction and zero there remain aleph_0 dark unit fractions. You do not deny that. Alas in confusing logic you claim counterfactually that all these remaining elements could be named.
>
> Regards, WM
>

Re: |N is a Peano set

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Subject: Re: |N is a Peano set
From: wpihug...@gmail.com (William)
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by: William - Sat, 5 Jun 2021 13:38 UTC

On Saturday, June 5, 2021 at 8:05:39 AM UTC-3, WM wrote:
> William schrieb am Freitag, 4. Juni 2021 um 23:15:05 UTC+2:

> > Note that the set (0,1] is "next to" 0 even though no element of the set is "next to" 0.
> Note that this is impossible

Nope the proof has been given.

>. Dark elements are required, and the proof of dark elements is easy because between every accessible unit fraction and zero there remain aleph_0 dark unit fractions. You do not deny that.

Of course not. You can always extend the ordering and put some set that you made up in between the unit fractions and 0. I do deny that, despite the misleading name you use for the set you made up, "dark unit fractions" are unit fractions. I also claim that between each definable unit fraction and 0 there is a set of definable unit fractions with cardinality aleph_0. This follow simply from the fact that |N (the set that you call the definable natural numbers) is a Peano set.

==
William Hughes

Re: |N is a Peano set

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Subject: Re: |N is a Peano set
From: wolfgang...@hs-augsburg.de (WM)
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by: WM - Sat, 5 Jun 2021 15:04 UTC

Re: |N is a Peano set

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Subject: Re: |N is a Peano set
From: wpihug...@gmail.com (William)
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by: William - Sat, 5 Jun 2021 15:41 UTC

Re: |N is a Peano set

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Subject: Re: |N is a Peano set
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by: Sergio - Sat, 5 Jun 2021 16:15 UTC

On 6/5/2021 10:04 AM, WM wrote:
> William schrieb am Samstag, 5. Juni 2021 um 15:38:05 UTC+2:
>
>> I also claim that between each definable unit fraction and 0 there is a set of definable unit fractions with cardinality aleph_0.
>
> A set with card aleph_0 has more than any finite number of elements.

duh.

>If all aleph_0 were definable, then they could be defined.

they are definable.

> But you cannot define so many (not because of lacking time - it would be sufficient to define the natnumber separating |N into two halves. All accessible sets have halves. If all |N was accessible, then half of it would be accessible too). Therefore your claim is wrong.

So cutting a set in half makes both parts not definable ? That does not
follow.

>
> Or do you claim that such an element proving your claim of aleph_0 definable elements is by definition undefinable? Only something like that will allow to maintain your axioms.

Axioms are first, your defining is after that, you have it backwards.

>
> Regards, WM
>
>
>

Re: |N is a Peano set

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Subject: Re: |N is a Peano set
From: wolfgang...@hs-augsburg.de (WM)
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by: WM - Sat, 5 Jun 2021 16:28 UTC

Re: |N is a Peano set

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Subject: Re: |N is a Peano set
From: wolfgang...@hs-augsburg.de (WM)
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by: WM - Sat, 5 Jun 2021 16:35 UTC

Re: |N is a Peano set

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Subject: Re: |N is a Peano set
From: wpihug...@gmail.com (William)
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by: William - Sat, 5 Jun 2021 16:49 UTC

Re: |N is a Peano set

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Subject: Re: |N is a Peano set
From: wpihug...@gmail.com (William)
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by: William - Sat, 5 Jun 2021 16:49 UTC

On Saturday, June 5, 2021 at 1:28:53 PM UTC-3, WM wrote:
> William schrieb am Samstag, 5. Juni 2021 um 17:41:20 UTC+2:
> > On Saturday, June 5, 2021 at 12:04:43 PM UTC-3, WM wrote:
> > > William schrieb am Samstag, 5. Juni 2021 um 15:38:05 UTC+2:
> > >
> > > > I also claim that between each definable unit fraction and 0 there is a set of definable unit fractions with cardinality aleph_0.
> > > A set with card aleph_0 has more than any finite number of elements. If all aleph_0 were definable, then they could be defined.
> > We have been through this. The set of *definable* elements is infinite, the set of *defined* elements is finite.
> Let us compromise: The collection |N_def of defined elements will remain finite in every system and for all times. But it has not upper bound.

