"Finally, as Gödel observes, his argument is restricted to countable vocabularies; Henkin proves the results for uncountable languages. [...] Henkin (Corollary 2) uses the uncountable vocabulary to deduce the full force of the Löwenheim-Skolem-Tarski theorem: a consistent first order theory has models in every infinite cardinality. [...] Henkin already points out that his proof (unlike Gödel's) generalizes easily to uncountable vocabularies. [...] McKinsey (also noting the uncountable application) and Heyting give straightforward accounts in Mathematical Reviews of the result of Henkin's papers on first order and theory of types respectively with no comments on the significance of the result. Still more striking, Ackermann's review of Henkin's proof gives a routine summary of the new argument and concludes with 'The reviewer can not follow the author when he speaks of an extension to an uncountable set of relation symbols, since such a system of notations can not exist'." [John Baldwin: "The explanatory power of a new proof: Henkin's completeness proof" (25 Feb 2017)]

Ackermann's clear and correct position seems "striking". Logic has really become a perverted subject – useful and interesting like chess competitions of mentally disabled persons!

Regards, WM

Well such vocabulary is easy.
Assume for every real number r e R a constant c_r.

WM schrieb am Samstag, 18. Dezember 2021 um 12:48:30 UTC+1:
> "Finally, as Gödel observes, his argument is restricted to countable vocabularies; Henkin proves the results for uncountable languages. [...] Henkin (Corollary 2) uses the uncountable vocabulary to deduce the full force of the Löwenheim-Skolem-Tarski theorem: a consistent first order theory has models in every infinite cardinality. [...] Henkin already points out that his proof (unlike Gödel's) generalizes easily to uncountable vocabularies. [...] McKinsey (also noting the uncountable application) and Heyting give straightforward accounts in Mathematical Reviews of the result of Henkin's papers on first order and theory of types respectively with no comments on the significance of the result. Still more striking, Ackermann's review of Henkin's proof gives a routine summary of the new argument and concludes with 'The reviewer can not follow the author when he speaks of an extension to an uncountable set of relation symbols, since such a system of notations can not exist'." [John Baldwin: "The explanatory power of a new proof: Henkin's completeness proof" (25 Feb 2017)]
>
> Ackermann's clear and correct position seems "striking". Logic has really become a perverted subject – useful and interesting like chess competitions of mentally disabled persons!
>
> Regards, WM

name dropping for mentally disabled persons!

On 12/18/2021 5:48 AM,

WM wrote:

> "Finally, as

Gödel observes, his argument is restricted to countable vocabularies;

Henkin proves the results for uncountable languages. [...]

Henkin (Corollary 2) uses the uncountable vocabulary to deduce the full force of the

Löwenheim-Skolem-Tarski theorem: a consistent first order theory has models in every infinite cardinality. [...]

Henkin already points out that his proof (unlike

Gödel's) generalizes easily to uncountable vocabularies. [...]

McKinsey (also noting the uncountable application) and

Heyting give straightforward accounts in Mathematical Reviews of the result of

Henkin's papers on first order and theory of types respectively with no comments on the significance of the result. Still more striking,

Ackermann's review of

Henkin's proof gives a routine summary of the new argument and concludes with '

The reviewer can not follow

the author when he speaks of an extension to an uncountable set of relation symbols, since such a system of notations can not exist'." [

John Baldwin: "The explanatory power of a new proof:

Henkin's completeness proof" (25 Feb 2017)]
>
>

Ackermann's clear and correct position seems "striking". Logic has really become a perverted subject – useful and interesting like chess competitions of
mentally disabled persons!
>
> Regards, WM
>

Mostowski Collapse schrieb am Samstag, 18. Dezember 2021 um 14:09:48 UTC+1:
> Well such vocabulary is easy.
> Assume for every real number r e R a constant c_r.

Then one key of the uncountable keyboard is pi. What is the key next to the key of pi?

