William schrieb am Montag, 23. August 2021 um 05:08:57 UTC+2:
> On Sunday, August 22, 2021 at 6:02:18 PM UTC-4, WM wrote:
> > William schrieb am Sonntag, 22. August 2021 um 21:04:03 UTC+2:
> > > On Sunday, August 22, 2021 at 2:46:49 PM UTC-4, WM wrote:
> > > > > > Here is the proof of incomplete
> > > > > The term "complete" is nonsense. it implies that there are "incomplete" sets,
> > > > No
> > > Yes. All sets contain what they contains. A set can be a proper subset but it cannot be "incomplete". If x is a FISON then x is an element of set set of FISONS.
> > > The set of FISONs is not "incomplete" because the set has no last element.
> > >
> > But a mapping can be incomplete.
> Nope,

It can be incomplete.

>a bijection is a set of ordered pairs.

a complete set.

> There is no such thing as an "incomplete" set. The set of FISONs has no last element but there is no FISON missing from the set.

There are not more FISONs than can be distinguished by FISONs, i.e., there are less than aleph_0 FISONs. Do you understand and accept quantifier logic?

~∃n ∈ ℕ: P(n) <==> ∀n ∈ ℕ: ~P(n)

Please apply this to

∀n ∈ ℕ: |ℕ \ {1, 2, 3, ..., n}| = ℵo.

Regards, WM

zelos...@gmail.com schrieb am Montag, 23. August 2021 um 06:51:10 UTC+2:
> torsdag 19 augusti 2021 kl. 14:19:46 UTC+2 skrev WM:
> > zelos...@gmail.com schrieb am Donnerstag, 19. August 2021 um 07:00:48 UTC+2:
> > > >The definable natural numbers have a discernible well-ordering
> > > Whatever "discernible" means.
> > It can be verified step by step.
> > >
> > > but N is well-ordered so again, N_def=N
> > >
> > No:
> > (1) When indexing the rationals all unit intervals can be completed.
> > (2) Not all unit intervals can be completed by one index because an index is issued to one rational of one interval only.
> > (3) So there are at least two indices required to complete all intervals.
> > (4) You cannot find out their order by their FISONs. If you could, you would complete aleph_0 rationals of one interval but you would need further indices. Contradiction. That is the true reason for dark numbers.

> mathematics doesn't work in "steps" like a fucking program.

The sequence of definable natural numbers works in steps. Every step can be analyzed.
>
> N is well-ordered and your N_def=N

Do you understand and accept quantifier logic?

~∃n ∈ ℕ: P(n) <==> ∀n ∈ ℕ: ~P(n)

Please apply this to

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

Regards, WM

Jim Burns schrieb am Montag, 23. August 2021 um 06:55:33 UTC+2:
> On 8/22/2021 6:00 PM, WM wrote:
> > Jim Burns schrieb
> > am Sonntag, 22. August 2021 um 21:45:36 UTC+2:
>
> >> *No rational completes any interval*
> >
> > Then there is no bijection and no cardinality.
> > For cardinality completion is required.
> >
> > There was no objection to a 'potential infinity' in the
> > form of an unending process, but an 'actual infinity' in
> > the form of a completed infinite set was harder to accept."
> > [H.B. Enderton: "Elements of set theory",
> > Academic Press, New York (1977) p. 14f]
> "p. 14f" would be the forward. Is that right?

Presently I have no acces to this book.
>
> Should I assume from your use of that quote that you do not
> accept the assumption that completed infinite sets exist?

No, I only showed William that potential infinity is a sensible notion.
>
> Perhaps you accept different assumptions.
> | A finite index is a finite index.
> | A positive rational is a positive rational.
> | Preserving truth preserves truth.
>
> Do you (WM) accept these assumptions?

Of course. Including the results. Do you understand and accept quantifier logic?

~∃n ∈ ℕ: P(n) <==> ∀n ∈ ℕ: ~P(n)

Please apply this to

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

In the interval (1000, 1001] the density of rationals is precisely the same as in the interval (0, 1]. Alas Cantor enumerates less than one promille.

When going from 1 to 0, then there is no unit fraction more later. Either, forced by the linearity of the sequence 1, 1/2, 1/3, ..., 0, a last unit fraction has been passed. Or there are dark numbers.

