William schrieb am Dienstag, 30. November 2021 um 23:11:44 UTC+1:
> On Tuesday, November 30, 2021 at 4:22:15 PM UTC-5, WM wrote:
> > William schrieb am Dienstag, 30. November 2021 um 16:55:27 UTC+1:
> > > On Tuesday, November 30, 2021 at 7:51:37 AM UTC-5, WM wrote:
> > > > William schrieb am Montag, 29. November 2021 um 18:03:21 UTC+1:
> > >
> > > > > Numbers that cannot be written down still have other properties
> > > > That may be
> > >
> > > So elements of the Peano set of natural numbers that cannot be written down still have other properties(e.g., an element of the set of natural numbers is the largest element of a FISON)
> > But all Peano numbers or largest elements of FISONs are followed by ℵ₀ dark elements.
> Nope, they are followed by a set of natural numbers with cardinality ℵ₀ (not the same set each time).

But a set that contains more than definable natural numbers. Otherwise not every definable number would be followed by them.

> The Peano set of natural numbers does not contain "dark" elements.

For all numbers in ℕ_P
we have
∀n ∈ ℕ_P: |ℕ \ {1, 2, 3, ..., n}| = ℵo.
Therefore |ℕ_P| < ℵo.

But better this can be seen by all infinite endsegments E(n) with n ∈ ℕ_P: All infinite endsegments have spared ℵo natnumbers for their contents (otherwise they could not be infinite). Hence they cannot apply ℵo natnumbers for their indices. Their indices are less than ℵo.

Regards, WM

zelos...@gmail.com schrieb am Mittwoch, 1. Dezember 2021 um 06:28:42 UTC+1:
> tisdag 30 november 2021 kl. 14:01:43 UTC+1 skrev WM:
> > zelos...@gmail.com schrieb am Dienstag, 30. November 2021 um 13:48:23 UTC+1:
> > > tisdag 30 november 2021 kl. 13:33:12 UTC+1 skrev WM:
> >
> > > >You can't. All you can belongs to ℕ_def but
> > > False, I can with N
> > >
> > > f:N->Z
> > > if x is even, f(x)=x/2
> > > if x is odd, f(x)=-(x+1)/2
> > > >∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo.
> > > making N_def=N
> > That is no individual definition. Note decimal representation!
> > > >You cannot remove the huge set of ℵo remainings.
> > > all elements are in N, none remaining :)
> > Note decimal representation!

> What about it? It is of no importance as it is just representation.

7 defines a number individually, x does not. My claim is that for individually definable numbers
∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo.

Hence |ℕ_def| < ℵo.

And you could at least try to understand that all infinite endsegments have spared ℵo natnumbers for their contents. Hence they cannot apply ℵo natnumbers for their indices. Their indices are less than ℵo.

If you cannot grasp this simple fact, further discussion is idle.

Regards, WM

zelos...@gmail.com schrieb am Mittwoch, 1. Dezember 2021 um 06:29:16 UTC+1:
> tisdag 30 november 2021 kl. 22:22:15 UTC+1 skrev WM:

> > But all Peano numbers or largest elements of FISONs are followed by ℵ₀ dark elements. So the set of numbers followed by ℵ₀ numbers cannot have cardinality ℵ₀.

> you have yet to define what "dark" means that doesn't make the set of them empty.

Dark means that no decimal representation is possible. You cannot count to dark numbers. They have no FISON. The are the ℵo-set resulting from
∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo .

Regards, WM

onsdag 1 december 2021 kl. 10:44:41 UTC+1 skrev WM:
> zelos...@gmail.com schrieb am Mittwoch, 1. Dezember 2021 um 06:28:42 UTC+1:
> > tisdag 30 november 2021 kl. 14:01:43 UTC+1 skrev WM:
> > > zelos...@gmail.com schrieb am Dienstag, 30. November 2021 um 13:48:23 UTC+1:
> > > > tisdag 30 november 2021 kl. 13:33:12 UTC+1 skrev WM:
> > >
> > > > >You can't. All you can belongs to ℕ_def but
> > > > False, I can with N
> > > >
> > > > f:N->Z
> > > > if x is even, f(x)=x/2
> > > > if x is odd, f(x)=-(x+1)/2
> > > > >∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo.
> > > > making N_def=N
> > > That is no individual definition. Note decimal representation!
> > > > >You cannot remove the huge set of ℵo remainings.
> > > > all elements are in N, none remaining :)
> > > Note decimal representation!
> > What about it? It is of no importance as it is just representation.
> 7 defines a number individually, x does not. My claim is that for individually definable numbers
> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo.
> Hence |ℕ_def| < ℵo.
>
> And you could at least try to understand that all infinite endsegments have spared ℵo natnumbers for their contents. Hence they cannot apply ℵo natnumbers for their indices. Their indices are less than ℵo.
>
> If you cannot grasp this simple fact, further discussion is idle.
>
> Regards, WM
7 REPRESENTS a natural number, it doesn't DEFINE it

