On Wednesday, January 4, 2023 at 10:36:00 AM UTC+1, Chris M. Thomasson wrote:

> Argh. A collection instead of a set?

What's your mathematical context? What is the _theory_ which allows for this(these) consideration(s)?

In "standard math" (i. e. classical mathematics) it is /set theory/. There "everything" is a set, even "collectons". (Though some approaches allow for so called proper classes and/or urelements.)

So what are theses collections of yours if they are no sets? Do you have a definition for them? And if you don't have a definition for them do you at least have a theory for them (where they appear as undefined notions)?

> A collection of all of the FISONS.

In the context of "classical mathematics": A set of all of the FISONs. The set of all FISONs:

Set_of_FISONs := {{m e IN : m <= n} : n e IN}.

> i[0] = { 0 } = { 0 }
> i[1] = { i[0], 1 } = { 0, 1 } // I guess you meant: ... = i[0] u { 1 } = { 0, 1 }
> i[2] = { i[1], 2 } = { 0, 1, 2 } // I guess you meant: ... = i[1] u { 2 } = { 0, 1, 2 }
> i[3] = { i[2], 3 } = { 0, 1, 2, 3 } // I guess you meant: ... = i[2] u { 3 } = { 0, 1, 2, 3 }
> :

Set_of_FISONs = {i[n] : n e IN} = { { 0 }, { 0, 1 }, { 0, 1, 2, 3 }, ... }

And yes, your function i can be recursively defined with the two (defining) equations:

i[0] = {0}
i[n+1] = i[n] u {n+1} (n e IN).

Hope this helps.

Fritz Feldhase schrieb am Mittwoch, 4. Januar 2023 um 00:16:33 UTC+1:

> IN_def = U_(n e IN) {1, 2, 3, ..., n} = IN.

You have unioned the successors too. You shall union only what you can identify as an individual. Then always infinitely many numbers remain outside.

Regards, WM

William schrieb am Mittwoch, 4. Januar 2023 um 02:58:07 UTC+1:
> On Tuesday, January 3, 2023 at 2:13:47 PM UTC-4, WM wrote:
> > William schrieb am Dienstag, 3. Januar 2023 um 17:08:53 UTC+1:
> >
> > > ∀n ∈ ℕ : |ℕ \ {1, 2, 3, ..., n}| = ℵo
> > Add *only these definable elements*
> no need to add anything. The statement is true for each element of |N,

If you add only those elements into a set ℕ_def then you will see that your statement is wrong because ℵo numbers are not added.

ℕ_def =/= ℕ.

But if you refuse, then you will stay in the darkness of matheology.

Regards, WM

On Wednesday, 4 January 2023 at 13:23:50 UTC+2, WM wrote:
> William schrieb am Mittwoch, 4. Januar 2023 um 02:58:07 UTC+1:
> > On Tuesday, January 3, 2023 at 2:13:47 PM UTC-4, WM wrote:
> > > William schrieb am Dienstag, 3. Januar 2023 um 17:08:53 UTC+1:
> > >
> > > > ∀n ∈ ℕ : |ℕ \ {1, 2, 3, ..., n}| = ℵo
> > > Add *only these definable elements*
> > no need to add anything. The statement is true for each element of |N,
> If you add only those elements into a set ℕ_def then you will see that your statement is wrong because ℵo numbers are not added.
>
> ℕ_def =/= ℕ.

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

How can ℕ_def not be equal to ℕ? The above statements are identical except for the names ℕ_def and ℕ.

>
> But if you refuse, then you will stay in the darkness of matheology.
>
> Regards, WM

On Wednesday, January 4, 2023 at 1:45:25 PM UTC+1, Eram semper recta wrote:

> ∀n ∈ ℕ : |ℕ \ {1, 2, 3, ..., n}| = ℵo and ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo
>
> How can ℕ_def not be equal to ℕ? The above statements are identical except for the names ℕ_def and ℕ.

So what?

IN_def may be empty: IN_def = { }. In this case

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

is still true.

Hint:

If we have

| Mückenheim is an idiot.

and

| John Gabriel is an idiot.

Does this imply

| Mückenheim = John Gabriel,

dumbo?