Regards, WM

On Monday, 23 August 2021 at 09:02:37 UTC-3, WM wrote:
> There are not more FISONs than can be distinguished by FISONs,

How perceptive!! Unfortunately, only an incorrigible moron like WM can think that the follow-on

> i.e., there are less than aleph_0 FISONs.

is a consequence of this remarkable insight.

> ~∃n ∈ ℕ: P(n) <==> ∀n ∈ ℕ: ~P(n)
>
> Please apply this to
>
> ∀n ∈ ℕ: |ℕ \ {1, 2, 3, ..., n}| = ℵo.

OK: ∀n ∈ ℕ: |ℕ \ {1, 2, 3, ..., n}| = ℵo <==> ~∃n ∈ ℕ: |ℕ \ {1, 2, 3, ..., n}| =/= ℵo.

Only WM seems to think that this is a revolutionary insight. It is not. Taking the difference of an infinite set and a finite set results in an infinite set. NOTHING TO SEE HERE. Interchanging limits involving the cardinality function is simply not allowed.

Muckenheim, you are too stupid to SHIT, let alone do mathematics.

On Monday, August 23, 2021 at 8:02:37 AM UTC-4, WM wrote:
> William schrieb am Montag, 23. August 2021 um 05:08:57 UTC+2:
> > On Sunday, August 22, 2021 at 6:02:18 PM UTC-4, WM wrote:
> > > William schrieb am Sonntag, 22. August 2021 um 21:04:03 UTC+2:
> > > > On Sunday, August 22, 2021 at 2:46:49 PM UTC-4, WM wrote:
> > > > > > > Here is the proof of incomplete
> > > > > > The term "complete" is nonsense. it implies that there are "incomplete" sets,
> > > > > No
> > > > Yes. All sets contain what they contains. A set can be a proper subset but it cannot be "incomplete". If x is a FISON then x is an element of set set of FISONS.
> > > > The set of FISONs is not "incomplete" because the set has no last element.
> > > >
> > > But a mapping can be incomplete.
> > Nope,
> It can be incomplete.
> >a bijection is a set of ordered pairs.
> a complete set.

"complete" is a nonsense term. There is no such thing as an "incomplete" set and since a mapping is a set, there is no such thing as an incomplete mapping (there are mappings that cannot be analysed by step by step methods)..

> > There is no such thing as an "incomplete" set. The set of FISONs has no last element but there is no FISON missing from the set.
> There are not more FISONs than can be distinguished by

the elements in the *set of FISONs * "A finite but unbounded set" is nonsense.
The set if FISONs is a Peano set and has cardinality aleph_0.
--
William Hughes

On Monday, August 23, 2021 at 2:02:37 PM UTC+2, WM wrote:

> there are less than aleph_0 FISONs.

Nope. I already told you that IN = Set_of_FISONs (in ZFC, due to von Neumann).

Hence there card(Set_of_FISONs) = card(IN) = aleph_0 (by definition).

So you are WRONG, Mückenheim.

Again, there are AS MANY FISONs AS NATURAL NUMBERS, since the set of natural numbers is identical with the set of FISONs. You know: 0 = {}, 1 = {0}, 2 = {0, 1}, 3 = {0, 1, 2}, ... Hence IN = {1, 2, 3, ...} = {{}, {0}, {0, 1}, {0, 1, 2}, ...} = Set_of_FISONs. Now there are aleph_0 natural numbers BY DEFINITION, hence there are aleph_0 FISONs too.

On Monday, August 23, 2021 at 2:08:57 PM UTC+2, WM wrote:
> Jim Burns schrieb am Montag, 23. August 2021 um 06:55:33 UTC+2:
> > >
> > > There was no objection to a 'potential infinity' in the
> > > form of an unending process, but an 'actual infinity' in
> > > the form of a completed infinite set was harder to accept."
> > > [H.B. Enderton: "Elements of set theory",
> > > Academic Press, New York (1977) p. 14f]
> > >
> > Should I assume from your use of that quote that you do not
> > accept the assumption that completed infinite sets exist?
> >
> No, I only showed William that potential infinity is a sensible notion.

lol. Yeah, "in the form of an unending process".

But a set is not a process you know. And talking about "potentially infinite" sets is just nonsense.

Hint: A set is either finite or infinite. Period.

