On Saturday, December 4, 2021 at 10:57:00 AM UTC+1, WM wrote:
> 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.

No, this is a trivial truth. Several proofs for this trivial set theoretic fact have been posted here, idiot.

Hint:

(1.) Each and every natural number has ℵ₀ successors. Hence (2.) the set of all natural numbers which have ℵ₀ successors is just the set of all natural numbers.

1. An e IN: card({m e IN : m > n}) = ℵ₀

2. {n e IN : card({m e IN : m > n}) = ℵ₀} = IN.

Trivial set theortic facts.

Since it seems that you are lacking basic logical knowledge, here's the general rule which has been applied:

[Ax e X: P(x)] ==> {x e X : P(X)} = X [for any set X and unary predicate P]

> In case of <bla>

Yeah, whatever, Mückenheim.

> > An e IN: Ex e ORD: n < x < ω.
> >
> Of course. [...}
> >
> > Beware of the quantifier shift!

Meaning: "Ex e ORD: An e IN: n < x < ω" does NOT hold.

Actually,

~Ex e ORD: An e IN: n < x < ω

is a theorem in set theory.

On Saturday, December 4, 2021 at 4:44:52 AM UTC-5, WM wrote:
> observed as individuals.

"observed as individuals" is yet another term you have for "can be written down". The fact that "most elements" of an infinite set cannot be written down does not mean they do not have other properties.

--
William Hughes

On Saturday, November 6, 2021 at 2:42:36 PM UTC-7, William wrote:
> The following statements of about the set |N_F are all true
>
> (i) Every element of |N_F is finite
>
> (ii) |N_F, like any Peano set, has cardinality aleph_0
>
> (iii) Between every element of |N_F and omega there are infinitely many natural numbers.
>
> (iv) There is no natural number between the set |N_F and omega.
>
> (iv) is true, not because there is an element of |N_F which is not separated from omega by an infinite set of natural numbers, but because there is no largest natural number.
>
> --
> William Hughes
>
>
> *i

So, you add a constant omega and omega numbers.

Now you have answered "if a set is infinite does it
have infinite members, yes, it does".

On 12/4/2021 3:38 AM, WM wrote:
> 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.

it is correct.

>
> 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.

no such thing as potential infinity

>
> 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.

no such thing as "definable" so your statement is bogus.

>
>> 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.

no. too many mistakes to correct.

>
>> 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.

"removed" is bogus. use Math statements.

>
> Regards, WM
>

On Saturday, December 4, 2021 at 5:30:54 AM UTC-8, Fritz Feldhase wrote:
> On Saturday, December 4, 2021 at 10:57:00 AM UTC+1, WM wrote:
> > 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.
> No, this is a trivial truth. Several proofs for this trivial set theoretic fact have been posted here, idiot.
>
> Hint:
>
> (1.) Each and every natural number has ℵ₀ successors. Hence (2.) the set of all natural numbers which have ℵ₀ successors is just the set of all natural numbers.
>
> 1. An e IN: card({m e IN : m > n}) = ℵ₀
>
> 2. {n e IN : card({m e IN : m > n}) = ℵ₀} = IN.
>
> Trivial set theortic facts.
>
> Since it seems that you are lacking basic logical knowledge, here's the general rule which has been applied:
>
> [Ax e X: P(x)] ==> {x e X : P(X)} = X [for any set X and unary predicate P]
>
> > In case of <bla>
>
> Yeah, whatever, Mückenheim.
> > > An e IN: Ex e ORD: n < x < ω.
> > >
> > Of course. [...}
> > >
> > > Beware of the quantifier shift!
> Meaning: "Ex e ORD: An e IN: n < x < ω" does NOT hold.
>
> Actually,
> ~Ex e ORD: An e IN: n < x < ω
> is a theorem in set theory.

This results after what is otherwise the predicative,
which is complete, which is the impredicative, which
follows via another deduction to that otherwise closed
induction, why it is only for example contingent.

Then the impredicativity result the quantifier ambiguity,
but also of course, the definition of the unambiguous cases
for the quantifier: when the transfer principle makes for
that there is the unambiguous case defined by the induction
and the unambiguous case defined by the deduction,
and also together for whatever sets, for example,
that the induction and deduction (like the usual deduction
that induction holds) result the "anti", some "anti-infinity at infinity".

