WM brought next idea :
> William schrieb am Freitag, 3. Dezember 2021 um 14:36:09 UTC+1:
>> On Friday, December 3, 2021 at 6:42:38 AM UTC-5, WM wrote:
>>> Collecting them cannot yield an ℵ₀-set because there are not two
>>> consecutive ℵ₀-sets in ℕ possible
>> Nope. The *elements* of the set have ℵ₀- successors. The *set* does not have
>> ℵ₀- successors.
>
> The elements are the set!

No, they are not! They do define the set however.

Fritz Feldhase schrieb am Donnerstag, 2. Dezember 2021 um 11:21:17 UTC+1:
> On Thursday, December 2, 2021 at 10:30:47 AM UTC+1, WM wrote:
>
> > we can collect all elements of ℕ which have ℵ₀ successors by the axiom of separation.
>
> Right.
>
> Let ℕ' = {n e ℕ : card({m e ℕ : m > n}) = ℵ₀}.
>
[Since] every element of ℕ has ℵ₀ successors [we get that]

We get a set ℕ without successors.

But for every collection ℕ_def of definable natural numbers n having a maximum n_max and ℵ₀ successors before ω we get the set of successors:
S = {m ∈ ℕ | n_max < m < ω} .

We can define the collection ℕ_def also by the intersection of endsegments
∩{n∈ ℕ_def} E_n = ∩{E_n | ∩{E_1, E_2, E_3, ..., E_n} =/= Ø} =/= Ø .

That means every addition of a definable individuals n yields the collection ℕ_def having a set S of successors between itself and ω.

But completing the gap S between ℕ_def and ℕ cannot be accomplished by definable individuals. That proves the existece of dark natural numbers.

Regards, WM

FromTheRafters schrieb am Freitag, 3. Dezember 2021 um 16:11:41 UTC+1:
> WM brought next idea :

> > The elements are the set!
> No, they are not! They do define the set however.

They are. All finite sets are nothing but the collection of their elements. {a, b, c} is a and b and c together - and nothing else. The properties can be read from the contents. Only infinite sets need some magic to be more in order to stultify matheologians.

Regards, WM

On 12/3/2021 9:26 AM, WM wrote:
> Fritz Feldhase schrieb am Donnerstag, 2. Dezember 2021 um 11:21:17 UTC+1:
>> On Thursday, December 2, 2021 at 10:30:47 AM UTC+1, WM wrote:
>>
>>> we can collect all elements of ℕ which have ℵ₀ successors by the axiom of separation.
>>
>> Right.
>>
>> Let ℕ' = {n e ℕ : card({m e ℕ : m > n}) = ℵ₀}.
>>

<snip crap>

nothing

WM used his keyboard to write :
> FromTheRafters schrieb am Freitag, 3. Dezember 2021 um 14:26:01 UTC+1:
>> WM has brought this to us :
>
>>> Simply answer the question: If there are all ℵ₀ natnumbers used,
>> By 'used' do you mean exhausted -- like they are no longer available?
>
> Every index of an endsegment is no longer available as contents of further
> endsegments.

Only because you are building a sequence of endsegments (calling them
sets) with each one excluding its least element to form the next set.
Seeing as there is no end to the sequence, these can be viewed as a
(just barely) proper subset of the immediate predecessor set.
>
>>> what is the first number of the infinite set of ℵ₀ successors?
>> It is n+1 for each n.
>
> The set of contents, if existing, is a fixed quantity.

Set of content?

> All n will get exhausted by indices.

So what?

> But a fixed quantity is claimed to remain in all endsegments -

A quantity, or a size?

> if all endsegments are infinite.

What do you mean by *if* when it was already defined as such? Why do
you insist upon injecting 'infinite' into various places in your
babbling?