On 8/23/2021 7:04 AM, WM wrote:
> zelos...@gmail.com schrieb am Montag, 23. August 2021 um 06:51:10 UTC+2:
>> torsdag 19 augusti 2021 kl. 14:19:46 UTC+2 skrev WM:
>>> zelos...@gmail.com schrieb am Donnerstag, 19. August 2021 um 07:00:48 UTC+2:
>>>>> The definable natural numbers have a discernible well-ordering
>>>> Whatever "discernible" means.
>>> It can be verified step by step.
>>>>
>>>> but N is well-ordered so again, N_def=N
>>>>
>>> No:
>>> (1) When indexing the rationals all unit intervals can be completed.
>>> (2) Not all unit intervals can be completed by one index because an index is issued to one rational of one interval only.
>>> (3) So there are at least two indices required to complete all intervals.
>>> (4) You cannot find out their order by their FISONs. If you could, you would complete aleph_0 rationals of one interval but you would need further indices. Contradiction. That is the true reason for dark numbers.
>
>> mathematics doesn't work in "steps" like a fucking program.
>
> The sequence of definable natural numbers works in steps. Every step can be analyzed.
>>
>> N is well-ordered and your N_def=N
>
> Do you understand and accept quantifier logic?
>
> ~∃n ∈ ℕ: P(n) <==> ∀n ∈ ℕ: ~P(n)
>
> Please apply this to
>
> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo.

why don't you apply it ? Do you not know how ? Or is ℕ_def a red herring ?

>
> Regards, WM
>

trolling since 2005

On 8/23/2021 7:02 AM, WM wrote:
> William schrieb am Montag, 23. August 2021 um 05:08:57 UTC+2:
>> On Sunday, August 22, 2021 at 6:02:18 PM UTC-4, WM wrote:
>>> William schrieb am Sonntag, 22. August 2021 um 21:04:03 UTC+2:
>>>> On Sunday, August 22, 2021 at 2:46:49 PM UTC-4, WM wrote:
>>>>>>> Here is the proof of incomplete
>>>>>> The term "complete" is nonsense. it implies that there are "incomplete" sets,
>>>>> No
>>>> Yes. All sets contain what they contains. A set can be a proper subset but it cannot be "incomplete". If x is a FISON then x is an element of set set of FISONS.
>>>> The set of FISONs is not "incomplete" because the set has no last element.
>>>>
>>> But a mapping can be incomplete.
>> Nope,
>
> It can be incomplete.

537 Incompetent Irrational Complement Cluster Ants
538 Incompletness theorem by completed Ants

>
>> a bijection is a set of ordered pairs.
>
> a complete set.

273 completed much earlier Ants
274 completed Ant infinity
275 Completed Infinity without any Potential Ants

913 Seriously Idiotic Ants
914 Set Ants
915 Set Ants that are Dalits
916 Set Bracket Ants
917 Set theorists Ants.
918 Set theorists are trying to cheat Ants
919 Set Ants that have been Declined
920 set of ants in linear order

>
>> There is no such thing as an "incomplete" set. The set of FISONs has no last element but there is no FISON missing from the set.
>
> There are not more FISONs than can be distinguished by FISONs, i.e., there are less than aleph_0 FISONs. Do you understand and accept quantifier logic?

58 Always Without Proof Ants
59 Always Wrong Ants
60 Alzheimer Ants

369 dishonest cheating Ants
370 distinguish between sense and nonsense Ants
371 Distinguishing Ants without Representation
372 Distracted Ants

446 FISON Ants that are definable therefore subject to Induction
447 FISONs captured Ants that are non-believers
448 FISONs Go Nowhere Ants

850 Quantifier Dyslexia Ants
851 Quantifier Shift Ants

>
> ~∃n ∈ ℕ: P(n) <==> ∀n ∈ ℕ: ~P(n)

11 |N with card aleph_0 Ants
12 ~Ex e IR: An e IN: 0<x<1/n Ants
13 $750 are enough ants

>
> Please apply this to
>
> ∀n ∈ ℕ: |ℕ \ {1, 2, 3, ..., n}| = ℵo.