Here there's usually the point at infinity or at least one infinity
that results having a limit point, it is absolutely in the scale of
all the finite numbers which establish a linear scale.

Infinity is introduced for any closure of arithmetic that results
what is not a number yet arbitrarily large, for example.

A sufficiently large estabishment of cases in induction makes
that instead of infinity is introduced for any closure of arithmetic
the results what is not a number yet arbitrarily large, here that
in the cases for deduction outside "that between zero and one,
the distances from zero to one and distances from one to zero
and symmetrical, and multiplying one by infinity preserves this
symmetry, and these also model the integers including infinite",
infinity suffices in terms then for restriction in "valid arithmetic
about infinity".

That formalism has demanding....

On Saturday, December 4, 2021 at 11:29:45 AM UTC-5, Ross A. Finlayson wrote:
>"if a set is infinite does it
> have infinite members, yes, it does".

Nope. Every element of a Peano set is finite (trivial induction), a Peano set has cardinality aleph_0, an Peano set exists (axiom of infinity)

--
William Hughes

On Saturday, December 4, 2021 at 9:11:44 AM UTC-8, William wrote:
> On Saturday, December 4, 2021 at 11:29:45 AM UTC-5, Ross A. Finlayson wrote:
> >"if a set is infinite does it
> > have infinite members, yes, it does".
> Nope. Every element of a Peano set is finite (trivial induction), a Peano set has cardinality aleph_0, an Peano set exists (axiom of infinity)
>
> --
> William Hughes

This results after what is otherwise the predicative,
which is complete, which is the impredicative, which
follows via another deduction to that otherwise closed
induction, why it is only for example contingent.

Then the impredicativity result the quantifier ambiguity,
but also of course, the definition of the unambiguous cases
for the quantifier: when the transfer principle makes for
that there is the unambiguous case defined by the induction
and the unambiguous case defined by the deduction,
and also together for whatever sets, for example,
that the induction and deduction (like the usual deduction
that induction holds) result the "anti", some "anti-infinity at infinity".

Here there's usually the point at infinity or at least one infinity
that results having a limit point, it is absolutely in the scale of
all the finite numbers which establish a linear scale.

Infinity is introduced for any closure of arithmetic that results
what is not a number yet arbitrarily large, for example.

A sufficiently large estabishment of cases in induction makes
that instead of infinity is introduced for any closure of arithmetic
the results what is not a number yet arbitrarily large, here that
in the cases for deduction outside "that between zero and one,
the distances from zero to one and distances from one to zero
and symmetrical, and multiplying one by infinity preserves this
symmetry, and these also model the integers including infinite",
infinity suffices in terms then for restriction in "valid arithmetic
about infinity".

That formalism has demanding....

Ross A. Finlayson submitted this idea :
> On Saturday, November 6, 2021 at 2:42:36 PM UTC-7, William wrote:
>> The following statements of about the set |N_F are all true
>>
>> (i) Every element of |N_F is finite
>>
>> (ii) |N_F, like any Peano set, has cardinality aleph_0
>>
>> (iii) Between every element of |N_F and omega there are infinitely many
>> natural numbers.
>>
>> (iv) There is no natural number between the set |N_F and omega.
>>
>> (iv) is true, not because there is an element of |N_F which is not separated
>> from omega by an infinite set of natural numbers, but because there is no
>> largest natural number.
>>
>> --
>> William Hughes
>>
>>
>> *i
>
> So, you add a constant omega and omega numbers.
>
> Now you have answered "if a set is infinite does it
> have infinite members, yes, it does".

The size of a set has nothing to do with the size of its elements.