On Wednesday, January 4, 2023 at 12:19:21 PM UTC+1, WM wrote:
> Fritz Feldhase schrieb am Mittwoch, 4. Januar 2023 um 00:16:33 UTC+1:
> >
> > IN_def = U_(n e IN) {1, 2, 3, ..., n} = IN.
> >
> You have unioned <bla>. You shall union only what you can identify as an individual.

IN_def = U_(n e IN) {n} = IN.

> Then always infinitely many numbers remain outside.

No (natural) number "remains outside", idiot.

IN_def = IN, once more.

I am more worried that the ordinal exponentiation ω^ω
cannot be intuitively grasped. This here is surely nonsense:

2, 4, 8, ...
3, 9, 27, ...
5, 25, 125, ...

If the carrier of ω^ω are the finite sequences of natural numbers,
and each row in the above depiction is a sequence from ω^ω,

then the silly square has two errors:

- Error 1: The rows need not be disjoint, for example one row could
be 6,7 and another row could be 7,8, the number 7 appearing in both rows.

- Error 2: The rows are not infinite, if you state that ω^ω are the finite
sequences, then it doesn't make sense to draw something with etc. dots ...

WM schrieb am Montag, 19. Dezember 2022 um 11:04:58 UTC+1:
> If the complete set ℕ of natural numbers exists, then it can be used to populate Hilbert's Hotel.
> Is it possible to add a natural number to the set ℕ?
>
> Regards, WM

On 1/4/2023 6:51 AM, Fritz Feldhase wrote:
> On Wednesday, January 4, 2023 at 1:45:25 PM UTC+1, Eram semper recta wrote:
>
>> ∀n ∈ ℕ : |ℕ \ {1, 2, 3, ..., n}| = ℵo and ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo
>>
>> How can ℕ_def not be equal to ℕ? The above statements are identical except for the names ℕ_def and ℕ.
>
> So what?
>
> IN_def may be empty: IN_def = { }. In this case
>
> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo
>
> is still true.
>
> Hint:
>
> If we have
>
> | Mückenheim is an idiot.
>
> and
>
> | John Gabriel is an idiot.
>
> Does this imply
>
> | Mückenheim = John Gabriel,
>
> dumbo?

list of bad logic here;

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

But maybe what WM wants to say. If we have a Hilbert Hotel,
we can connect Guests. So if we connect Guest 2 and Guest 3,
this might be the same as Guest 6, since 2*3. So in this silly

square, the poor Guest 6 was doomed redundant, and is not
listed anymore, since we can connect Guest 2 and Guest 3
instead:

2, 4, 8, ...
3, 9, 27, ...
5, 25, 125, ...
....

We can do this for every natural number, connect some
nodes from the square. And 6 = 2^1*3^1.so we can also
say 6 = <1,1> i.e. the finite list <1,1>. So we find

every finite list in the square, possibly filled with some
zeros at the beginning, so that the n-th element of the
finite list is also the multiplicity of the n-th prime number.

Now if we allow connecting infinitely many Guests,
we can indeed also encode infinite lists of natural numbers,
and hence we can encode real numbers, for

example by adopting continued fractions. So the silly
square "contains" indeed the real line, if we look at all
infinite connecting patterns.

LoL

Mostowski Collapse schrieb am Mittwoch, 4. Januar 2023 um 15:11:20 UTC+1:
> I am more worried that the ordinal exponentiation ω^ω
> cannot be intuitively grasped. This here is surely nonsense:
>
> 2, 4, 8, ...
> 3, 9, 27, ...
> 5, 25, 125, ...
>
> If the carrier of ω^ω are the finite sequences of natural numbers,
> and each row in the above depiction is a sequence from ω^ω,
>
> then the silly square has two errors:
>
> - Error 1: The rows need not be disjoint, for example one row could
> be 6,7 and another row could be 7,8, the number 7 appearing in both rows.
>
> - Error 2: The rows are not infinite, if you state that ω^ω are the finite
> sequences, then it doesn't make sense to draw something with etc. dots ....
>
> WM schrieb am Montag, 19. Dezember 2022 um 11:04:58 UTC+1:
> > If the complete set ℕ of natural numbers exists, then it can be used to populate Hilbert's Hotel.
> > Is it possible to add a natural number to the set ℕ?
> >
> > Regards, WM