674 Mückenhausen Ant virus
675 Mumbo Jumbo Ants with Dementia
676 My Way or the Highway Ants

>
> Regards, WM
>

652 McAffee's Ants that Fart worldwide
653 McDuck’s Ants that are QuACkErS
654 Mental Defect Ants

William schrieb am Montag, 23. August 2021 um 15:11:06 UTC+2:
> On Monday, August 23, 2021 at 8:02:37 AM UTC-4, WM wrote:
> > William schrieb am Montag, 23. August 2021 um 05:08:57 UTC+2:
> > > On Sunday, August 22, 2021 at 6:02:18 PM UTC-4, WM wrote:
> > > > William schrieb am Sonntag, 22. August 2021 um 21:04:03 UTC+2:
> > > > > On Sunday, August 22, 2021 at 2:46:49 PM UTC-4, WM wrote:
> > > > > > > > Here is the proof of incomplete
> > > > > > > The term "complete" is nonsense. it implies that there are "incomplete" sets,
> > > > > > No
> > > > > Yes. All sets contain what they contains. A set can be a proper subset but it cannot be "incomplete". If x is a FISON then x is an element of set set of FISONS.
> > > > > The set of FISONs is not "incomplete" because the set has no last element.
> > > > >
> > > > But a mapping can be incomplete.
> > > Nope,
> > It can be incomplete.
> > >a bijection is a set of ordered pairs.
> > a complete set.
> "complete" is a nonsense term. There is no such thing as an "incomplete" set and since a mapping is a set, there is no such thing as an incomplete mapping

Here is an incomplete mapping: In the interval (1000, 1001] the density of rationals is precisely the same as in the interval (0, 1]. Alas Cantor enumerates less than one promille. This is an incomplete mapping.

Regards, WM

zelos...@gmail.com wrote:

>> Regards, WM
> mathematics doesn't work in "steps" like a fucking program.
> N is well-ordered and your N_def=N

dementia hit, but he thinks he is still a doctor:

Biden calls on U.S. companies to mandate the COVID-19 vaccine for
employees

Greg Cunt schrieb am Montag, 23. August 2021 um 15:40:12 UTC+2:

> But a set is not a process you know.

The well-ordered set (1, 2, 3, ...) can be analyzed step by step. As long as definable elements are available. This is is a process.

> And talking about "potentially infinite" sets is just nonsense.
>
Here is an incomplete mapping, i.e., an incomplete set: In the interval (1000, 1001] the density of rationals is precisely the same as in the interval (0, 1]. Alas Cantor enumerates less than one promille. This is an incomplete mapping. A possible bijection does not change this result.

Regards, WM

zelos...@gmail.com wrote:

>> (4) You cannot find out their order by their FISONs.
>> (5) But you claim that only indices having FISONs are existing.
>> Regards, WM
> your "dark numbers" have no meaning, they do not exist cause you cannot
> meaningfully define them.

you too, why would you want trangeder to woman and become "vulnerable".
Just stay gay.

On Monday, 23 August 2021 at 15:36:31 UTC-3, WM wrote:
[...]
> Here is an incomplete mapping: In the interval (1000, 1001] the density of rationals is precisely the same as in the interval (0, 1]. Alas Cantor enumerates less than one promille. This is an incomplete mapping.

Oh, Boohoo. The only thing incomplete about it is your understanding (as usual). Actually, no, your understanding is not incomplete, it is nonexistent, more precisely. You don't even bother to describe a mapping; you just toss out a few words, hoping that, like spaghetti, they will stick to the wall, or to anything. You are a demented crackpot.

Cantor's mapping itself, taken as a mapping of (all of) IN onto the (entire) set of positive rationals is complete and exhausts *ALL* positive rationals. It might take some intervals a bit longer to get going, but the mapping never runs out of integers to map, nor of rationals to map to. Infinity is a lot bigger than you will ever understand!

After 16+ years one shouldn't expect more out of you than your usual bullshit, but it is still disheartening that you are allowed to teach at an institution of higher learning, even if it is just the Technical University of Augsburg. They are committing a fraud against their students, year after year.

On Monday, August 23, 2021 at 8:39:52 PM UTC+2, WM wrote:
> Greg Cunt schrieb am Montag, 23. August 2021 um 15:40:12 UTC+2:
> >
> > a set is not a process, you know.
> >
> The well-ordered set (1, 2, 3, ...)

(1, 2, 3, ...) is not a well ordered set, but a SEQUENCE.

> can be analyzed step by step. [...] This is a process.

Yes, you blithering idiot: THE stepwise ANALYSIS is a PROCESS. This does not mean that the set IN or the sequence (1, 2, 3, ...) or the well-orderd set (IN, <) is a PROCESS too.

Anyway,

> > talking about "potentially infinite" sets is just nonsense.

Now...

> less than one promille.

Are you sure? You really should't drink and write!