On Saturday, December 4, 2021 at 11:30:29 AM UTC-8, FromTheRafters wrote:
> Ross A. Finlayson submitted this idea :
> > On Saturday, November 6, 2021 at 2:42:36 PM UTC-7, William wrote:
> >> The following statements of about the set |N_F are all true
> >>
> >> (i) Every element of |N_F is finite
> >>
> >> (ii) |N_F, like any Peano set, has cardinality aleph_0
> >>
> >> (iii) Between every element of |N_F and omega there are infinitely many
> >> natural numbers.
> >>
> >> (iv) There is no natural number between the set |N_F and omega.
> >>
> >> (iv) is true, not because there is an element of |N_F which is not separated
> >> from omega by an infinite set of natural numbers, but because there is no
> >> largest natural number.
> >>
> >> --
> >> William Hughes
> >>
> >>
> >> *i
> >
> > So, you add a constant omega and omega numbers.
> >
> > Now you have answered "if a set is infinite does it
> > have infinite members, yes, it does".
> The size of a set has nothing to do with the size of its elements.

Or "why be wrong just because the infinite is antinomous to the
finite and the discrete is antinomous to the continuous".

Or vice versa....

Why not be not wrong?

(There is set size in cardinals and ordinals.)

The whole point and entire statement of the transfer principle is
that what is so for the elements is so for the set.

I.e. that is a usual property that is not a usual property.
But, it is so, for at least one set: infinity.

I wrote this measure and length assignment for mathematics.

Fritz Feldhase schrieb am Samstag, 4. Dezember 2021 um 14:30:54 UTC+1:
> On Saturday, December 4, 2021 at 10:57:00 AM UTC+1, WM wrote:
> > 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.
> No, this is a trivial truth.

Each and every definable natural number has ℵo successors. But all natural numbers together are the set of all natural numbers. It has no successors before ω.

1) The union of FISONs is same as the union of all definable natnumbers: ℕ_def.
The union of FISONs does not change when all FISONs smaller than other FISONs are omitted. Can you understand that? Therefore the union of FISONs is same as one FISON. And that is finite and not an ℵ₀-set of like ℕ.

2) Assume that the union of all FISONs is ℕ. Then all FISONs smaller than |ℕ| = ℵ can be omitted from the union of FISONs. Do you agree? If not, why not? Can you show a first FISON not to be omitted from the union?

Regards, WM

William schrieb am Samstag, 4. Dezember 2021 um 17:14:28 UTC+1:
> On Saturday, December 4, 2021 at 4:44:52 AM UTC-5, WM wrote:
> > observed as individuals.
>
> "observed as individuals" is yet another term you have for "can be written down".

Its meaning is this: n can be observed as an individual if a FISON F(n) exists.
> The fact that "most elements" of an infinite set cannot be written down does not mean they do not have other properties.

So it is. But they have the property that they cannot be written down or communicated otherwise or thought of as individuals and therefore they cannot be attached to other individuals.

They are not sufficient to form the set ℕ.

Assume that the union of all FISONs is ℕ. Then all FISONs smaller than |ℕ| = ℵ can be omitted from the union of FISONs. Do you agree? If not, why not? Can you show a first FISON not to be omitted from the union?

Regards, WM

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

> Beware of the quantifier shift!

Why? If every definable natnumber has ℵo successors, then the collection [*] of *these* natnumbers has ℵo successors. In what way should this simple property defining the collected elements disappear?

[*] Simply collect by the axiom of separation all natnumbers having ℵo successors. The result is a collection having ℵo successors. Beware of collecting more! (But it is impossible to collect more natnumbers individually.)

Regards, WM

On 12/5/2021 2:06 AM, WM wrote:
> Fritz Feldhase schrieb am Samstag, 4. Dezember 2021 um 14:30:54 UTC+1:
>> On Saturday, December 4, 2021 at 10:57:00 AM UTC+1, WM wrote:
>>> 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.
>> No, this is a trivial truth.
>
> Each and every definable natural number has ℵo successors. But all natural numbers together are the set of all natural numbers. It has no successors before ω.
>
> 1) The union of FISONs is same as the union of all definable natnumbers: ℕ_def.
[...]

Any natural number is able to be defined? No?