Chris M. Thomasson explained on 1/4/2023 :
> On 1/4/2023 1:17 AM, Chris M. Thomasson wrote:
>> On 1/4/2023 1:02 AM, Fritz Feldhase wrote:
>>> On Wednesday, January 4, 2023 at 6:56:24 AM UTC+1, Eram semper recta
>>> wrote:
>>>
>>>> ℕ contains ALL its elements.
>>>
>>> Yeah! It's hard to imagine a set which does not contain ALL its elements.
>>> :-)
>>
>> Define a symbol to represent an infinite set, say N for the naturals.
>> That's all.
>>
>
> Argh. A collection instead of a set? A collection of all of the FISONS.

They're sets, don't let WM fool you.

> i[0] = { 0 } = { 0 }
> i[1] = { i[0], 1 } = { 0, 1 }
> i[2] = { i[1], 2 } = { 0, 1, 2 }
> i[3] = { i[2], 3 } = { 0, 1, 2, 3 }
> ...

Isn't the emptyset a FISON? Is E(1) (a.k.a. |N) an endsegment? These
sets above all have nonzero cardinality. Do you need to add one more,
or change hotels? Have they finished building the Continuum Hotel
across the street yet? Not only infinitely many rooms, but double,
triple, even infinite occupancy in each room.

On 1/3/2023 4:06 PM, WM wrote:
> Fritz Feldhase schrieb am Dienstag, 3. Januar 2023 um 19:22:05 UTC+1:
>> On Tuesday, January 3, 2023 at 7:07:57 PM UTC+1, WM wrote:
>>
>>> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo
>>> Add *only these definable elements* into a set ℕ_def.
>>> Is that possible?
>> It is nonsense. Try again.
>
> It is possible for the first FISONs.
> Du you think there is a point where you could not continue?
>
> Regards, WM

what happened to endsegments ? did you forget about them ?

since ℕ_def changes it is not a set, so what does this statement really say ?

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

obviously;

(QuaCK!!)^(QuACk!!)

On 1/4/2023 8:11 AM, Mostowski Collapse wrote:
> I am more worried that the ordinal exponentiation ω^ω
> cannot be intuitively grasped. This here is surely nonsense:
>
> 2, 4, 8, ...
> 3, 9, 27, ...
> 5, 25, 125, ...
>
> If the carrier of ω^ω are the finite sequences of natural numbers,
> and each row in the above depiction is a sequence from ω^ω,
>
> then the silly square has two errors:
>
> - Error 1: The rows need not be disjoint, for example one row could
> be 6,7 and another row could be 7,8, the number 7 appearing in both rows.
>
> - Error 2: The rows are not infinite, if you state that ω^ω are the finite
> sequences, then it doesn't make sense to draw something with etc. dots ...
>
> WM schrieb am Montag, 19. Dezember 2022 um 11:04:58 UTC+1:
>> If the complete set ℕ of natural numbers exists, then it can be used to populate Hilbert's Hotel.
>> Is it possible to add a natural number to the set ℕ?
>>
>> Regards, WM

On Wednesday, January 4, 2023 at 7:23:50 AM UTC-4, WM wrote:
> William schrieb am Mittwoch, 4. Januar 2023 um 02:58:07 UTC+1:
> > On Tuesday, January 3, 2023 at 2:13:47 PM UTC-4, WM wrote:
> > > William schrieb am Dienstag, 3. Januar 2023 um 17:08:53 UTC+1:
> > >
> > > > ∀n ∈ ℕ : |ℕ \ {1, 2, 3, ..., n}| = ℵo
> > > Add *only these definable elements*
> > no need to add anything. The statement is true for each element of |N,
> If you add only those elements into a set ℕ_def

N_def is not a set it is something that changes. If I collect all n that satisfy the statement I get the set (∀n ∈ ℕ) which is of course the set |N

On Wednesday, January 4, 2023 at 6:32:01 PM UTC+1, William wrote:

> > |ℕ \ {1, 2, 3, ..., n}| = ℵo
> >
> If I collect all [natural numbers] n that satisfy the statement I get the set (∀n ∈ ℕ)

Huh?! Highly non-standard notation.