On 12/3/2021 9:30 AM, WM wrote:
> FromTheRafters schrieb am Freitag, 3. Dezember 2021 um 16:11:41 UTC+1:
>> WM brought next idea :
>
>>> The elements are the set!
>> No, they are not! They do define the set however.
>
> They are. All finite sets are nothing but the collection of their elements. {a, b, c} is a and b and c together - and nothing else. The properties can be read from the contents. Only infinite sets need some magic to be more in order to stultify matheologians.
>
> Regards, WM
>

WM pushing BAD MATH

On 12/3/2021 9:01 AM, WM wrote:
> FromTheRafters schrieb am Freitag, 3. Dezember 2021 um 14:26:01 UTC+1:
>> WM has brought this to us :
>
>>> Simply answer the question: If there are all ℵ₀ natnumbers used,
>> By 'used' do you mean exhausted -- like they are no longer available?
>
> Every index of an endsegment is no longer available as contents of further endsegments.

so, 10 + 10 is not possable because 1 and 0 are both used or exhausted ?

is that what you are saying ??

WM explained on 12/3/2021 :
> Fritz Feldhase schrieb am Donnerstag, 2. Dezember 2021 um 11:21:17 UTC+1:
>> On Thursday, December 2, 2021 at 10:30:47 AM UTC+1, WM wrote:
>>
>>> we can collect all elements of ℕ which have ℵ₀ successors by the axiom of
>>> separation.
>>
>> Right.
>>
>> Let ℕ' = {n e ℕ : card({m e ℕ : m > n}) = ℵ₀}.
>>
> [Since] every element of ℕ has ℵ₀ successors [we get that]
>
> We get a set ℕ without successors.

Every natural number has one unique successor, so stop lying.
>
> But for every collection ℕ_def of definable natural numbers n having a
> maximum n_max and ℵ₀ successors before ω we get the set of successors: S = {m
> ∈ ℕ | n_max < m < ω} .
>
> We can define the collection ℕ_def also by the intersection of endsegments
> ∩{n∈ ℕ_def} E_n = ∩{E_n | ∩{E_1, E_2, E_3, ..., E_n} =/= Ø} =/= Ø .
>
> That means every addition of a definable individuals n yields the collection
> ℕ_def having a set S of successors between itself and ω.
>
> But completing the gap S between ℕ_def and ℕ cannot be accomplished by
> definable individuals. That proves the existece of dark natural numbers.

Thanks for not saying 'infinite' in this babbling.

FromTheRafters schrieb am Freitag, 3. Dezember 2021 um 16:34:20 UTC+1:
> WM used his keyboard to write :
> > FromTheRafters schrieb am Freitag, 3. Dezember 2021 um 14:26:01 UTC+1:
> >> WM has brought this to us :
> >
> >>> Simply answer the question: If there are all ℵ₀ natnumbers used,
> >> By 'used' do you mean exhausted -- like they are no longer available?
> >
> > Every index of an endsegment is no longer available as contents of further
> > endsegments.
> Only because you are building a sequence of endsegments (calling them
> sets) with each one excluding its least element to form the next set.
> Seeing as there is no end to the sequence, these can be viewed as a
> (just barely) proper subset of the immediate predecessor set.
> >
> >>> what is the first number of the infinite set of ℵ₀ successors?
> >> It is n+1 for each n.
> >
> > The set of contents, if existing, is a fixed quantity.
> Set of content?
> > All n will get exhausted by indices.
> So what?
> > But a fixed quantity is claimed to remain in all endsegments -
> A quantity, or a size?

An infinite set is claimed to be in every endsegment. That means a quantitity.

> > if all endsegments are infinite.
> What do you mean by *if* when it was already defined as such?

Because it is wrong like the definition of a triangular circle. I only stress it as a wake up call.

> Why do
> you insist upon injecting 'infinite' into various places

Because even a lost matheologian cannot claim an infinite ℵ₀-set followed by an infinite ℵ₀-set in ℕ. The set of infinite endsegments has not ℵ₀ natnumbers as indices available.

Regards, WM

On Friday, December 3, 2021 at 9:08:48 AM UTC-5, WM wrote:

> The (more or less sudden)

There is nothing "more or less sudden" about a change from a finite set to an infinite set.
The natural numbers that can be written down are a finite set. The set of natural numbers is a Peano set with cardinality aleph_0.