Correct, and the set of *definable* elements has cardinality aleph_0

> >
> > Do you ever intend to tell us what your misleadingly named "dark unit fractions" are?
> The complement of the collection of |N_def is dark: |N \ |N_def.

This does not work. Your misleadingly named "dark natural numbers" are not natural numbers, so they cannot be elements of |N.

--
William Hughes

Re: |N is a Peano set

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Subject: Re: |N is a Peano set
From: horand.g...@gmail.com (Gus Gassmann)
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by: Gus Gassmann - Sat, 5 Jun 2021 16:50 UTC

Re: |N is a Peano set

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Subject: Re: |N is a Peano set
From: wolfgang...@hs-augsburg.de (WM)
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by: WM - Sat, 5 Jun 2021 17:22 UTC

William schrieb am Samstag, 5. Juni 2021 um 18:49:40 UTC+2:
> On Saturday, June 5, 2021 at 1:28:53 PM UTC-3, WM wrote:
> > William schrieb am Samstag, 5. Juni 2021 um 17:41:20 UTC+2:
> > > On Saturday, June 5, 2021 at 12:04:43 PM UTC-3, WM wrote:
> > > > William schrieb am Samstag, 5. Juni 2021 um 15:38:05 UTC+2:
> > > >
> > > > > I also claim that between each definable unit fraction and 0 there is a set of definable unit fractions with cardinality aleph_0.
> > > > A set with card aleph_0 has more than any finite number of elements. If all aleph_0 were definable, then they could be defined.
> > > We have been through this. The set of *definable* elements is infinite, the set of *defined* elements is finite.
> > Let us compromise: The collection |N_def of defined elements will remain finite in every system and for all times. But it has not upper bound.
> Correct, and the set of *definable* elements has cardinality aleph_0

Ok, if you like that phrase.
> > >
> > > Do you ever intend to tell us what your misleadingly named "dark unit fractions" are?
> > The complement of the collection of |N_def is dark: |N \ |N_def.
> This does not work. Your misleadingly named "dark natural numbers" are not natural numbers, so they cannot be elements of |N.

You agreed hat the collection |N_def nof defined elements is finite, hence it is a subset of the set |N which you call the set of definable elements. Hence the difference |N \ |N_def is not empty but infinite. What does not work?

Regards, WM

Re: |N is a Peano set

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Subject: Re: |N is a Peano set
From: wolfgang...@hs-augsburg.de (WM)
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by: WM - Sat, 5 Jun 2021 17:22 UTC

Re: |N is a Peano set

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Subject: Re: |N is a Peano set
From: wolfgang...@hs-augsburg.de (WM)
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by: WM - Sat, 5 Jun 2021 17:29 UTC

Re: |N is a Peano set

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Subject: Re: |N is a Peano set
From: franz.fr...@gmail.com (Greg Cunt)
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by: Greg Cunt - Sat, 5 Jun 2021 17:59 UTC

Re: |N is a Peano set

<b9b5b12d-d7f3-eb4e-e3be-6729cc8169bc@att.net>

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From: james.g....@att.net (Jim Burns)
Newsgroups: sci.math
Subject: Re: |N is a Peano set
Date: Sat, 5 Jun 2021 14:30:38 -0400
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by: Jim Burns - Sat, 5 Jun 2021 18:30 UTC

On 6/3/2021 3:44 PM, WM wrote:
> Jim Burns schrieb
> am Donnerstag, 3. Juni 2021 um 08:02:04 UTC+2:
>> On 6/2/2021 8:00 AM, WM wrote:

>>> But we start with excluding teleportation from the scratch.

You are describing Achilles' path.

>> How do we say "No teleporting allowed"?
>>
>> We say
>> | The path on which Achilles chases and passes the tortoise
>> | is _continuous_ on a _connected_ time-domain.
>
> No time required to see that the path is without gaps.

I am describing Achilles' path.
| For a _time interval_ from "Achilles starts running" to
| "Achilles passes the tortoise", Achilles is somewhere, at a
| unique point in space.

We should say more about the path.
| We exclude teleporting.

Good. And how do we say that?

Seeing that the path is without gaps is less useful than saying.
(I won't mention seeing _all_ paths, a supertask.)