>x does not.

Neither defines, both represents a natural number. First one is a specific one, second is a generic one.

>My claim is that for individually definable numbers

which is meaningless, "individually definable" has no mathematical meaning.

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

applies to all natural numbers.

>Hence |ℕ_def| < ℵo.

False, the fact it applies to all natural numbers means N_def=N

>And you could at least try to understand that all infinite endsegments have spared ℵo natnumbers for their contents. Hence they cannot apply ℵo natnumbers for their indices. Their indices are less than ℵo.

If you want me to understand, then use the language of mathematics because your semantic weaseling is making it increadibly difficult to know what the fuck you're talking about.

onsdag 1 december 2021 kl. 10:47:17 UTC+1 skrev WM:
> zelos...@gmail.com schrieb am Mittwoch, 1. Dezember 2021 um 06:29:16 UTC+1:
> > tisdag 30 november 2021 kl. 22:22:15 UTC+1 skrev WM:
>
> > > But all Peano numbers or largest elements of FISONs are followed by ℵ₀ dark elements. So the set of numbers followed by ℵ₀ numbers cannot have cardinality ℵ₀.
> > you have yet to define what "dark" means that doesn't make the set of them empty.
> Dark means that no decimal representation is possible. You cannot count to dark numbers. They have no FISON. The are the ℵo-set resulting from
> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo .
>
> Regards, WM

>Dark means that no decimal representation is possible.

That means the set of dark numbersi s empty.

>You cannot count to dark numbers.

Ergo no natural number is dark numbers, hence empty.

>They have no FISON.

Which applies to NO natural numbers, hence empty.

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

is true for ALL natural numbers, making your "dark numbers" set empty. Hence none exists

After serious thinking WM wrote :
> zelos...@gmail.com schrieb am Mittwoch, 1. Dezember 2021 um 06:28:42 UTC+1:
>> tisdag 30 november 2021 kl. 14:01:43 UTC+1 skrev WM:
>>> zelos...@gmail.com schrieb am Dienstag, 30. November 2021 um 13:48:23
>>> UTC+1:
>>>> tisdag 30 november 2021 kl. 13:33:12 UTC+1 skrev WM:
>>>>> You can't. All you can belongs to ℕ_def but
>>>> False, I can with N
>>>>
>>>> f:N->Z
>>>> if x is even, f(x)=x/2
>>>> if x is odd, f(x)=-(x+1)/2
>>>>> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo.
>>>> making N_def=N That is no individual definition. Note decimal
>>>> representation!
>>>>> You cannot remove the huge set of ℵo remainings.
>>>> all elements are in N, none remaining :)
>>> Note decimal representation!
>
>> What about it? It is of no importance as it is just representation.
>
> 7 defines a number individually, x does not. My claim is that for
> individually definable numbers ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo.
>
> Hence |ℕ_def| < ℵo.
>
> And you could at least try to understand that all infinite endsegments have
> spared ℵo natnumbers for their contents.

Not so easy if it was a seven ten split.

On Wednesday, December 1, 2021 at 4:41:01 AM UTC-5, WM wrote:
element of a FISON)
> > > But all Peano numbers or largest elements of FISONs are followed by ℵ₀ dark elements.
> > Nope, they are followed by a set of natural numbers with cardinality ℵ₀ (not the same set each time).
> But a set that contains more than definable natural numbers.

Corrrect. But |N_def is not the set of natural numbers

Otherwise not every definable number would be followed by them.
> > The Peano set of natural numbers does not contain "dark" elements.
> For all numbers in ℕ_P
> we have

> ∀n ∈ ℕ_P: |[ℕ_P] \ {1, 2, 3, ..., n}| = ℵo.