On Monday, August 23, 2021 at 8:39:52 PM UTC+2, WM wrote:
> Greg Cunt schrieb am Montag, 23. August 2021 um 15:40:12 UTC+2:
> >
> > a set is not a process, you know.
> >
> The well-ordered set (1, 2, 3, ...)

(1, 2, 3, ...) is not a well-ordered set, but a SEQUENCE.

> can be analyzed step by step. [...] This is a process.

Yes, you blithering idiot: THE stepwise ANALYSIS is a PROCESS. This does not mean that the set IN or the sequence (1, 2, 3, ...) or the well-orderd set (IN, <=) is a PROCESS too.

Anyway,

> > talking about "potentially infinite" sets is just nonsense.

Now...

> less than one promille.

Are you sure? You really shouldn't drink and write!

On 8/23/2021 1:39 PM, Muccio Grande wrote:
> zelos...@gmail.com wrote:
>
>>> Regards, WM
>> mathematics doesn't work in "steps" like a fucking program.
>> N is well-ordered and your N_def=N
>
> dementia hit, but he thinks he is still a doctor:
>
> Biden calls on U.S. companies to mandate the COVID-19 vaccine for
> employees
>

that is it! Biden needs to be N_def-ed, defined as a number, he is still in the dark!

Gus Gassmann schrieb am Montag, 23. August 2021 um 21:01:25 UTC+2:
> On Monday, 23 August 2021 at 15:36:31 UTC-3, WM wrote:
> [...]
> > Here is an incomplete mapping: In the interval (1000, 1001] the density of rationals is precisely the same as in the interval (0, 1]. Alas Cantor enumerates less than one promille. This is an incomplete mapping.
>
> Cantor's mapping itself, taken as a mapping of (all of) IN onto the (entire) set of positive rationals is complete and exhausts *ALL* positive rationals. It might take some intervals a bit longer to get going,

I have taken the limit. There we have the ratio less than 1/1000.

> but the mapping never runs out of integers to map, nor of rationals to map to.

In the limit the mapping is completed.

Regards, WM

Greg Cunt schrieb am Montag, 23. August 2021 um 21:06:14 UTC+2:
> On Monday, August 23, 2021 at 8:39:52 PM UTC+2, WM wrote:
> > Greg Cunt schrieb am Montag, 23. August 2021 um 15:40:12 UTC+2:
> > >
> > > a set is not a process, you know.
> > >
> > The well-ordered set (1, 2, 3, ...)
> (1, 2, 3, ...) is not a well-ordered set, but a SEQUENCE.

Everey sequence without repetitions is a well-ordered set.
>
> > can be analyzed step by step. [...] This is a process.
>
> THE stepwise ANALYSIS is a PROCESS. This does not mean that the set IN or the sequence (1, 2, 3, ...) or the well-orderd set (IN, <=) is a PROCESS too.

It means just this for a non-platonist. But the sequence cannot be analyzed stepby step. That is only possible for the non-dark part.

> > less than one promille.
>
> Are you sure?

Yes, it is an incomplete mapping: In the interval (1000, 1001] the density of rationals is precisely the same as in the interval (0, 1]. Alas Cantor enumerates less than one promille. This is an incomplete mapping.

Regards, WM

On Monday, 23 August 2021 at 16:21:45 UTC-3, WM wrote:
> Gus Gassmann schrieb am Montag, 23. August 2021 um 21:01:25 UTC+2:
> > On Monday, 23 August 2021 at 15:36:31 UTC-3, WM wrote:
> > [...]
> > > Here is an incomplete mapping: In the interval (1000, 1001] the density of rationals is precisely the same as in the interval (0, 1]. Alas Cantor enumerates less than one promille. This is an incomplete mapping.
> >
> > Cantor's mapping itself, taken as a mapping of (all of) IN onto the (entire) set of positive rationals is complete and exhausts *ALL* positive rationals. It might take some intervals a bit longer to get going,
> I have taken the limit. There we have the ratio less than 1/1000.

As usual, you took the *WRONG* fucking limit, you imbecile. You took the limit of the ratios of cardinalities, which tells you absolutely *NOTHING* about the problem that you are interested in, which is "Did the mapping cover all every positive rational number". Since it is easy to bound the position at which every positive rational n/m is covered (namely never more than 0..5*(n+m)^2), the answer to that question is an obvious "Yes". Nothing else matters. You have been barking up the wrong tree for the last 16 years, and you are getting less and less adept at it.

> > but the mapping never runs out of integers to map, nor of rationals to map to.
> In the limit the mapping is completed.