Say what you have seen and we will wave truth-preserving
inferences over it, so that more true-for-all statements
will appear. Now we're back at
| How do we say "No teleporting"?

We say
| The path on which Achilles chases and passes the tortoise
| is _continuous_ on a _connected_ time-domain.

That is not the way that you describe his path.
But the way that you describe his path has serious problems.
You think it has problems, too.

<WM<Greg Gassmann>>
>
>> Zeno's arrow is the one that *doesn't move*.
>
> That's why the path between these points is missing.
>
Date: Fri, 4 Jun 2021 05:30:29 -0700 (PDT)
Message-ID: <22652321-c929-41b0-bf7b-c6315e7990e1n@googlegroups.com>

The path between your chosen points is missing
_from your description_

You should describe the path differently, so that the path
is NOT missing from your description. You know, "description"?

You present dark points as the solution to your problem,
but all you say is that they're the missing points.
Other than giving them a cute name, all you've done is say
"Yes, my description misses big chunks of the path".

----
Here is my description of Achilles' path.

| Each time-point in the time-interval [0,1] ==
| [ when-Achilles-starts-running, when-Achilles-passes-tortoise ]
| is paired with no more than one and no less than one space point.

Formally, we can describe this as a crowd of (x,t) pairs
fulfilling certain conditions.

| _At least_ finite-length decimal fractions of 1 exist in [0,1].
| I call these the bright points.

I stole your term "bright points" to describe them. There are more
bright points, the way you use the term, but, with the next condition,
all the other bright points and much more are included, so
mentioning more seems like an unnecessary complication.

| No teleporting. This condition has two parts.
| | For each space-time point (x,t) on the path, changes to the
| x-coordinate will be less than epsilon, for any epsilon > 0,
| if changes to the t-coordinate are less than delta, for some
| delta > 0. The familiar _continuity_
| | For each cut B,A of the points in [0,1], each point in B
| before each point in A, a point c exists between them.
| Without this condition, we have not excluded teleporting.

If there is any cut B,A without c between them, Achilles has a
_continuous_ path standing at the start through the time-points
in B, teleporting, then standing at the tortoise through the
the time-points in A.

I like to think of the bright points as a wire armature for a
sculpture of a geometric line. Adding the points between cuts
daubs clay onto the armature and keeps the sculpture from
falling apart.

"Bright points, no teleporting" implies more points exist than
definitions. That seems to be a concern of yours.
However, I don't see how points in a continuum (no teleporting!)
would be well-described as points in a sequence (neighbors before
and after with none between).

Re: |N is a Peano set

<s9gg70$6kc$1@gioia.aioe.org>

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From: inva...@invalid.com (Sergio)
Newsgroups: sci.math
Subject: Re: |N is a Peano set
Date: Sat, 5 Jun 2021 13:36:17 -0500
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by: Sergio - Sat, 5 Jun 2021 18:36 UTC

On 6/5/2021 11:35 AM, WM wrote:
> Sergio schrieb am Samstag, 5. Juni 2021 um 18:15:46 UTC+2:
>
>> So cutting a set in half makes both parts not definable ?
>
> No. If all elements of a set are definable, then we can define half of them.
>
>> Axioms are first,
>
> No, mathematics comes first.

gad !!!!!

Your demonstrated time sequence of Math Knowledge is quickly diminishing
and approaching the final Null Set !!

WM(1) = 1
WM(k+1) = WM(k)*(0.9)

k increases by 1 every 3 months.

(of course Algebra, calculus, and higher math areas was never included
in WM(1) sequence)

Better STOP now!

no wonder your Dark Numbers are catching up with you.

>Then some people try to concentrate it. In history always axioms have been devised after mathematics.

100% wrong. No wonder you never took algebra.

>
> Regards, WM
>

Re: |N is a Peano set

<572ec7de-ad89-4d19-9015-3c174abec5b6n@googlegroups.com>

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Subject: Re: |N is a Peano set
From: wpihug...@gmail.com (William)
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by: William - Sat, 5 Jun 2021 18:49 UTC

On Saturday, June 5, 2021 at 2:22:14 PM UTC-3, WM wrote:
> William schrieb am Samstag, 5. Juni 2021 um 18:49:15 UTC+2:

> > The set of *definable* numbers is infinite. The term "half" has no meaning for infinite sets.
> The term definable element has meaning. For every definable natnumber 2n also n is definable.