Regards, WM

On Tuesday, December 21, 2021 at 1:12:33 PM UTC+1, WM wrote:
> Mostowski Collapse schrieb am Samstag, 18. Dezember 2021 um 14:09:48 UTC+1:
> > Well such vocabulary is easy.
> > Assume for every real number r e R a constant c_r.
> Then one key of the uncountable keyboard is pi. What is the key next to the key of pi?
>
> Regards, WM

What's the real number next to pi, nincompoop?
What do keyboards have to do with it?
Did Plato have a keyboard?
I claim no knowledge nor understanding of these issues, but I do understand that your
critique is nonsense, as usual.

On 12/21/2021 7:52 AM, JVR wrote:
> On Tuesday, December 21, 2021 at 1:12:33 PM UTC+1, WM wrote:
>> Mostowski Collapse schrieb am Samstag, 18. Dezember 2021 um 14:09:48 UTC+1:
>>> Well such vocabulary is easy.
>>> Assume for every real number r e R a constant c_r.
>> Then one key of the uncountable keyboard is pi. What is the key next to the key of pi?
>>
>> Regards, WM
>
> What's the real number next to pi, nincompoop?
> What do keyboards have to do with it?
> Did Plato have a keyboard?
>
> I claim no knowledge nor understanding of these issues, but I do understand that your
> critique is nonsense, as usual.

Keyboard Ants
key "NextTo" Ants

key next to key Ants of the Eternally Confused

On Tuesday, December 21, 2021 at 4:12:33 AM UTC-8, WM wrote:
> Mostowski Collapse schrieb am Samstag, 18. Dezember 2021 um 14:09:48 UTC+1:
> > Well such vocabulary is easy.
> > Assume for every real number r e R a constant c_r.
> Then one key of the uncountable keyboard is pi. What is the key next to the key of pi?

What is a key between 's' and 'd' ?

JVR schrieb am Dienstag, 21. Dezember 2021 um 14:52:33 UTC+1:
> On Tuesday, December 21, 2021 at 1:12:33 PM UTC+1, WM wrote:
> > Mostowski Collapse schrieb am Samstag, 18. Dezember 2021 um 14:09:48 UTC+1:
> > > Well such vocabulary is easy.
> > > Assume for every real number r e R a constant c_r.
> > Then one key of the uncountable keyboard is pi. What is the key next to the key of pi?
> >
> What's the real number next to pi,

Note that uncountably many letters must be appear in an uncountable alphabet. Using only the next available real numbers will never yield more than countable many.

> What do keyboards have to do with it?

Try to figure it out for yourself.

> Did Plato have a keyboard?

How many letters did he use?
>
> I claim no knowledge nor understanding of these issues,

Trying to imitate Socrates?

> but I do understand that your
> critique is nonsense, as usual.

Quite wrong, as usual. Those who claim to use uncountable alphabets should be imprisoned in lunatics asylums. So much stupidity is forbidden among humans.

Regards, WM

After serious thinking WM wrote :
> Mostowski Collapse schrieb am Samstag, 18. Dezember 2021 um 14:09:48 UTC+1:
>> Well such vocabulary is easy.
>> Assume for every real number r e R a constant c_r.
>
> Then one key of the uncountable keyboard is pi. What is the key next to the
> key of pi?

Lime.

On Tuesday, December 21, 2021 at 4:59:12 PM UTC+1, WM wrote:
> JVR schrieb am Dienstag, 21. Dezember 2021 um 14:52:33 UTC+1:
> > On Tuesday, December 21, 2021 at 1:12:33 PM UTC+1, WM wrote:
> > > Mostowski Collapse schrieb am Samstag, 18. Dezember 2021 um 14:09:48 UTC+1:
> > > > Well such vocabulary is easy.
> > > > Assume for every real number r e R a constant c_r.
> > > Then one key of the uncountable keyboard is pi. What is the key next to the key of pi?
> > >
> > What's the real number next to pi,
> Note that uncountably many letters must be appear in an uncountable alphabet. Using only the next available real numbers will never yield more than countable many.
> > What do keyboards have to do with it?
> Try to figure it out for yourself.
> > Did Plato have a keyboard?
> How many letters did he use?
> >
> > I claim no knowledge nor understanding of these issues,
> Trying to imitate Socrates?
> > but I do understand that your
> > critique is nonsense, as usual.
> Quite wrong, as usual. Those who claim to use uncountable alphabets should be imprisoned in lunatics asylums. So much stupidity is forbidden among humans.
>
> Regards, WM