--
William Hughes

William schrieb am Freitag, 3. Dezember 2021 um 16:43:19 UTC+1:
> On Friday, December 3, 2021 at 9:08:48 AM UTC-5, WM wrote:
>
> > The (more or less sudden)
> There is nothing "more or less sudden" about a change from a finite set to an infinite set.
> The natural numbers that can be written down are a finite set. The set of natural numbers is a Peano set with cardinality aleph_0.
>
Anyhow, the loss of successors cannot be observed - whatever you try.
This shows that something is existing between finite and infinite that cannot be observed.
Whether you call it Peano or not is irrelevant.

Regards, WM

WM explained :
> FromTheRafters schrieb am Freitag, 3. Dezember 2021 um 16:11:41 UTC+1:
>> WM brought next idea :
>
>>> The elements are the set!
>> No, they are not! They do define the set however.
>
> They are. All finite sets are nothing but the collection of their elements.
> {a, b, c} is a and b and c together - and nothing else.

> The properties can be read from the contents.

How many subsets does it have?

> Only infinite sets need some magic to be more in order to stultify
> matheologians.

Stop lying.

It happens that WM formulated :
> FromTheRafters schrieb am Freitag, 3. Dezember 2021 um 16:34:20 UTC+1:
>> WM used his keyboard to write :
>>> FromTheRafters schrieb am Freitag, 3. Dezember 2021 um 14:26:01 UTC+1:
>>>> WM has brought this to us :
>>>
>>>>> Simply answer the question: If there are all ℵ₀ natnumbers used,
>>>> By 'used' do you mean exhausted -- like they are no longer available?
>>>
>>> Every index of an endsegment is no longer available as contents of further
>>> endsegments.
>> Only because you are building a sequence of endsegments (calling them
>> sets) with each one excluding its least element to form the next set.
>> Seeing as there is no end to the sequence, these can be viewed as a
>> (just barely) proper subset of the immediate predecessor set.
>>>
>>>>> what is the first number of the infinite set of ℵ₀ successors?
>>>> It is n+1 for each n.
>>>
>>> The set of contents, if existing, is a fixed quantity. Set of content?
>>> All n will get exhausted by indices. So what?
>>> But a fixed quantity is claimed to remain in all endsegments -
>> A quantity, or a size?
>
> An infinite set is claimed to be in every endsegment. That means a
> quantitity.

To me, it means a size.

>>> if all endsegments are infinite.
>> What do you mean by *if* when it was already defined as such?
>
> Because it is wrong

Stop lying.

On Friday, December 3, 2021 at 4:48:59 PM UTC+1, WM wrote:

> Anyhow, the loss of successors cannot be observed

Look closer!

https://en.wikipedia.org/wiki/Observation

On Friday, December 3, 2021 at 3:08:48 PM UTC+1, WM wrote:

> Anyhow, this shows that something here cannot be obeserved.
> Do you agree that something here cannot be observed?

Maybe because it is dark?

"Dark matter is called dark because it does not appear to interact with the electromagnetic field, which means it does not absorb, reflect or emit electromagnetic radiation, and is therefore difficult to detect."

See: https://en.wikipedia.org/wiki/Dark_matter

On Friday, December 3, 2021 at 10:48:59 AM UTC-5, WM wrote:
> William schrieb am Freitag, 3. Dezember 2021 um 16:43:19 UTC+1:
> > On Friday, December 3, 2021 at 9:08:48 AM UTC-5, WM wrote:
> >
> > > The (more or less sudden)
> > There is nothing "more or less sudden" about a change from a finite set to an infinite set.
> > The natural numbers that can be written down are a finite set. The set of natural numbers is a Peano set with cardinality aleph_0.
> >
> Anyhow, the loss of successors cannot be observed -

Meaningless drivel A finite set has successors, An infinite set does not,

--
William Hughes

On 12/3/2021 8:08 AM, WM wrote:
> William schrieb am Freitag, 3. Dezember 2021 um 14:36:09 UTC+1:
>> On Friday, December 3, 2021 at 6:42:38 AM UTC-5, WM wrote:
>>> Collecting them cannot yield an ℵ₀-set because there are not two consecutive ℵ₀-sets in ℕ possible
>> Nope. The *elements* of the set have ℵ₀- successors. The *set* does not have ℵ₀- successors.
>

<snip fowled thinking>

> The (more or less sudden) change cannot be observed.
> Anyhow, this shows that something here cannot be obeserved.
> Do you agree that something here cannot be observed?

yes, one cannot observe any Math in your posts, only goofy vaguatuities.