Yes, Note the {n|n definable element of |N) and {2n| n a definable element of N} are not disjoint. However, we can decompose |N into disjoint N_1 and N_2, where both are infinite. What we cannot do is find an m element of |N such that the sets n<,m and n>=m
are both infinite. The first is always finite. So it makes no sense to talk about the second half of an infinite set.

> If there were aleph_0 definable elements, then the set of their halves would have card aleph_0 too
> But obviously its elements are not completely definable.
> This shows that the second half

Meaningless for infinite sets.

==
William Hughes

Re: |N is a Peano set

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Subject: Re: |N is a Peano set
From: wpihug...@gmail.com (William)
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by: William - Sat, 5 Jun 2021 19:11 UTC

On Saturday, June 5, 2021 at 2:22:08 PM UTC-3, WM wrote:
> William schrieb am Samstag, 5. Juni 2021 um 18:49:40 UTC+2:
> > On Saturday, June 5, 2021 at 1:28:53 PM UTC-3, WM wrote:
> > > William schrieb am Samstag, 5. Juni 2021 um 17:41:20 UTC+2:
>
> > > > Do you ever intend to tell us what your misleadingly named "dark unit fractions" are?
> > > The complement of the collection of |N_def is dark: |N \ |N_def.
> > This does not work. Your misleadingly named "dark natural numbers" are not natural numbers, so they cannot be elements of |N.
> You agreed hat the collection |N_def nof defined elements is finite, hence it is a subset of the set |N which you call the set of definable elements.. Hence the difference |N \ |N_def is not empty but infinite. What does not work?

|N is a set that contains only natural numbers. So things that are not natural numbers, e.g. your misleadingly named "dark natural numbers", cannot be elements of |N and thus cannot be elements of |N \ |N_def. Thus no "dark natural number" is an element of |N \ |N_def.

--
William Hughes

Re: |N is a Peano set

<3bf95a34-5f31-4111-a29f-02cb7c594dbdn@googlegroups.com>

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Subject: Re: |N is a Peano set
From: wolfgang...@hs-augsburg.de (WM)
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by: WM - Sat, 5 Jun 2021 20:26 UTC

William schrieb am Samstag, 5. Juni 2021 um 21:11:06 UTC+2:
> On Saturday, June 5, 2021 at 2:22:08 PM UTC-3, WM wrote:
> > William schrieb am Samstag, 5. Juni 2021 um 18:49:40 UTC+2:
> > > On Saturday, June 5, 2021 at 1:28:53 PM UTC-3, WM wrote:
> > > > William schrieb am Samstag, 5. Juni 2021 um 17:41:20 UTC+2:
> >
> > > > > Do you ever intend to tell us what your misleadingly named "dark unit fractions" are?
> > > > The complement of the collection of |N_def is dark: |N \ |N_def.
> > > This does not work. Your misleadingly named "dark natural numbers" are not natural numbers, so they cannot be elements of |N.
> > You agreed hat the collection |N_def of defined elements is finite, hence it is a subset of the set |N which you call the set of definable elements. Hence the difference |N \ |N_def is not empty but infinite. What does not work?
> |N is a set that contains only natural numbers. So things that are not natural numbers, e.g. your misleadingly named "dark natural numbers", cannot be elements of |N and thus cannot be elements of |N \ |N_def. Thus no "dark natural number" is an element of |N \ |N_def.

But you have agreed that |N is infinite and |N_def, the set of ever defined natnumbers is finite.
Therefore |N \ |N_def consists of natural number which never will be defined, therefore they cannot be defined, although you call them definable natnumbers. This is a license in order to aim at the central question: What is your idea of |N \ |N_def ? Is it infinite, finite, empty?