I guess you mean the set {n e IN : |ℕ \ {1, 2, 3, ..., n}| = ℵo}

> which is of course the set |N.

Of course.

On 1/4/2023 2:31 AM, Fritz Feldhase wrote:
> On Wednesday, January 4, 2023 at 10:36:00 AM UTC+1, Chris M. Thomasson wrote:
>
>> Argh. A collection instead of a set?
>
> What's your mathematical context? What is the _theory_ which allows for this(these) consideration(s)?
>
> In "standard math" (i. e. classical mathematics) it is /set theory/. There "everything" is a set, even "collectons". (Though some approaches allow for so called proper classes and/or urelements.)
>
> So what are theses collections of yours if they are no sets? Do you have a definition for them? And if you don't have a definition for them do you at least have a theory for them (where they appear as undefined notions)?
>
>> A collection of all of the FISONS.
>
> In the context of "classical mathematics": A set of all of the FISONs. The set of all FISONs:
>
> Set_of_FISONs := {{m e IN : m <= n} : n e IN}.
>
>> i[0] = { 0 } = { 0 }
>> i[1] = { i[0], 1 } = { 0, 1 } // I guess you meant: ... = i[0] u { 1 } = { 0, 1 }
>> i[2] = { i[1], 2 } = { 0, 1, 2 } // I guess you meant: ... = i[1] u { 2 } = { 0, 1, 2 }
>> i[3] = { i[2], 3 } = { 0, 1, 2, 3 } // I guess you meant: ... = i[2] u { 3 } = { 0, 1, 2, 3 }
>> :
>
> Set_of_FISONs = {i[n] : n e IN} = { { 0 }, { 0, 1 }, { 0, 1, 2, 3 }, ... }

You missed one FISON: { 0, 1, 2 }

{ { 0 }, { 0, 1 }, { 0, 1, 2 }, { 0, 1, 2, 3 }, ... }

;^)

> And yes, your function i can be recursively defined with the two (defining) equations:
>
> i[0] = {0}
> i[n+1] = i[n] u {n+1} (n e IN).
>
> Hope this helps.

I am quite fond of recursion, even though it scares me a bit in the land
of programming wrt recursing too deep and causing an error.

On 1/4/2023 6:37 AM, FromTheRafters wrote:
> Chris M. Thomasson explained on 1/4/2023 :
>> On 1/4/2023 1:17 AM, Chris M. Thomasson wrote:
>>> On 1/4/2023 1:02 AM, Fritz Feldhase wrote:
>>>> On Wednesday, January 4, 2023 at 6:56:24 AM UTC+1, Eram semper recta
>>>> wrote:
>>>>
>>>>> ℕ contains ALL its elements.
>>>>
>>>> Yeah! It's hard to imagine a set which does not contain ALL its
>>>> elements. :-)
>>>
>>> Define a symbol to represent an infinite set, say N for the naturals.
>>> That's all.
>>>
>>
>> Argh. A collection instead of a set? A collection of all of the FISONS.
>
> They're sets, don't let WM fool you.
>
>> i[0] = { 0 } = { 0 }
>> i[1] = { i[0], 1 } = { 0, 1 }
>> i[2] = { i[1], 2 } = { 0, 1, 2 }
>> i[3] = { i[2], 3 } = { 0, 1, 2, 3 }
>> ...
>
> Isn't the emptyset a FISON? Is E(1) (a.k.a. |N) an endsegment? These
> sets above all have nonzero cardinality. Do you need to add one more, or
> change hotels? Have they finished building the Continuum Hotel across
> the street yet? Not only infinitely many rooms, but double, triple, even
> infinite occupancy in each room.

There is infinite room here: Welcome.