It happens that Chris M. Thomasson formulated :
> On 12/5/2021 2:06 AM, WM wrote:
>> Fritz Feldhase schrieb am Samstag, 4. Dezember 2021 um 14:30:54 UTC+1:
>>> On Saturday, December 4, 2021 at 10:57:00 AM UTC+1, WM wrote:
>>>> 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.
>>> No, this is a trivial truth.
>>
>> Each and every definable natural number has ℵo successors. But all natural
>> numbers together are the set of all natural numbers. It has no successors
>> before ω.
>>
>> 1) The union of FISONs is same as the union of all definable natnumbers:
>> ℕ_def.
> [...]
>
> Any natural number is able to be defined? No?

Every natural number is already defined.

On 12/5/2021 4:06 AM, WM wrote:
> William schrieb am Samstag, 4. Dezember 2021 um 17:14:28 UTC+1:
>> On Saturday, December 4, 2021 at 4:44:52 AM UTC-5, WM wrote:
>>> observed as individuals.
>>
>> "observed as individuals" is yet another term you have for "can be written down".
>
> Its meaning is this: n can be observed as an individual if a FISON F(n) exists.

what do you mean by "exists" ?

>
>> The fact that "most elements" of an infinite set cannot be written down does not mean they do not have other properties.
>
> So it is. But they have the property that they cannot be written down or communicated otherwise or thought of as individuals and therefore they cannot be attached to other individuals.

wrong. they do not have such property.

>
> They are not sufficient to form the set ℕ.

what do you mean by "They" ?

>
> Assume that the union of all FISONs is ℕ.

ok

> Then all FISONs smaller than |ℕ| = ℵ can be omitted from the union of FISONs.

Then it is not the union of FISONs anymore.

> Do you agree? If not, why not? Can you show a first FISON not to be omitted from the union?

Im not going to do math for you. Do it your self, you all grown up now.

>
> Regards, WM
>

On 12/5/2021 4:06 AM, WM wrote:
> Fritz Feldhase schrieb am Samstag, 4. Dezember 2021 um 14:30:54 UTC+1:
>> On Saturday, December 4, 2021 at 10:57:00 AM UTC+1, WM wrote:
>>> 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.
>> No, this is a trivial truth.
>
> Each and every definable natural number has ℵo successors. But all natural numbers together are the set of all natural numbers. It has no successors before ω.
>
> 1) The union of FISONs is same as the union of all definable natnumbers: ℕ_def.

then Prove it.

> The union of FISONs does not change when all FISONs smaller than other FISONs are omitted.

wrong, because it is not the union of all FISONs anymore.

Show Equations!

> Can you understand that? Therefore the union of FISONs is same as one FISON.

lies

> And that is finite and not an ℵ₀-set of like ℕ.

lies

>
> 2) Assume that the union of all FISONs is ℕ. Then all FISONs smaller than |ℕ| = ℵ can be omitted from the union of FISONs.

no, because it is not the union of all FISONs anymore.

Do you agree? If not, why not? Can you show a first FISON not to be omitted from the union?

this is your mess, you do it, but use MATH

>
> Regards, WM
>

On 12/5/2021 4:15 AM, WM wrote:
> Fritz Feldhase schrieb am Freitag, 3. Dezember 2021 um 19:33:36 UTC+1:
>
>> Beware of the quantifier shift!
>
> Why? If every definable natnumber has ℵo successors, then the collection [*] of *these* natnumbers has ℵo successors. In what way should this simple property defining the collected elements disappear?

that is meaningless. the following are your vague words that are not defined in math: collected, definable, disappear, defining

fail.

On Sunday, December 5, 2021 at 5:07:04 AM UTC-5, WM wrote

A repeat of a silly argument
> Assume that the union of all FISONs is [|N_P]

No assumption necessary. For any natural number n, the union of the set of FISONS, S_n, the FISONS with largest element greater than n,
is the natural numbers, |N_P, Note that while we can make the smallest element of S_n as large as we want. the smallest element must be finite so the cardintality of S_n is aleph_0 The fact that we can make the smallest element of S_n as large as we want does not mean that we can make S_n, the empty set.