Very likely literally nobody "claims to use uncountable alphabets". Certainly you haven't found anybody.
On the other hand, every mathematician explicitly or implicitly uses uncountable sets, including uncountable
index sets, which in your infantile imagination might amount to using "uncountable alphabets". Without
continuity there is no Euclidean geometry, no function sin(x) etc, etc.

Also lautet ein Beschluß,
Daß Mückemaus was lernen muß. -
Nicht allein das Abc
Bringt den Menschen in die Höh';
Nicht allein in Schreiben, Lesen
Übt sich ein vernünftig Wesen;
Nicht allein in Rechnungssachen
Soll der Mensch sich Mühe machen,
Sondern auch der Weisheit Lehren
Muß man mit Vergnügen hören. -

JVR schrieb am Dienstag, 21. Dezember 2021 um 17:23:17 UTC+1:
> On Tuesday, December 21, 2021 at 4:59:12 PM UTC+1, WM wrote:
> > Those who claim to use uncountable alphabets should be imprisoned in lunatics asylums. So much stupidity is forbidden among humans.

> Very likely literally nobody "claims to use uncountable alphabets".

You are misinformed as usual. You should read https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf to fill your gaps.

> On the other hand, every mathematician explicitly or implicitly uses uncountable sets, including uncountable

Threre are no uncountable sets! But that nonsense is not as severe as uncountable alphabets.

> Without continuity there is no Euclidean geometry, no function sin(x) etc, etc.

Continuity proves dark numbers.
>
> Also lautet ein Beschluß,

***** On uncountable alphabets *****

"By an alphabet A we mean a nonempty set of symbols." [H.-D. Ebbinghaus, J. Flum, W. Thomas: "Mathematical logic", 2nd ed., Springer, New York (1994) p. 11]

That is true but insufficient to describe an alphabet that can be applied to form words and expressions. An alphabet is an alphabetically or lexically ordered string or list of symbols, where "lexically ordered" means that an algorithm exists to find a desired word within finite time.

"The set {c_r | r ∈ ℝ }, which contains a symbol c_r for every real number r, is an example for an uncountable alphabet." [loc cit p. 13]

That is impossible because the set of symbols is countable – at least if transmitted through the internet, printed, or carved of matter within the universe.

"Ref. kann dem Verf. aber nicht folgen, wenn er von der Möglichkeit einer mehr als abzählbaren Menge von primitiven Symbolen spricht, da es ein derartiges Bezeichnungssystem doch nicht geben kann." [W. Ackerman: Review of "L. Henkin: 'The completeness of the first-order functional calculus'", Journal of Symbolic Logic 15 (1950) p. 68]

Ackermann's clear and correct position seems "striking". Logic has really become a perverted subject – useful and interesting like chess competitions of mentally disabled persons!

"Roughly, a formal language could be completely mastered by a suitable machine, without any understanding. (This needs qualification where the formal language has an uncountable alphabet: In such case it is not clear that the formal language could be completely mastered by anything.)" [Geoffrey Hunter: "Metalogic", 6th ed., University of California Press (1996) p. 4f]

"this, of course, allows formulas which are not 'writable' in the sense of being finite strings of symbols. Also, in the book, no notions of recursiveness, admissibility, or even definability are applied to the formulas or sets of formulas considered." [W.P. Hanf: "Review of M.A. Dickmann: 'Large infinitary languages, model theory'", Bull. Am. Math. Soc. 83,2 (1977) p. 184]