On 1/4/2023 6:43 AM, Sergi o wrote:
> On 1/3/2023 4:06 PM, WM wrote:
>> Fritz Feldhase schrieb am Dienstag, 3. Januar 2023 um 19:22:05 UTC+1:
>>> On Tuesday, January 3, 2023 at 7:07:57 PM UTC+1, WM wrote:
>>>
>>>> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo
>>>> Add *only these definable elements* into a set ℕ_def.
>>>> Is that possible?
>>> It is nonsense. Try again.
>>
>> It is possible for the first FISONs.
>> Du you think there is a point where you could not continue?
>>
>> Regards, WM
>
>
> what happened to endsegments ?   did you forget about them ?

Since infinity never ends, there are no endsegments.

>
>
>
> since ℕ_def changes it is not a set, so what does this statement really
> say ?
>
> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo

On Wednesday, 4 January 2023 at 14:51:22 UTC+2, Fritz Feldhase wrote:
> On Wednesday, January 4, 2023 at 1:45:25 PM UTC+1, Eram semper recta wrote:
>
> > ∀n ∈ ℕ : |ℕ \ {1, 2, 3, ..., n}| = ℵo and ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ...., n}| = ℵo
> >
> > How can ℕ_def not be equal to ℕ? The above statements are identical except for the names ℕ_def and ℕ.
> So what?
>
> IN_def may be empty: IN_def = { }. In this case
> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo
> is still true.
>
> Hint:
>
> If we have
>
> | Mückenheim is an idiot.
>
> and
>
> | John Gabriel is an idiot.
>
> Does this imply
>
> | Mückenheim = John Gabriel,
>
> dumbo?

Hey moron.

A = B and B = C means A = C.

William schrieb am Mittwoch, 4. Januar 2023 um 18:32:01 UTC+1:
> On Wednesday, January 4, 2023 at 7:23:50 AM UTC-4, WM wrote:
> > William schrieb am Mittwoch, 4. Januar 2023 um 02:58:07 UTC+1:
> > > On Tuesday, January 3, 2023 at 2:13:47 PM UTC-4, WM wrote:
> > > > William schrieb am Dienstag, 3. Januar 2023 um 17:08:53 UTC+1:
> > > >
> > > > > ∀n ∈ ℕ : |ℕ \ {1, 2, 3, ..., n}| = ℵo
> > > > Add *only these definable elements*
> > > no need to add anything. The statement is true for each element of |N,
> > If you add only those elements into a set ℕ_def
> N_def is not a set it is something that changes. If I collect all n that satisfy the statement I get the set (∀n ∈ ℕ) which is of course the set |N

You have to add only those numbers which leave ℵo numbers outside. Please obey this order! Then not all numbers can be inside.

Regards, WM

Fritz Feldhase schrieb am Mittwoch, 4. Januar 2023 um 20:06:45 UTC+1:
> On Wednesday, January 4, 2023 at 6:32:01 PM UTC+1, William wrote:
>
> > > |ℕ \ {1, 2, 3, ..., n}| = ℵo
> > >
> > If I collect all [natural numbers] n that satisfy the statement I get the set (∀n ∈ ℕ)
>
> Huh?! Highly non-standard notation.
>
> I guess you mean the set {n e IN : |ℕ \ {1, 2, 3, ..., n}| = ℵo}
>
> > which is of course the set |N.
>
> Of course.

Nonsense. You have to add only those numbers which leave ℵo numbers outside. Please obey this order! Then not all numbers of ℕ can be inside.

Regards, WM

On Wednesday, January 4, 2023 at 6:37:46 PM UTC-4, WM wrote:
> William schrieb am Mittwoch, 4. Januar 2023 um 18:32:01 UTC+1:
> > On Wednesday, January 4, 2023 at 7:23:50 AM UTC-4, WM wrote:
> > > William schrieb am Mittwoch, 4. Januar 2023 um 02:58:07 UTC+1:
> > > > On Tuesday, January 3, 2023 at 2:13:47 PM UTC-4, WM wrote:
> > > > > William schrieb am Dienstag, 3. Januar 2023 um 17:08:53 UTC+1:
> > > > >
> > > > > > ∀n ∈ ℕ : |ℕ \ {1, 2, 3, ..., n}| = ℵo
> > > > > Add *only these definable elements*
> > > > no need to add anything. The statement is true for each element of |N,
> > > If you add only those elements into a set ℕ_def
> > N_def is not a set it is something that changes. If I collect all n that satisfy the statement I get the set (∀n ∈ ℕ) which is of course the set |N
> You have to add only those numbers which leave ℵo numbers outside..
I am adding those numbers. Since each element of |N leaves a set of ℵo numbers outside I add each element of |N.