--
William Hughes

On 12/5/2021 4:47 AM, FromTheRafters wrote:
> It happens that Chris M. Thomasson formulated :
>> On 12/5/2021 2:06 AM, WM wrote:
>>> Fritz Feldhase schrieb am Samstag, 4. Dezember 2021 um 14:30:54 UTC+1:
>>>> On Saturday, December 4, 2021 at 10:57:00 AM UTC+1, WM wrote:
>>>>> 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.
>>>> No, this is a trivial truth.
>>>
>>> Each and every definable natural number has ℵo successors. But all
>>> natural numbers together are the set of all natural numbers. It has
>>> no successors before ω.
>>>
>>> 1) The union of FISONs is same as the union of all definable
>>> natnumbers: ℕ_def.
>> [...]
>>
>> Any natural number is able to be defined? No?
>
> Every natural number is already defined.

Touche! :^D

Chris M. Thomasson was thinking very hard :
> On 12/5/2021 4:47 AM, FromTheRafters wrote:
>> It happens that Chris M. Thomasson formulated :
>>> On 12/5/2021 2:06 AM, WM wrote:
>>>> Fritz Feldhase schrieb am Samstag, 4. Dezember 2021 um 14:30:54 UTC+1:
>>>>> On Saturday, December 4, 2021 at 10:57:00 AM UTC+1, WM wrote:
>>>>>> 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.
>>>>> No, this is a trivial truth.
>>>>
>>>> Each and every definable natural number has ℵo successors. But all
>>>> natural numbers together are the set of all natural numbers. It has no
>>>> successors before ω.
>>>>
>>>> 1) The union of FISONs is same as the union of all definable natnumbers:
>>>> ℕ_def.
>>> [...]
>>>
>>> Any natural number is able to be defined? No?
>>
>> Every natural number is already defined.
>
> Touche! :^D

But things are different in muckymath. The first two axioms seem to be
the "Axiom Schema of Because I said so" and the "Axiom of Repetition".

fredag 3 december 2021 kl. 13:24:51 UTC+1 skrev WM:
> Fritz Feldhase schrieb am Freitag, 3. Dezember 2021 um 13:19:35 UTC+1:
> > On Friday, December 3, 2021 at 10:37:52 AM UTC+1, WM wrote:
> >
> > > Here we have only a finite collection
> > No.
> Where does the infinite set of successors start?
> > > because only elements with ℵ₀ successors are collected.
> > Non sequitur.
> Simply answer the question: If there are all ℵ₀ natnumbers used, what is the first number of the infinite set of ℵ₀ successors?
> >
> > Hint: Let IN' = {n e IN : card({m e IN : m > n}) = ℵ₀}. Then card(IN') = ℵ₀.
> Obviously matheologians have never thought about the nonsense they proclaim.
>
> Regards, WM

There is no such thing as "matheologians", there are CRANKS however, and YOU are one!

fredag 3 december 2021 kl. 15:08:48 UTC+1 skrev WM:
> 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!
>
> Collecting natural numbers yields this set. Collecting only definable natural numbers yields finite sets with ℵo successors before ω. You will agree that this cannot be changed.
>
> Collecting all natnumbers however yields a set with no successors before ω.
>
> 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?
>
> Regards, WM

Your set of "definable natural numbers" is not finite

fredag 3 december 2021 kl. 16:26:11 UTC+1 skrev WM:
> 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
why would N have a sauccessor?

William schrieb am Sonntag, 5. Dezember 2021 um 19:13:36 UTC+1:

> Note that while we can make the smallest element of S_n as large as we want. the smallest element must be finite so the cardinality of S_n is aleph_0 The fact that we can make the smallest element of S_n as large as we want

That means the preceding collection is potentially infinite.

> does not mean that we can make S_n, the empty set.

On the contrary, the set of never addressed natnumbers, so called dark numbers, is actually infinite.

Regards, WM

zelos...@gmail.com schrieb am Montag, 6. Dezember 2021 um 06:23:22 UTC+1:
> fredag 3 december 2021 kl. 15:08:48 UTC+1 skrev WM:

> > Collecting natural numbers yields this set. Collecting only definable natural numbers yields finite sets with ℵo successors before ω. You will agree that this cannot be changed.
> >
> > Collecting all natnumbers however yields a set with no successors before ω.
> >
> > 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?
> >
> Your set of "definable natural numbers" is not finite

It is potentially infinite. But it cannot be an ℵo-set, because it is followed by an ℵo-set.

Regards, WM