This is a correct statement about the elements of the set of natural numbers, N_P. I follows trivially form the fact that N_P has no last element.

> Therefore |ℕ_P| < ℵo.

This is a false statement about the set or natural numbers N_P A Peano set has cardinality ℵo. No element of N_P is a set without last element. N_P is a set without last element.

--
William Hughes

On 12/1/2021 3:47 AM, WM wrote:
> zelos...@gmail.com schrieb am Mittwoch, 1. Dezember 2021 um 06:29:16 UTC+1:
>> tisdag 30 november 2021 kl. 22:22:15 UTC+1 skrev WM:
>
>>> But all Peano numbers or largest elements of FISONs are followed by ℵ₀ dark elements. So the set of numbers followed by ℵ₀ numbers cannot have cardinality ℵ₀.
>
>> you have yet to define what "dark" means that doesn't make the set of them empty.
>
> Dark means that no decimal representation is possible. You cannot count to dark numbers. They have no FISON.

there are no such thing as dark numbers.

> The are the ℵo-set resulting from
> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo .

_def is meaningless

>
> Regards, WM
>

On 12/1/2021 3:44 AM, WM wrote:
> zelos...@gmail.com schrieb am Mittwoch, 1. Dezember 2021 um 06:28:42 UTC+1:
>> tisdag 30 november 2021 kl. 14:01:43 UTC+1 skrev WM:
>>> zelos...@gmail.com schrieb am Dienstag, 30. November 2021 um 13:48:23 UTC+1:
>>>> tisdag 30 november 2021 kl. 13:33:12 UTC+1 skrev WM:
>>>
>>>>> You can't. All you can belongs to ℕ_def but
>>>> False, I can with N
>>>>
>>>> f:N->Z
>>>> if x is even, f(x)=x/2
>>>> if x is odd, f(x)=-(x+1)/2
>>>>> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo.
>>>> making N_def=N
>>> That is no individual definition. Note decimal representation!
>>>>> You cannot remove the huge set of ℵo remainings.
>>>> all elements are in N, none remaining :)
>>> Note decimal representation!
>
>> What about it? It is of no importance as it is just representation.
>
> 7 defines a number individually, x does not. My claim is that for individually definable numbers
> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo.

by your own thinking, n is not an individually definable number, so your equation is moot.

>
> Hence |ℕ_def| < ℵo.

_def is meaningless.

>
> And you could at least try to understand that all infinite endsegments have spared ℵo natnumbers for their contents. Hence they cannot apply ℵo natnumbers for their indices. Their indices are less than ℵo.

wrong, here are your mistakes;
1. all endsegments are fixed sets of infinite number of elements by definition.
2. sets do not "spare" elements,
3. they do not "apply" numbers
4. Sets do not have "contents", they have "elements"
5. Their number of indices are ℵo

>
> If you cannot grasp this simple fact, further discussion is idle.

your facts are poopus

>
> Regards, WM
>

zelos...@gmail.com schrieb am Mittwoch, 1. Dezember 2021 um 11:38:58 UTC+1:

> 7 REPRESENTS a natural number, it doesn't DEFINE it

7 defines a place on the number line.
>
> >x does not.
>
> Neither defines, both represents a natural number.

No. 7 is in trichonomy with every natural number, x is not in trichotomy with any number.

> >My claim is that for individually definable numbers
> which is meaningless, "individually definable" has no mathematical meaning.

It has. Learn the meaning of trichotomy.

> >∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo.
> applies to all natural numbers.
>
> >Hence |ℕ_def| < ℵo.
>
> False, the fact it applies to all natural numbers means N_def=N

Learn that two consecutive ℵo-sets cannot exist in the natural order of ℕ.

Regards, WM

William schrieb am Mittwoch, 1. Dezember 2021 um 14:50:24 UTC+1:
> On Wednesday, December 1, 2021 at 4:41:01 AM UTC-5, WM wrote:

> > For all numbers in ℕ_P
> > we have
> > ∀n ∈ ℕ_P: |[ℕ_P] \ {1, 2, 3, ..., n}| = ℵo.
>
> This is a correct statement about the elements of the set of natural numbers, N_P. I follows trivially form the fact that N_P has no last element.