Good. So at last we agree on that. Just to be sure: the mapping is completed after covering every positive rational number, in particular aleph_0 of them in the interval (0,1], aleph_0 in the interval (1, 2], aleph_0 in the interval (1000, 1001], and so on.

On 8/23/2021 2:21 PM, WM wrote:
> Gus Gassmann schrieb am Montag, 23. August 2021 um 21:01:25 UTC+2:
>> On Monday, 23 August 2021 at 15:36:31 UTC-3, WM wrote:
>> [...]
>>> Here is an incomplete mapping: In the interval (1000, 1001] the density of rationals is precisely the same as in the interval (0, 1]. Alas Cantor enumerates less than one promille. This is an incomplete mapping.
>>
>> Cantor's mapping itself, taken as a mapping of (all of) IN onto the (entire) set of positive rationals is complete and exhausts *ALL* positive rationals. It might take some intervals a bit longer to get going,
>
> I have taken the limit. There we have the ratio less than 1/1000.

Wrong, and stupid, you are trolling using stupidity.

(0, 0.1] <=> (1000000000, 1000000001] is that now a ratio of 1/10000000000 ? Big Time Fail.

How many rationals are their in any size interval ?

Hint;

"Aleph-null bottles of beer on the wall, Aleph-null bottles of beer, Take one down, and pass it around, Aleph-null bottles of beer on the wall" (repeat).

>
>> but the mapping never runs out of integers to map, nor of rationals to map to.
>
> In the limit the mapping is completed.

what limit ? what mapping, what do you mean by completed ?

you are missing context, and you need to use math.

>
> Regards, WM
>

you do not understand infinity, troll.

Gus Gassmann wrote:
> On Monday, 23 August 2021 at 15:36:31 UTC-3, WM wrote:
> [...]
>> Here is an incomplete mapping: In the interval (1000, 1001] the density of rationals is precisely the same as in the interval (0, 1]. Alas Cantor enumerates less than one promille. This is an incomplete mapping.
>
> Oh, Boohoo. The only thing incomplete about it is your understanding (as usual). Actually, no, your understanding is not incomplete, it is nonexistent, more precisely. You don't even bother to describe a mapping; you just toss out a few words, hoping that, like spaghetti, they will stick to the wall, or to anything. You are a demented crackpot.
>
> Cantor's mapping itself, taken as a mapping of (all of) IN onto the (entire) set of positive rationals is complete and exhausts *ALL* positive rationals. It might take some intervals a bit longer to get going, but the mapping never runs out of integers to map, nor of rationals to map to. Infinity is a lot bigger than you will ever understand!
>
> After 16+ years one shouldn't expect more out of you than your usual bullshit, but it is still disheartening that you are allowed to teach at an institution of higher learning, even if it is just the Technical University of Augsburg. They are committing a fraud against their students, year after year.

They are. Definitely. But they evade on the ground of "academic freedom"
you know?

On Tuesday, August 24, 2021 at 1:21:21 AM UTC+2, Python wrote:
> Gus Gassmann wrote:
> >
> > [...] it is still disheartening that you are allowed to teach at an institution of higher learning,
> > even if it is just the Technical University of Augsburg. They are committing a fraud against
> > their students, year after year.
> >
> They are. Definitely. But they evade on the ground of "academic freedom"
> you know?

Yes. But I'm wondering, would they allow someome to teach that there was no WWII? - "Acdemic Freedom"? Or that the moon consist of green chese? So why do the accept this clown to teach that, say, ZFC is "inconsistent", etc. I'm mean that is clearly counterfactual - in no way "open to debate".

Same with claims that "set theory" is nonsensical, of no value in math, etc, etc. Clearly nonsenical and counterfactual claims. Actually, lies. Sure, there are alternative approaches, still "classical mathematics" is based on set theory.

"Classical mathematics refers generally to the mainstream approach to mathematics, which is based on classical logic and ZFC set theory."