Regards, WM

Re: |N is a Peano set

<798e9f1c-a013-4655-9065-cc31356216b8n@googlegroups.com>

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Subject: Re: |N is a Peano set
From: wolfgang...@hs-augsburg.de (WM)
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by: WM - Sat, 5 Jun 2021 20:30 UTC

William schrieb am Samstag, 5. Juni 2021 um 20:50:04 UTC+2:
> On Saturday, June 5, 2021 at 2:22:14 PM UTC-3, WM wrote:
> > William schrieb am Samstag, 5. Juni 2021 um 18:49:15 UTC+2:
>
> > > The set of *definable* numbers is infinite. The term "half" has no meaning for infinite sets.
> > The term definable element has meaning. For every definable natnumber 2n also n is definable.
> Yes, Note the {n|n definable element of |N) and {2n| n a definable element of N} are not disjoint. However, we can decompose |N into disjoint N_1 and N_2, where both are infinite. What we cannot do is find an m element of |N such that the sets n<,m and n>=m
> are both infinite. The first is always finite. So it makes no sense to talk about the second half of an infinite set.

It makes sense because the set of defined natnumbers (I called them definable above, but you don't like that word, so I will say defined) is always finite. Of course the second is also always finite.

> > If there were aleph_0 definable elements, then the set of their halves would have card aleph_0 too
> > But obviously its elements are not completely definable.
> > This shows that the second half
> Meaningless for infinite sets.

Maybe. Why? Because really , i.e., actually infinite sets are mainly not defined. For defined elements it would not be meaningless to talk about the first half and the second half.

Regards, WM

Re: |N is a Peano set

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Subject: Re: |N is a Peano set
From: wolfgang...@hs-augsburg.de (WM)
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by: WM - Sat, 5 Jun 2021 20:33 UTC

Re: |N is a Peano set

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Subject: Re: |N is a Peano set
From: wolfgang...@hs-augsburg.de (WM)
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by: WM - Sat, 5 Jun 2021 20:45 UTC

Jim Burns schrieb am Samstag, 5. Juni 2021 um 20:30:48 UTC+2:
> On 6/3/2021 3:44 PM, WM wrote:

> | We exclude teleporting.
>
> Good. And how do we say that?

We could use the continuity condition of real analysis, but in order to keep the noise low, we only say that the path contains all frames with finite index.
>
> | How do we say "No teleporting"?
>
> We say
> | The path on which Achilles chases and passes the tortoise
> | is _continuous_ on a _connected_ time-domain.
>
> That is not the way that you describe his path.
> But the way that you describe his path has serious problems.

There is no problem.

> You think it has problems, too.
>
> The path between your chosen points is missing
> _from your description_

That does not play a role.

> You present dark points as the solution to your problem,
> but all you say is that they're the missing points.

I say that all frames Achilles can remember are a finite set while set theory says that all frames are an infinite set.

> Here is my description of Achilles' path.
>
> | Each time-point in the time-interval [0,1] ==
> | [ when-Achilles-starts-running, when-Achilles-passes-tortoise ]
> | is paired with no more than one and no less than one space point.
>
> Formally, we can describe this as a crowd of (x,t) pairs
> fulfilling certain conditions.
>
> | _At least_ finite-length decimal fractions of 1 exist in [0,1].
> | I call these the bright points.
>
> I stole your term "bright points" to describe them. There are more
> bright points, the way you use the term, but, with the next condition,
> all the other bright points and much more are included, so
> mentioning more seems like an unnecessary complication.
>
> | No teleporting. This condition has two parts.
> |
> | For each space-time point (x,t) on the path, changes to the
> | x-coordinate will be less than epsilon, for any epsilon > 0,
> | if changes to the t-coordinate are less than delta, for some
> | delta > 0. The familiar _continuity_
> |
> | For each cut B,A of the points in [0,1], each point in B
> | before each point in A, a point c exists between them.
> | Without this condition, we have not excluded teleporting.
>
> If there is any cut B,A without c between them, Achilles has a
> _continuous_ path standing at the start through the time-points
> in B, teleporting, then standing at the tortoise through the
> the time-points in A.
>
> I like to think of the bright points as a wire armature for a
> sculpture of a geometric line. Adding the points between cuts
> daubs clay onto the armature and keeps the sculpture from
> falling apart.
>
> "Bright points, no teleporting" implies more points exist than
> definitions. That seems to be a concern of yours.
> However, I don't see how points in a continuum (no teleporting!)
> would be well-described as points in a sequence (neighbors before
> and after with none between).