"A well-known theorem of Mills asserts that there is a model of Peano Arithmetic M in an uncountable language such that M has no elementary end extension (e.e.e.). I ask whether every complete extension of PA in an uncountable language can have models of every cardinality with arbitrary large e.e.e.'s." [shahram: "Elementary end extensions of models of Peano arithmetic in uncountable languages", MathOverflow (11 Sep 2013)]

"You simply name all the submanifolds of your manifold and work with an uncountable language." [user 1686 in "Algebraic description of compact smooth manifolds?", MathOverflow (19 Nov 2009)]

"For uncountable languages, the logic is not complete with respect to the standard algebra [0, 1]G, and completeness fails already for propositional logic." [Emil Jeřábek in "Compactness and completeness in Gödel logic", MathOverflow (12 Mar 2014)]

"Extending Shelah's main gap to non first-order or even to first-order theories in an uncountable language is a major hard open problem." [Rami Grossberg in "Main gap phenomenon", MathOverflow (19 Mar 2015)]

"Anyway, it's impossible to countably axiomatize a (consistent) theory in an uncountable language, unless all but countably many of the symbols are left completely arbitrary by the axioms." [Emil Jeřábek in "Is the following theory countably axiomatizable?", MathOverflow (13 Dec 2016)]

"TΩ is not countably axiomatizable for trivial reasons: it has an uncountable language, and says non-trivial things about each symbol in that language." [Noah Schweber, loc cit]

"You may also find it helpful to think of a computer whose keyboard has, not the usual 101 keys, but infinitely many keys; [...] In this book we'll study the 'sentences' that can be typed on such a keyboard [...] Also for simplicity of notation, we have chosen an alphabet that is only countably infinite. That alphabet is adequate for most applications of logic, but some logicians prefer to allow uncountable alphabets as well. (Imagine an even larger infinite computer keyboard, with real numbers written on the key caps!)" [Eric Schechter: "Classical and nonclassical logics", Princeton Univ. Press (2005) p. 207f]

"How could the manufacturer write a real number except the few which have their own names like 2 or √3 or 1/4 or π? [...] the real numbers written on the key caps have to be individuals, i.e., it is not sufficient to distinguish each one from some 'given' real numbers but each one must differ from all other real numbers. How can that be possible by finite strings of symbols on the key caps? I assume consent that infinite strings of symbols don't carry any information that could be with sufficient completeness conveyed to the typist." [Wilhelm: "How to write an individual real number?", MathEducators.StackExchange (27 Nov 2017)]

"It seems like it needs comment that this is only a hypothetical/theoretical thought experiment. No such keyboard is actually constructable. Not even the simpler countably infinite one." [Daniel R. Collins, loc cit]

"This is not a hypothetical question since a rather big group of logicians claims that uncountable alphabets can be used in formal languages. If so, we should know a means how to use them." [Wilhelm, loc cit]

"You want to use a finite/countable alphabet of symbols to construct the language of real numbers. That is simply not possible, so why do you keep on trying. You need an uncountably big alphabet to have any chance at all, and then you could simply make ℝ itself your alphabet." [Dirk Liebhold, loc cit]

"The point you're missing is that, if you take the real numbers to be your alphabet, they are the very symbols you use to write with. [...] You know what symbol corresponds to what number because the symbol is the number." [Hurkyl, loc cit]

"A symbol is not a number. An alphabet is an alphabetically or lexically ordered set, namely a list or sequence of symbols that serve to form words while almost all real numbers cannot be defined let alone be chosen. The real numbers are definitely not an alphabet." [Wilhelm, loc cit]

"Zero probability does not mean impossibility. The magnitude of the real number defines it and this will have position." [Jeffery Thompson, loc cit]

"But it is hard to touch a selected number if the probability is zero. Further it is impossible to write a word containing two neighbouring numbers because neighbours are not defined on the real line. Nevertheless there are no gaps either." [Wilhelm, loc cit]

It is really deplorable how the deepest level of intellect is undercut by these poor newbies.