>
> Regards, WM
>

On 12/3/2021 9:41 AM, WM wrote:
> FromTheRafters schrieb am Freitag, 3. Dezember 2021 um 16:34:20 UTC+1:
>> WM used his keyboard to write :
>>> FromTheRafters schrieb am Freitag, 3. Dezember 2021 um 14:26:01 UTC+1:
>>>> WM has brought this to us :
>>>
>>>>> Simply answer the question: If there are all ℵ₀ natnumbers used,
>>>> By 'used' do you mean exhausted -- like they are no longer available?
>>>
>>> Every index of an endsegment is no longer available as contents of further
>>> endsegments.
>> Only because you are building a sequence of endsegments (calling them
>> sets) with each one excluding its least element to form the next set.
>> Seeing as there is no end to the sequence, these can be viewed as a
>> (just barely) proper subset of the immediate predecessor set.
>>>
>>>>> what is the first number of the infinite set of ℵ₀ successors?
>>>> It is n+1 for each n.
>>>
>>> The set of contents, if existing, is a fixed quantity.
>> Set of content?
>>> All n will get exhausted by indices.
>> So what?
>>> But a fixed quantity is claimed to remain in all endsegments -
>> A quantity, or a size?
>
> An infinite set is claimed to be in every endsegment. That means a quantitity.

Wrong,

an Endsegment is defined as a fixed infinite set, there is no "claimed".

the quantitity is infinite, which is in the definition, or did you miss that part ?

On Friday, December 3, 2021 at 4:30:25 PM UTC+1, WM wrote:
> FromTheRafters schrieb am Freitag, 3. Dezember 2021 um 16:11:41 UTC+1:
> > WM brought next idea :
> > >
> > > The elements are the set!

THIS again. *sigh*

> > No, they are not! They do define the set however.
> >
> They are. All finite sets are nothing but the collection of their elements.

Holy shit! Yeah, all [non empty] sets "are [...] the collection of their elements"

An incredible insight!

> {a, b, c} is a and b and c together - and nothing else.

You are an idiot - and nothing else!

Hint: {a, b, c} is neither a, nor b, nor c: {a, b, c} =/= a, {a, b, c} =/= b, {a, b, c} =/= c.

Due to Mückenheim: {a} is a - and ​nothing else? Even though there is no such set in ZFC? Huh?!

You see if your claim were true then Zermelo's set of natural number Z, would just be the empty set, since {{}} = {}, {{{}}} = {{}} = {}, {{{{}}}} = {{{}}} = {{}} = {}, etc. Hence Z = {{}, {{}}, {{{}}}, ...} = {{}, {}, {}, ...} = {{}} = {}.

Not that reasonable, to say the least, you fucking asshole full of shit.

And {} is WHAT in your psychotic kingdom?

Hint: {} does not have any elements, you know.

Can you PLEEEZE stop this avalanche of nonsense, man?!

On Friday, December 3, 2021 at 3:08:48 PM UTC+1, WM wrote:
> William schrieb am Freitag, 3. Dezember 2021 um 14:36:09 UTC+1:
> > On Friday, December 3, 2021 at 6:42:38 AM UTC-5, WM wrote:
> > >
> > > Collecting them cannot yield an ℵ₀-set because <bla>

Nonsense.

The set of all natural numbers which have ℵ₀ successors is just the set of all natural numbers, SINCE each and every natural number has ℵ₀ successors.

The set all natural numbers, IN, is a "ℵ₀-set", in other words card(IN) = ℵ₀.

> Collecting all natnumbers [...] yields a set with no successors before ω.