Source: https://en.wikipedia.org/wiki/Classical_mathematics

måndag 23 augusti 2021 kl. 14:04:53 UTC+2 skrev WM:
> zelos...@gmail.com schrieb am Montag, 23. August 2021 um 06:51:10 UTC+2:
> > torsdag 19 augusti 2021 kl. 14:19:46 UTC+2 skrev WM:
> > > zelos...@gmail.com schrieb am Donnerstag, 19. August 2021 um 07:00:48 UTC+2:
> > > > >The definable natural numbers have a discernible well-ordering
> > > > Whatever "discernible" means.
> > > It can be verified step by step.
> > > >
> > > > but N is well-ordered so again, N_def=N
> > > >
> > > No:
> > > (1) When indexing the rationals all unit intervals can be completed.
> > > (2) Not all unit intervals can be completed by one index because an index is issued to one rational of one interval only.
> > > (3) So there are at least two indices required to complete all intervals.
> > > (4) You cannot find out their order by their FISONs. If you could, you would complete aleph_0 rationals of one interval but you would need further indices. Contradiction. That is the true reason for dark numbers.
> > mathematics doesn't work in "steps" like a fucking program.
> The sequence of definable natural numbers works in steps. Every step can be analyzed.
> >
> > N is well-ordered and your N_def=N
> Do you understand and accept quantifier logic?
>
> ~∃n ∈ ℕ: P(n) <==> ∀n ∈ ℕ: ~P(n)
>
> Please apply this to
> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo.
>
> Regards, WM

>The sequence of definable natural numbers works in steps. Every step can be analyzed.

THere are no steps in it in the way you want there to be.

>There are not more FISONs than can be distinguished by FISONs, i.e., there are less than aleph_0 FISONs.

False, given FISONs are in bijection with N, that means there is aleph_0 FISONs

On 8/23/2021 8:08 AM, WM wrote:
> Jim Burns schrieb
> am Montag, 23. August 2021 um 06:55:33 UTC+2:
>> On 8/22/2021 6:00 PM, WM wrote:
>>> Jim Burns schrieb
>>> am Sonntag, 22. August 2021 um 21:45:36 UTC+2:

>>>> *No rational completes any interval*
>>>
>>> Then there is no bijection and no cardinality.
>>> For cardinality completion is required.
>>>
>>> There was no objection to a 'potential infinity' in the
>>> form of an unending process, but an 'actual infinity' in
>>> the form of a completed infinite set was harder to accept."
>>> [H.B. Enderton: "Elements of set theory",
>>> Academic Press, New York (1977) p. 14f]
>>
>> "p. 14f" would be the forward. Is that right?
>
> Presently I have no acces to this book.

A preview on Google Books shows the quote starting at the end
of page 14. It's in "Historical Notes", the last section of
Chapter 1, "Introduction".

<Enderton, 14-15>

Cantor's work was well received by some of the prominent
mathematicians of the day, such as Richard Dedekind.
But his willingness to regard infinite sets as objects
to be treated in much the same way as finite sets was
attacked bitterly by others, particularly Kronecker.
There was no objection to a "potential infinity" in the
form of an unending process, but an "actual infinity" in
the form of a completed infinite set was harder to
accept.

</Enderton>

Executive summary of "Historical Notes", Ch. 1:

The concept of a set has been used since ancient times.

Georg Cantor was led to the study of set theory by way
of the study of Fourier series. In 1871 he realized that
certain operations on sets of real numbers could be
iterated more than a finite number of times. In 1879
and subsequent years, he published a series of papers
setting forth the general concepts of abstract sets
and "transfinite numbers".

About the turn of the century, attempts were made to
present the principles of set theory as being
self-evident truths of deductive thought. But the
simplicity and directness of Russell's paradox seemed
to utterly destroy the attempt to base mathematics
on that sort of set theory.

End summary.

----
I was interested in where Enderton _began_ his explanation
of set theory, so I read his Chapter 1 in Google Books.

This is something I've been picking at for a while now:
when is it (pedagogically, as material builds on earlier
material) that posters (such as yourself, WM) with such
divergent views first begin to diverge? This has given
me a greater interest in beginnings that I had ages ago
when I first met these concepts in the classroom.

Enderton's first section is "Baby Set Theory", wherein
he quickly goes over the concepts "t is a member of A"
and "t is not a member of A". In my opinion, he should
have started a little bit earlier, with the concepts
"t" and "A", that is, with _variables_

With variables, we can partially describe _one of_ the
steps in an unending process, making claims which are
true of that step _no matter which_ step is referred
to. No steps are exempt from that description. But
no step is last; that's part of its description.

I hear about bitter attacks from Kronecker, and I
suspect that this is the sort of thing he attacked.
We, finite beings, claiming to describe _all_ of an
unending process.

And we do claim that. It doesn't have to take infinitely
long to say something true about infinitely-many.