That means there are always many elements remaining after ℕ_P. But after the set ℕ nothing is remaining.
>
> > Therefore |ℕ_P| < ℵo.
>
> This is a false statement about the set or natural numbers N_P A Peano set has cardinality ℵo. No element of N_P is a set without last element. N_P is a set without last element.

Sets are defined by the elements they have, not be elements which are missing. Correct is: There are always ℵo elements remaining after all elements of ℕ_P. But after the set ℕ nothing is remaining. That shows a difference.

Regards, WM

On Wednesday, December 1, 2021 at 5:00:29 PM UTC-5, WM wrote:

>There are always ℵo elements remaining after [each] element [s] of ℕ_P. But after the set [ℕ_P] nothing is remaining. That shows a difference.

The difference is that no *element* of N_P is a set without last element, but the *set* N_P is a set without last element.

--
William Hughes

WM presented the following explanation :
> zelos...@gmail.com schrieb am Mittwoch, 1. Dezember 2021 um 11:38:58 UTC+1:
>
>> 7 REPRESENTS a natural number, it doesn't DEFINE it
>
> 7 defines a place on the number line.
>>
>>> x does not.
>>
>> Neither defines, both represents a natural number.
>
> No. 7 is in trichonomy with every natural number, x is not in trichotomy with
> any number.

So x is zero (or one) then? Why not just say so?

>>> My claim is that for individually definable numbers
>> which is meaningless, "individually definable" has no mathematical meaning.
>
> It has. Learn the meaning of trichotomy.

It's the surgical removal of your tricho.

On 12/1/2021 4:28 PM, FromTheRafters wrote:
> WM presented the following explanation :
>> zelos...@gmail.com schrieb am Mittwoch, 1. Dezember 2021 um 11:38:58 UTC+1:
>>
>>> 7 REPRESENTS a natural number, it doesn't DEFINE it
>>
>> 7 defines a place on the number line.
>>>
>>>> x does not.
>>>
>>> Neither defines, both represents a natural number.
>>
>> No. 7 is in trichonomy with every natural number, x is not in trichotomy with any number.
>
> So x is zero (or one) then? Why not just say so?
>
>>>> My claim is that for individually definable numbers
>>> which is meaningless, "individually definable" has no mathematical meaning.
>>
>> It has. Learn the meaning of trichotomy.
>
> It's the surgical removal of your tricho.

you can cut them out with a sharp exacto knife, the squeaker in a plastic dog bone toy, even though it has not been individually definable.

Extend WMs thinking into the real world => it is 2D, only what you see, nothing is defined behind that computer screen... When you look away, it is
gone, dissapeared, undefined, not real anymore...

onsdag 1 december 2021 kl. 22:54:28 UTC+1 skrev WM:
> zelos...@gmail.com schrieb am Mittwoch, 1. Dezember 2021 um 11:38:58 UTC+1:
>
> > 7 REPRESENTS a natural number, it doesn't DEFINE it
> 7 defines a place on the number line.
> >
> > >x does not.
> >
> > Neither defines, both represents a natural number.
> No. 7 is in trichonomy with every natural number, x is not in trichotomy with any number.
> > >My claim is that for individually definable numbers
> > which is meaningless, "individually definable" has no mathematical meaning.
> It has. Learn the meaning of trichotomy.
> > >∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo.
> > applies to all natural numbers.
> >
> > >Hence |ℕ_def| < ℵo.
> >
> > False, the fact it applies to all natural numbers means N_def=N
> Learn that two consecutive ℵo-sets cannot exist in the natural order of ℕ.
>
> Regards, WM

>7 defines a place on the number line.

Nope, it REPRESENTS a SPECIFIC number in N

>No. 7 is in trichonomy with every natural number, x is not in trichotomy with any number.

False, both are

x is greater than, less than or equal to n, just like 7

>It has. Learn the meaning of trichotomy.

I know the meaning of it, it has nothing to do with "individually definable"

>Learn that two consecutive ℵo-sets cannot exist in the natural order of ℕ.

I have never said it nor does anything imply it. Why do you keep spewing this garbage claim?