Please define what you mean be a /successor of a set/. Then prove your claim.

Hint: There is no ordinal number which is between ALL natural numbers and ω:

~Ex e ORD: An e IN: n < x < ω.

Though, on the other hand, for each and ever natural numbers n there's an ordinal number which is between n and ω.

An e IN: Ex e ORD: n < x < ω.

Beware of the quantifier shift!

Fritz Feldhase schrieb am Freitag, 3. Dezember 2021 um 19:33:36 UTC+1:

> The set of all natural numbers which have ℵ₀ successors is just the set of all natural numbers, SINCE each and every natural number has ℵ₀ successors.

That is wrong.

In case of potential infinity, there are infinitely many natnumbers beyond every natnumber, but there is no completion denoted by ℵo. There is only oo.

In case of actual infinity there is the complete set ℕ. Every definable natnumber has ℵo successors which cannot be removed from the set individually but can be removed from the set collectively.

> Hint: There is no ordinal number which is between ALL natural numbers and ω:

There is no definable ordinal number because every definable ordinal number belongs to the potentially infinite collection the elements of which can be removed individually. But there are ℵo successors of this collection which can be removed collectively only.

> An e IN: Ex e ORD: n < x < ω.

Of course. That is the property of actual infinity.

> Beware of the quantifier shift!

No quantifier shift but simple fact shows: Almost all natnumbers cannot be removed individually but can be removed.

Regards, WM

William schrieb am Freitag, 3. Dezember 2021 um 17:56:04 UTC+1:
> On Friday, December 3, 2021 at 10:48:59 AM UTC-5, WM wrote:
> > William schrieb am Freitag, 3. Dezember 2021 um 16:43:19 UTC+1:
> > > On Friday, December 3, 2021 at 9:08:48 AM UTC-5, WM wrote:
> > >
> > > > The (more or less sudden)
> > > There is nothing "more or less sudden" about a change from a finite set to an infinite set.
> > > The natural numbers that can be written down are a finite set. The set of natural numbers is a Peano set with cardinality aleph_0.
> > >
> > Anyhow, the loss of successors cannot be observed -
> A finite set has successors, An infinite set does not,

The elements of a finite set (or of a finite subset of an infinite set) can be observed as individuals. Most elements of an infinite set cannot be observed as individuals but can be treated collectively. Do you agree?

Regards, WM

Fritz Feldhase schrieb am Freitag, 3. Dezember 2021 um 19:33:36 UTC+1:

> The set of all natural numbers which have ℵ₀ successors is just the set of all natural numbers, SINCE each and every natural number has ℵ₀ successors.

That is wrong.

In case of potential infinity, there are infinitely many natnumbers beyond every natnumber, but there is no completion denoted by ℵo. There is only oo.

In case of actual infinity there is the complete set ℕ. Every definable natnumber has ℵo successors which cannot be removed from the set individually but can be removed from the set collectively.

> Hint: There is no ordinal number which is between ALL natural numbers and ω:

There is no *definable* ordinal number because every definable ordinal number belongs to the potentially infinite collection the elements of which can be removed individually. But there are ℵo successors of this collection which can be removed collectively only.

> An e IN: Ex e ORD: n < x < ω.

Of course. That is the property of potential infinity.

> Beware of the quantifier shift!

No quantifier shift but simple fact shows: Almost all natnumbers cannot be removed individually but can be removed.

Regards, WM

On 12/4/2021 1:56 AM, WM wrote:
> ordinal number

Every infinite set can be indexed. Right?

[0] = 3
[1] = [0] + .1
[2] = [1] + .04
[3] = [2] + .001
[4] = [3] + .0005
[5] = [4] + .00009
....

for pi...

WM expressed precisely :
> Fritz Feldhase schrieb am Freitag, 3. Dezember 2021 um 19:33:36 UTC+1:
>
>> The set of all natural numbers which have ℵ₀ successors is just the set of
>> all natural numbers, SINCE each and every natural number has ℵ₀ successors.
>
> That is wrong.

Stop lying.