Your description is too complicated. For the purpose of proving dark frames we need not exclude teleportation between the frames but only make sure that every frame is passed.

Regards, WM

Re: |N is a Peano set

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Subject: Re: |N is a Peano set
From: wpihug...@gmail.com (William)
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by: William - Sat, 5 Jun 2021 21:26 UTC

On Saturday, June 5, 2021 at 5:30:08 PM UTC-3, WM wrote:
> William schrieb am Samstag, 5. Juni 2021 um 20:50:04 UTC+2:
> > On Saturday, June 5, 2021 at 2:22:14 PM UTC-3, WM wrote:
> > > William schrieb am Samstag, 5. Juni 2021 um 18:49:15 UTC+2:
> >
> > > > The set of *definable* numbers is infinite. The term "half" has no meaning for infinite sets.
> > > The term definable element has meaning. For every definable natnumber 2n also n is definable.
> > Yes, Note the {n|n definable element of |N) and {2n| n a definable element of N} are not disjoint. However, we can decompose |N into disjoint N_1 and N_2, where both are infinite. What we cannot do is find an m element of |N such that the sets n<,m and n>=m
> > are both infinite. The first is always finite. So it makes no sense to talk about the second half of an infinite set.
> It makes sense because the set of defined natnumbers (I called them definable above, but you don't like that word, so I will say defined)

No, we went through this. The definable natural numbers and the defined natural numbers are two different sets. The set of definable natural numbers is infinite. The set of defined natural numbers is finite.

--
William Hughes

Re: |N is a Peano set

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Subject: Re: |N is a Peano set
From: wpihug...@gmail.com (William)
Injection-Date: Sat, 05 Jun 2021 21:29:12 +0000
Content-Type: text/plain; charset="UTF-8"
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by: William - Sat, 5 Jun 2021 21:29 UTC

On Saturday, June 5, 2021 at 5:26:55 PM UTC-3, WM wrote:
> William schrieb am Samstag, 5. Juni 2021 um 21:11:06 UTC+2:
> > On Saturday, June 5, 2021 at 2:22:08 PM UTC-3, WM wrote:
> > > William schrieb am Samstag, 5. Juni 2021 um 18:49:40 UTC+2:
> > > > On Saturday, June 5, 2021 at 1:28:53 PM UTC-3, WM wrote:
> > > > > William schrieb am Samstag, 5. Juni 2021 um 17:41:20 UTC+2:
> > >
> > > > > > Do you ever intend to tell us what your misleadingly named "dark unit fractions" are?
> > > > > The complement of the collection of |N_def is dark: |N \ |N_def.
> > > > This does not work. Your misleadingly named "dark natural numbers" are not natural numbers, so they cannot be elements of |N.
> > > You agreed hat the collection |N_def of defined elements is finite, hence it is a subset of the set |N which you call the set of definable elements. Hence the difference |N \ |N_def is not empty but infinite. What does not work?
> > |N is a set that contains only natural numbers. So things that are not natural numbers, e.g. your misleadingly named "dark natural numbers", cannot be elements of |N and thus cannot be elements of |N \ |N_def. Thus no "dark natural number" is an element of |N \ |N_def.
> But you have agreed that |N is infinite and |N_def, the set of ever defined natnumbers is finite.
> Therefore |N \ |N_def consists of natural number which never will be defined, therefore they cannot be defined,

Nope, we went through this. All natural numbers are definable. Though only a finite set will be defined there is no element of |N (the set you call the definable natural numbers) that cannot be in this set. In any case |N \ |N_def contains only natural numbers so things that are not natural numbers cannot be elements of this set.

> although you call them definable natnumbers. This is a license in order to aim at the central question: What is your idea of |N \ |N_def ?

A set of Natural numbers (the fact that they are definable follows from the definition of |N as a Peano set) of cardinality aleph_0. An "undefinable natural number" does not satisfy the definition of natural number, and, despite the name, is not a natural number. You want to talk about a set that can contain "undefinable" elements. Come up with a new name.