William schrieb am Mittwoch, 1. Dezember 2021 um 23:15:44 UTC+1:
> On Wednesday, December 1, 2021 at 5:00:29 PM UTC-5, WM wrote:
>
> >There are always ℵo elements remaining after [each] element [s] of ℕ_P. But after the set [ℕ_P] nothing is remaining. That shows a difference.
>
> The difference is that no *element* of N_P is a set without last element, but the *set* N_P is a set without last element.

This can be expressed a bit more concise: Every element of ℕ_P has ℵ₀ successors. That means we can collect all elements of ℕ_P which have ℵ₀ successors by the axiom of separation. Now the question is: Does this collection have ℵ₀ successors too? If the axiom of separation can be applied to form sets, then the answer is yes.

ℕ has not ℵ₀ successors.

Regards, WM

FromTheRafters schrieb am Mittwoch, 1. Dezember 2021 um 23:28:46 UTC+1:
> WM presented the following explanation :
> > zelos...@gmail.com schrieb am Mittwoch, 1. Dezember 2021 um 11:38:58 UTC+1:
> >
> >> 7 REPRESENTS a natural number, it doesn't DEFINE it
> >
> > 7 defines a place on the number line.
> >>
> >>> x does not.
> >>
> >> Neither defines, both represents a natural number.
> >
> > No. 7 is in trichonomy with every natural number, x is not in trichotomy with
> > any number.
> So x is zero (or one) then?

1 is in trichotomy with all definable numbers.

> Why not just say so?

Because x can also be meant to represent a number that cannot be defined and is not in trichotomy with others.

Regards, WM

zelos...@gmail.com schrieb am Donnerstag, 2. Dezember 2021 um 06:21:53 UTC+1:
> onsdag 1 december 2021 kl. 22:54:28 UTC+1 skrev WM:

> > > False, the fact it applies to all natural numbers means N_def=N
> > Learn that two consecutive ℵo-sets cannot exist in the natural order of ℕ.
> >
Have you done your exercise?
>
> >7 defines a place on the number line.
> Nope, it REPRESENTS a SPECIFIC number in N

That means a place on the number line.

> >No. 7 is in trichonomy with every natural number, x is not in trichotomy with any number.
> False, both are
>
> x is greater than, less than or equal to n, just like 7

For 7 and 3 we need no "or". For x and 3 we need it.

> >It has. Learn the meaning of trichotomy.
> I know the meaning of it, it has nothing to do with "individually definable"

No, you know not even the difference between a constant, like 7, and a variable, like x.

> >Learn that two consecutive ℵo-sets cannot exist in the natural order of ℕ.
> I have never said it nor does anything imply it

Learn that an infinite set of infinite endsegments would imply it.

Regards, WM

It happens that WM formulated :
> FromTheRafters schrieb am Mittwoch, 1. Dezember 2021 um 23:28:46 UTC+1:
>> WM presented the following explanation :
>>> zelos...@gmail.com schrieb am Mittwoch, 1. Dezember 2021 um 11:38:58 UTC+1:
>>>
>>>> 7 REPRESENTS a natural number, it doesn't DEFINE it
>>>
>>> 7 defines a place on the number line.
>>>>
>>>>> x does not.
>>>>
>>>> Neither defines, both represents a natural number.
>>>
>>> No. 7 is in trichonomy with every natural number, x is not in trichotomy
>>> with any number.
>> So x is zero (or one) then?
>
> 1 is in trichotomy with all definable numbers.
>
>> Why not just say so?
>
> Because x can also be meant to represent a number that cannot be defined and
> is not in trichotomy with others.

Not so if you have declared x to be in the set of naturals in their
natural order.

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]

ℕ' = ℕ.

Proof: ℕ' c ℕ by definition. For all n e ℕ: card({m e ℕ : m > n}) = ℵ₀. Hence for all n e ℕ: n e ℕ'. Hence ℕ c ℕ'. Hence ℕ' = ℕ. qed

> Now the question is: Does this [set] have ℵ₀ successors too?

Please specify FORMALLY (i. e. in the FORMAL language of set theory) what it means for a set to have ℵ₀ successors:

X has ℵ₀ successors :<-> ___ X ___ [<< please fill in the blanks using the FORMAL language of set theory]

> ℕ has not ℵ₀ successors.

Please specify FORMALLY (i. e. in the FORMAL language of set theory) what it means for a set to have ℵ₀ successors.

X has ℵ₀ successors :<-> ___ X ___ [<< please fill in the blanks using the FORMAL language of set theory]

Then PROVE the claim: It's not the case that ℕ has ℵ₀ successors.