Regards, WM

On 12/21/2021 10:14 AM, FromTheRafters wrote:
> After serious thinking WM wrote :
>> Mostowski Collapse schrieb am Samstag, 18. Dezember 2021 um 14:09:48 UTC+1:
>>> Well such vocabulary is easy. Assume for every real number r e R a constant c_r.
>>
>> Then one key of the uncountable keyboard is pi. What is the key next to the key of pi?
>
> Lime.

yes, I thought it was West, but no, it is Lime for sure

On Tuesday, December 21, 2021 at 7:00:40 PM UTC+1, WM wrote:
> JVR schrieb am Dienstag, 21. Dezember 2021 um 17:23:17 UTC+1:
> > On Tuesday, December 21, 2021 at 4:59:12 PM UTC+1, WM wrote:
> > > Those who claim to use uncountable alphabets should be imprisoned in lunatics asylums. So much stupidity is forbidden among humans.
> > Very likely literally nobody "claims to use uncountable alphabets".
> You are misinformed as usual. You should read https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf to fill your gaps.
> > On the other hand, every mathematician explicitly or implicitly uses uncountable sets, including uncountable
> Threre are no uncountable sets! But that nonsense is not as severe as uncountable alphabets.
> > Without continuity there is no Euclidean geometry, no function sin(x) etc, etc.
> Continuity proves dark numbers.
> >
> > Also lautet ein Beschluß,
>
> ***** On uncountable alphabets *****
>
> "By an alphabet A we mean a nonempty set of symbols." [H.-D. Ebbinghaus, J. Flum, W. Thomas: "Mathematical logic", 2nd ed., Springer, New York (1994) p. 11]
>
> That is true but insufficient to describe an alphabet that can be applied to form words and expressions. An alphabet is an alphabetically or lexically ordered string or list of symbols, where "lexically ordered" means that an algorithm exists to find a desired word within finite time.
>
> "The set {c_r | r ∈ ℝ }, which contains a symbol c_r for every real number r, is an example for an uncountable alphabet." [loc cit p.. 13]
>
> That is impossible because the set of symbols is countable – at least if transmitted through the internet, printed, or carved of matter within the universe.
>
> "Ref. kann dem Verf. aber nicht folgen, wenn er von der Möglichkeit einer mehr als abzählbaren Menge von primitiven Symbolen spricht, da es ein derartiges Bezeichnungssystem doch nicht geben kann." [W. Ackerman: Review of "L. Henkin: 'The completeness of the first-order functional calculus'", Journal of Symbolic Logic 15 (1950) p. 68]
> Ackermann's clear and correct position seems "striking". Logic has really become a perverted subject – useful and interesting like chess competitions of mentally disabled persons!
> "Roughly, a formal language could be completely mastered by a suitable machine, without any understanding. (This needs qualification where the formal language has an uncountable alphabet: In such case it is not clear that the formal language could be completely mastered by anything.)" [Geoffrey Hunter: "Metalogic", 6th ed., University of California Press (1996) p. 4f]
>
> "this, of course, allows formulas which are not 'writable' in the sense of being finite strings of symbols. Also, in the book, no notions of recursiveness, admissibility, or even definability are applied to the formulas or sets of formulas considered." [W.P. Hanf: "Review of M.A. Dickmann: 'Large infinitary languages, model theory'", Bull. Am. Math. Soc. 83,2 (1977) p. 184]
>
> "A well-known theorem of Mills asserts that there is a model of Peano Arithmetic M in an uncountable language such that M has no elementary end extension (e.e.e.). I ask whether every complete extension of PA in an uncountable language can have models of every cardinality with arbitrary large e.e.e.'s." [shahram: "Elementary end extensions of models of Peano arithmetic in uncountable languages", MathOverflow (11 Sep 2013)]
>
> "You simply name all the submanifolds of your manifold and work with an uncountable language." [user 1686 in "Algebraic description of compact smooth manifolds?", MathOverflow (19 Nov 2009)]