--
William H

Subject	Author
\|N is a Peano set	William
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	William
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	William
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	Python
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	Meritocracy
Re: \|N is a Peano set	William
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	William
Re: \|N is a Peano set	William
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	William
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	William
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	Meritocracy
Re: \|N is a Peano set	William
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	William
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	William
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	Meritocracy
Re: \|N is a Peano set	Quantum Bubbles
Re: \|N is a Peano set	Jim Burns
Re: \|N is a Peano set	William
Re: \|N is a Peano set	William
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	Jim Burns
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	Meritocracy
Re: \|N is a Peano set	Jim Burns
Re: \|N is a Peano set	FromTheRafters
Re: \|N is a Peano set	Gus Gassmann
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	William
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	FromTheRafters
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	FromTheRafters
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	William
Re: \|N is a Peano set	FromTheRafters
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	Meritocracy
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	Gus Gassmann
Re: \|N is a Peano set	FromTheRafters
Re: \|N is a Peano set	Meritocracy
Re: \|N is a Peano set	Meritocracy
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	Meritocracy
Re: \|N is a Peano set	Gus Gassmann
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	Sergio
Re: \|N is a Peano set	Gus Gassmann
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	Jim Burns
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	Sergio
Re: \|N is a Peano set	Gus Gassmann
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	Greg Cunt
Re: \|N is a Peano set	William
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	FromTheRafters
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	William
Re: \|N is a Peano set	William
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	William
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	William
Re: \|N is a Peano set	Greg Cunt
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	Sergio
Re: \|N is a Peano set	William
Re: \|N is a Peano set	Greg Cunt
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	FromTheRafters
Re: \|N is a Peano set	William
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	Sergio
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	William
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	William
Re: \|N is a Peano set	William
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	Sergio
Re: \|N is a Peano set	William
Re: \|N is a Peano set	William
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	Sergio
Re: \|N is a Peano set	Jim Burns
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	William
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	William
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	Greg Cunt
Re: \|N is a Peano set	William
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	William
Re: \|N is a Peano set	Greg Cunt
Re: \|N is a Peano set	Greg Cunt
Re: \|N is a Peano set	Greg Cunt
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	William
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	William
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	William
Re: \|N is a Peano set	Greg Cunt
Re: \|N is a Peano set	Greg Cunt
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	zelos...@gmail.com
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	zelos...@gmail.com
Re: \|N is a Peano set	William
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	William
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	William
Re: \|N is a Peano set	Gus Gassmann
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	William
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	William
Re: \|N is a Peano set	zelos...@gmail.com
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	William
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	William
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	William
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	William
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	William
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	William
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	Greg Cunt
Re: \|N is a Peano set	William
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	William
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	William
Re: \|N is a Peano set	Greg Cunt
Re: \|N is a Peano set	Greg Cunt
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	Greg Cunt
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	Greg Cunt
Re: \|N is a Peano set	William
Re: \|N is a Peano set	Greg Cunt
Re: \|N is a Peano set	Greg Cunt
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	Greg Cunt
Re: \|N is a Peano set	Greg Cunt
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	FromTheRafters
Re: \|N is a Peano set	FromTheRafters
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	Meritocracy
Re: \|N is a Peano set	William
Re: \|N is a Peano set	Gus Gassmann
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	Gus Gassmann
Re: \|N is a Peano set	Meritocracy
Re: \|N is a Peano set	William
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	William
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	William
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	William
Re: \|N is a Peano set	William
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	William
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	William
Re: \|N is a Peano set	Greg Cunt
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	Greg Cunt
Re: \|N is a Peano set	Greg Cunt
Re: \|N is a Peano set	William
Re: \|N is a Peano set	Greg Cunt
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	WM
Re: \|N is a Peano set	Meritocracy
Re: \|N is a Peano set	Gus Gassmann
Re: \|N is a Peano set	Meritocracy
Re: \|N is a Peano set	zelos...@gmail.com
Re: \|N is a Peano set	Jim Burns
Re: \|N is a Peano set	Gus Gassmann
Re: \|N is a Peano set	zelos...@gmail.com
Re: \|N is a Peano set	Eram semper recta
Re: \|N is a Peano set	Frank Gorsuch

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tech / sci.math / Re: |N is a Peano set

tech / sci.math / Re: |N is a Peano set

19 May, 2024: Line wrapping has been changed to be more consistent with Usenet standards.
If you find that it is broken please let me know here rocksolid.nodes.help