Using symbol: ~(ℕ has ℵ₀ successors).

On Thursday, December 2, 2021 at 10:33:23 AM UTC+1, WM wrote:

Concerning /trichotomy/.

> x can also be meant to represent a number that [...] is not [...]

No. For each and every x,y e IN: either x < y, or x = y, or x > y.

See: https://en.wikipedia.org/wiki/Trichotomy_(mathematics)

You are dumb like shit, man.

torsdag 2 december 2021 kl. 10:37:58 UTC+1 skrev WM:
> zelos...@gmail.com schrieb am Donnerstag, 2. Dezember 2021 um 06:21:53 UTC+1:
> > onsdag 1 december 2021 kl. 22:54:28 UTC+1 skrev WM:
>
> > > > False, the fact it applies to all natural numbers means N_def=N
> > > Learn that two consecutive ℵo-sets cannot exist in the natural order of ℕ.
> > >
> Have you done your exercise?
> >
> > >7 defines a place on the number line.
> > Nope, it REPRESENTS a SPECIFIC number in N
> That means a place on the number line.
> > >No. 7 is in trichonomy with every natural number, x is not in trichotomy with any number.
> > False, both are
> >
> > x is greater than, less than or equal to n, just like 7
> For 7 and 3 we need no "or". For x and 3 we need it.
> > >It has. Learn the meaning of trichotomy.
> > I know the meaning of it, it has nothing to do with "individually definable"
> No, you know not even the difference between a constant, like 7, and a variable, like x.
> > >Learn that two consecutive ℵo-sets cannot exist in the natural order of ℕ.
> > I have never said it nor does anything imply it
> Learn that an infinite set of infinite endsegments would imply it.
>
> Regards, WM

>That means a place on the number line.

Numberline is a visual concept of it, we just talk about numbers. and 7 represents one of hte elements in it.

>For 7 and 3 we need no "or". For x and 3 we need it.

"7 is greater than 3 or 7 is equal to 3 or 7 is less than three"

is still a valid sentence even in logic :) So it still applies.

>No, you know not even the difference between a constant, like 7, and a variable, like x.

I know what a constant is, but you clearly do not. 7 is a specific element. A constant is somethign that doesn't changes. A variable changes.

x is not a variable in all contexts, in some it is a singular but generic element.

y=kx+m, k and m are constants, x is a variable

>Learn that an infinite set of infinite endsegments would imply it.

It implies nothing of the sort you crank.

On Thursday, December 2, 2021 at 4:30:47 AM UTC-5, WM wrote:>

> Every element of ℕ_P has ℵ₀ successors. That means we can collect all elements of ℕ_P which have ℵ₀ successors by the axiom of separation.
> Now the question is: Does this collection have ℵ₀ successors too?

The answer is no. The axiom of separation does not imply that if every *element* of a set has property P then the *set* has property P.

--
William Hughes

William schrieb am Donnerstag, 2. Dezember 2021 um 17:36:02 UTC+1:
> On Thursday, December 2, 2021 at 4:30:47 AM UTC-5, WM wrote:>
>
> > Every element of ℕ_P has ℵ₀ successors. That means we can collect all elements of ℕ_P which have ℵ₀ successors by the axiom of separation.
> > Now the question is: Does this collection have ℵ₀ successors too?
> The answer is no. The axiom of separation does not imply that if every *element* of a set has property P then the *set* has property P.

Here we have only a finite collection because only elements with ℵ₀ successors are collected. Before the successors have been reduced below ℵ₀, the collection cannot increase to infinity. But we explicitly are collecting only elements of ℕ_P having ℵ₀ successors. This proves that all elements in this collection have ℵ₀ successors because by definition only those have been chosen.

Regards, WM

Fritz Feldhase schrieb am Donnerstag, 2. Dezember 2021 um 11:29:09 UTC+1:
> On Thursday, December 2, 2021 at 10:33:23 AM UTC+1, WM wrote:
>
> Concerning /trichotomy/.

> No. For each and every x,y e IN: either x < y, or x = y, or x > y.

For 7 we can fix 7 > 3. For x we cannot fix x > 3. That's an important difference.
Hint: x is called a variable, meaning not being fixed.

Regards, WM