>
> "For uncountable languages, the logic is not complete with respect to the standard algebra [0, 1]G, and completeness fails already for propositional logic." [Emil Jeřábek in "Compactness and completeness in Gödel logic", MathOverflow (12 Mar 2014)]
>
> "Extending Shelah's main gap to non first-order or even to first-order theories in an uncountable language is a major hard open problem." [Rami Grossberg in "Main gap phenomenon", MathOverflow (19 Mar 2015)]
>
> "Anyway, it's impossible to countably axiomatize a (consistent) theory in an uncountable language, unless all but countably many of the symbols are left completely arbitrary by the axioms." [Emil Jeřábek in "Is the following theory countably axiomatizable?", MathOverflow (13 Dec 2016)]
>
> "TΩ is not countably axiomatizable for trivial reasons: it has an uncountable language, and says non-trivial things about each symbol in that language." [Noah Schweber, loc cit]
>
> "You may also find it helpful to think of a computer whose keyboard has, not the usual 101 keys, but infinitely many keys; [...] In this book we'll study the 'sentences' that can be typed on such a keyboard [...] Also for simplicity of notation, we have chosen an alphabet that is only countably infinite. That alphabet is adequate for most applications of logic, but some logicians prefer to allow uncountable alphabets as well. (Imagine an even larger infinite computer keyboard, with real numbers written on the key caps!)" [Eric Schechter: "Classical and nonclassical logics", Princeton Univ. Press (2005) p. 207f]
>
> "How could the manufacturer write a real number except the few which have their own names like 2 or √3 or 1/4 or π? [...] the real numbers written on the key caps have to be individuals, i.e., it is not sufficient to distinguish each one from some 'given' real numbers but each one must differ from all other real numbers. How can that be possible by finite strings of symbols on the key caps? I assume consent that infinite strings of symbols don't carry any information that could be with sufficient completeness conveyed to the typist." [Wilhelm: "How to write an individual real number?", MathEducators.StackExchange (27 Nov 2017)]
>
> "It seems like it needs comment that this is only a hypothetical/theoretical thought experiment. No such keyboard is actually constructable. Not even the simpler countably infinite one." [Daniel R. Collins, loc cit]
>
> "This is not a hypothetical question since a rather big group of logicians claims that uncountable alphabets can be used in formal languages. If so, we should know a means how to use them." [Wilhelm, loc cit]
>
> "You want to use a finite/countable alphabet of symbols to construct the language of real numbers. That is simply not possible, so why do you keep on trying. You need an uncountably big alphabet to have any chance at all, and then you could simply make ℝ itself your alphabet." [Dirk Liebhold, loc cit]
>
> "The point you're missing is that, if you take the real numbers to be your alphabet, they are the very symbols you use to write with. [...] You know what symbol corresponds to what number because the symbol is the number." [Hurkyl, loc cit]
>
> "A symbol is not a number. An alphabet is an alphabetically or lexically ordered set, namely a list or sequence of symbols that serve to form words while almost all real numbers cannot be defined let alone be chosen. The real numbers are definitely not an alphabet." [Wilhelm, loc cit]
>
> "Zero probability does not mean impossibility. The magnitude of the real number defines it and this will have position." [Jeffery Thompson, loc cit]
>
> "But it is hard to touch a selected number if the probability is zero. Further it is impossible to write a word containing two neighbouring numbers because neighbours are not defined on the real line. Nevertheless there are no gaps either." [Wilhelm, loc cit]
>
> It is really deplorable how the deepest level of intellect is undercut by these poor newbies.
>
> Regards, WM