William schrieb am Mittwoch, 4. Januar 2023 um 23:44:25 UTC+1:
> On Wednesday, January 4, 2023 at 6:37:46 PM UTC-4, WM wrote:
> > William schrieb am Mittwoch, 4. Januar 2023 um 18:32:01 UTC+1:
> > > On Wednesday, January 4, 2023 at 7:23:50 AM UTC-4, WM wrote:
> > > > William schrieb am Mittwoch, 4. Januar 2023 um 02:58:07 UTC+1:
> > > > > On Tuesday, January 3, 2023 at 2:13:47 PM UTC-4, WM wrote:
> > > > > > William schrieb am Dienstag, 3. Januar 2023 um 17:08:53 UTC+1:
> > > > > >
> > > > > > > ∀n ∈ ℕ : |ℕ \ {1, 2, 3, ..., n}| = ℵo
> > > > > > Add *only these definable elements*
> > > > > no need to add anything. The statement is true for each element of |N,
> > > > If you add only those elements into a set ℕ_def
> > > N_def is not a set it is something that changes. If I collect all n that satisfy the statement I get the set (∀n ∈ ℕ) which is of course the set |N
> > You have to add only those numbers which leave ℵo numbers outside.
> I am adding those numbers. Since each element of |N leaves a set of ℵo numbers outside I add each element of |N.

What is remaining outside?
As soon as it shrinks below ℵo natnumbers stop and reverse the last addition.

Regards, WM

On Wednesday, January 4, 2023 at 6:50:42 PM UTC-4, WM wrote:
> I am adding those numbers. Since each element of |N leaves a set of ℵo numbers outside I add each element of |N.
> What is remaining outside?

The set of numbers greater than the given element of |N.

> As soon as it shrinks below ℵo

This does not happen. The cardinality of the set is ℵo for every element of |N.

"Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:

> Since infinity never ends, there are no endsegments.

I fear you have drunk the cool-aid though I don't know whose! Not WM's
cool aid since there are endsegments even in WMaths.

In ordinary mathematics, E_n = { k ∈ N | k >= n } is a perfectly fine
set of natural numbers. ∀n, |E_n| = aleph_0. ∩_{n∈N} E_n = {}.

And elsewhere you stated that N, being infinite, can't include "all"
naturals. I suspect you think of a definition like

1 ∈ N
n ∈ N => s(n) ∈ N

as a "process", but it just means that N is closed under the successor
function s.

--
Ben.

WM <askasker48@gmail.com> writes:
(AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des
Unendlichen" at Hochschule Augsburg.)

> Ben Bacarisse schrieb am Dienstag, 3. Januar 2023 um 02:45:17 UTC+1:
>> A valid (and complete) room
>> assignment is available both for the old set of guests and for the new
>> set of guests. No room need to be added.
>>
> Hilbert's hotel can easily be misinterpreted by unsharp minds.

You, fortunately, did not misinterpret it though you have chosen to hide
your agreement that both sets of guests can be accommodated by perfectly
valid complete room assignments.

> Compared with infinitely many guests one more guest is nearly same as
> no more guest.

Great. We agree that both

h0: N->N with h0(n) = n and h1: ({-1} U N)->N with h1(n) = n+1 are valid
room assignments in proper mathematics as well as WMaths.

(In case you want to change your mind, I remind you that you have
already agreed to this.)

> The better argument is this: The number of not indexed fractions
> remains |ℕ|*(|ℕ|-1) for all definable terms of the sequence 1/1, 1/2,
> 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 2/4, 3/3, 4/2, 5/1, 1/6,
> 2/5, 3/4, 4/3, 5/2, 6/1, ... . Hence |ℕ|*(|ℕ|-1) fractions cannot be
> indexed by definable indices.

Oops! You want to talk about something else because you had to agree
that no new room is needed to accommodate the new guest!

--
Ben.