The time has come,' the Walrus said,
To talk of many things:
Of shoes — and ships — and sealing-wax —
Of cabbages — and kings —
And why the sea is boiling hot —
And whether pigs have wings.'

On 12/21/2021 12:00 PM, WM wrote:
> JVR schrieb am Dienstag, 21. Dezember 2021 um 17:23:17 UTC+1:
>> On Tuesday, December 21, 2021 at 4:59:12 PM UTC+1, WM wrote:
>>> Those who claim to use uncountable alphabets should be imprisoned in lunatics asylums. So much stupidity is forbidden among humans.
>
>> Very likely literally nobody "claims to use uncountable alphabets".
>
> You are misinformed as usual. You should read https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf to fill your gaps.
>
>> On the other hand, every mathematician explicitly or implicitly uses uncountable sets, including uncountable
>
> Threre are no uncountable sets! But that nonsense is not as severe as uncountable alphabets.
>
>> Without continuity there is no Euclidean geometry, no function sin(x) etc, etc.
>
> Continuity proves dark numbers.
>>
>> Also lautet ein Beschluß,
>
> ***** On uncountable alphabets *****
>
> "By an alphabet A we mean a nonempty set of symbols." [H.-D. Ebbinghaus, J. Flum, W. Thomas: "Mathematical logic", 2nd ed., Springer, New York (1994) p. 11]
>
> That is true but insufficient to describe an alphabet that can be applied to form words and expressions. An alphabet is an alphabetically or lexically ordered string or list of symbols, where "lexically ordered" means that an algorithm exists to find a desired word within finite time.
>
> "The set {c_r | r ∈ ℝ }, which contains a symbol c_r for every real number r, is an example for an uncountable alphabet." [loc cit p. 13]
>
> That is impossible because the set of symbols is countable – at least if transmitted through the internet, printed, or carved of matter within the universe.
>
> "Ref. kann dem Verf. aber nicht folgen, wenn er von der Möglichkeit einer mehr als abzählbaren Menge von primitiven Symbolen spricht, da es ein derartiges Bezeichnungssystem doch nicht geben kann." [W. Ackerman: Review of "L. Henkin: 'The completeness of the first-order functional calculus'", Journal of Symbolic Logic 15 (1950) p. 68]
>
> Ackermann's clear and correct position seems "striking". Logic has really become a perverted subject – useful and interesting like chess competitions of mentally disabled persons!
>
> "Roughly, a formal language could be completely mastered by a suitable machine, without any understanding. (This needs qualification where the formal language has an uncountable alphabet: In such case it is not clear that the formal language could be completely mastered by anything.)" [Geoffrey Hunter: "Metalogic", 6th ed., University of California Press (1996) p. 4f]
>
> "this, of course, allows formulas which are not 'writable' in the sense of being finite strings of symbols. Also, in the book, no notions of recursiveness, admissibility, or even definability are applied to the formulas or sets of formulas considered." [W.P. Hanf: "Review of M.A. Dickmann: 'Large infinitary languages, model theory'", Bull. Am. Math. Soc. 83,2 (1977) p. 184]
>
> "A well-known theorem of Mills asserts that there is a model of Peano Arithmetic M in an uncountable language such that M has no elementary end extension (e.e.e.). I ask whether every complete extension of PA in an uncountable language can have models of every cardinality with arbitrary large e.e.e.'s." [shahram: "Elementary end extensions of models of Peano arithmetic in uncountable languages", MathOverflow (11 Sep 2013)]
>
> "You simply name all the submanifolds of your manifold and work with an uncountable language." [user 1686 in "Algebraic description of compact smooth manifolds?", MathOverflow (19 Nov 2009)]
>
> "For uncountable languages, the logic is not complete with respect to the standard algebra [0, 1]G, and completeness fails already for propositional logic." [Emil Jeřábek in "Compactness and completeness in Gödel logic", MathOverflow (12 Mar 2014)]
>
> "Extending Shelah's main gap to non first-order or even to first-order theories in an uncountable language is a major hard open problem." [Rami Grossberg in "Main gap phenomenon", MathOverflow (19 Mar 2015)]
>
> "Anyway, it's impossible to countably axiomatize a (consistent) theory in an uncountable language, unless all but countably many of the symbols are left completely arbitrary by the axioms." [Emil Jeřábek in "Is the following theory countably axiomatizable?", MathOverflow (13 Dec 2016)]
>
> "TΩ is not countably axiomatizable for trivial reasons: it has an uncountable language, and says non-trivial things about each symbol in that language." [Noah Schweber, loc cit]
>
> "You may also find it helpful to think of a computer whose keyboard has, not the usual 101 keys, but infinitely many keys; [...] In this book we'll study the 'sentences' that can be typed on such a keyboard [...] Also for simplicity of notation, we have chosen an alphabet that is only countably infinite. That alphabet is adequate for most applications of logic, but some logicians prefer to allow uncountable alphabets as well. (Imagine an even larger infinite computer keyboard, with real numbers written on the key caps!)" [Eric Schechter: "Classical and nonclassical logics", Princeton Univ. Press (2005) p. 207f]
>
> "How could the manufacturer write a real number except the few which have their own names like 2 or √3 or 1/4 or π? [...] the real numbers written on the key caps have to be individuals, i.e., it is not sufficient to distinguish each one from some 'given' real numbers but each one must differ from all other real numbers. How can that be possible by finite strings of symbols on the key caps? I assume consent that infinite strings of symbols don't carry any information that could be with sufficient completeness conveyed to the typist." [Wilhelm: "How to write an individual real number?", MathEducators.StackExchange (27 Nov 2017)]
>
> "It seems like it needs comment that this is only a hypothetical/theoretical thought experiment. No such keyboard is actually constructable. Not even the simpler countably infinite one." [Daniel R. Collins, loc cit]
>
> "This is not a hypothetical question since a rather big group of logicians claims that uncountable alphabets can be used in formal languages. If so, we should know a means how to use them." [Wilhelm, loc cit]
>
> "You want to use a finite/countable alphabet of symbols to construct the language of real numbers. That is simply not possible, so why do you keep on trying. You need an uncountably big alphabet to have any chance at all, and then you could simply make ℝ itself your alphabet." [Dirk Liebhold, loc cit]
>
> "The point you're missing is that, if you take the real numbers to be your alphabet, they are the very symbols you use to write with. [...] You know what symbol corresponds to what number because the symbol is the number." [Hurkyl, loc cit]
>
> "A symbol is not a number. An alphabet is an alphabetically or lexically ordered set, namely a list or sequence of symbols that serve to form words while almost all real numbers cannot be defined let alone be chosen. The real numbers are definitely not an alphabet." [Wilhelm, loc cit]
>
> "Zero probability does not mean impossibility. The magnitude of the real number defines it and this will have position." [Jeffery Thompson, loc cit]
>
> "But it is hard to touch a selected number if the probability is zero. Further it is impossible to write a word containing two neighbouring numbers because neighbours are not defined on the real line. Nevertheless there are no gaps either." [Wilhelm, loc cit]
>
> It is really deplorable how the deepest level of intellect is undercut by these poor newbies.
>
> Regards, WM
>

no wonder you remain so confused, believing vague and misleading stuff posted in mathoverflow.

If you want to Learn Math, stick to the Professionally reviewed Journals, or standard Math Books. many of the self published books are vague and
misleading also.

tisdag 21 december 2021 kl. 16:59:12 UTC+1 skrev WM:
> JVR schrieb am Dienstag, 21. Dezember 2021 um 14:52:33 UTC+1:
> > On Tuesday, December 21, 2021 at 1:12:33 PM UTC+1, WM wrote:
> > > Mostowski Collapse schrieb am Samstag, 18. Dezember 2021 um 14:09:48 UTC+1:
> > > > Well such vocabulary is easy.
> > > > Assume for every real number r e R a constant c_r.
> > > Then one key of the uncountable keyboard is pi. What is the key next to the key of pi?
> > >
> > What's the real number next to pi,
> Note that uncountably many letters must be appear in an uncountable alphabet. Using only the next available real numbers will never yield more than countable many.
> > What do keyboards have to do with it?
> Try to figure it out for yourself.
> > Did Plato have a keyboard?
> How many letters did he use?
> >
> > I claim no knowledge nor understanding of these issues,
> Trying to imitate Socrates?
> > but I do understand that your
> > critique is nonsense, as usual.
> Quite wrong, as usual. Those who claim to use uncountable alphabets should be imprisoned in lunatics asylums. So much stupidity is forbidden among humans.
>
> Regards, WM
Nope, the one that is insane here is you mate