Discussion:
Is Gödel's 1931 incompleteness paper actually an invalid reasoning, since there should have been a formalized definition for the informally perceived but crucially needed unary "magnitude(n)" function - in his encoding scheme?
(too old to reply)
Khong Dong
2020-03-28 18:12:30 UTC
Permalink
The title is also that of the Quora thread:

https://qr.ae/pNvAdF

which should be a *summary* of why Gödel's 1931 incompleteness paper is
an invalid argument (1st Incompleteness statement anyway).

There are more reasons than those detailed in the summary, but if you'd like to correctly, technically counter argue the summary as it is, you could of course
make yourself be heard.
Peter Percival
2020-03-28 18:38:44 UTC
Permalink
Post by Khong Dong
https://qr.ae/pNvAdF
which should be a *summary* of why Gödel's 1931 incompleteness paper is
an invalid argument (1st Incompleteness statement anyway).
There are more reasons than those detailed in the summary, but if you'd like to correctly, technically counter argue the summary as it is, you could of course
make yourself be heard.
Comprehensive reply from Alex Eustis, Ph.D. Mathematics, University of
California, San Diego (2013)

No, that's not possible. Even without fully understanding the substance
of your complaint, I can surmise that you're wrong, and with as much
certainty as it is ever possible to have about anything.

If all of Gödel's results happen to be complete bunk because he wrote <
instead of >

or something like that, then there is absolutely no chance that nearly a
century's worth of scrutiny by the whole worldwide Mathematical
community somehow missed this — and yet you, some random dude with a
B.A. in Mathematics and an inflated ego, are the first person to notice
this error. I mean, how arrogant is that?

This paper was a breakthrough — it more or less kicked off the entire
study of Incompleteness, Independence, provability logic, and so forth.
But just as we can do Galois Theory without having to refer back to
Galois’ original notebooks that he frantically scribbled the night
before his fatal duel, we can today do Incompleteness Theory and prove
the independence of various statements from certain axiomatic systems
without directly citing Gödel's 1931 paper.

For instance, according to Wikipedia,

Stephen Cole Kleene (1943) presented a proof of Gödel's
incompleteness theorem using basic results of computability theory.

Franzén (2005, p. 73) explains how Matiyasevich's solution to
Hilbert's 10th problem can be used to obtain a proof to Gödel's first
incompleteness theorem.

Smorynski (1977, p. 842) shows how the existence of inseparable
sets can be used to prove the first incompleteness theorem.

Chaitin's incompleteness theorem gives a different method of
producing independent sentences, based on Kolmogorov complexity.

So even if you happen to be right (which you aren't), and there is some
fundamental, uncorrectable error with Gödel numbering (which there
isn't) — can you find an ingenius rebuttal for all these proofs, and
more? Can you single-handedly overturn a century of Mathematics?
Khong Dong
2020-03-28 18:50:44 UTC
Permalink
Post by Peter Percival
Post by Khong Dong
https://qr.ae/pNvAdF
which should be a *summary* of why Gödel's 1931 incompleteness paper is
an invalid argument (1st Incompleteness statement anyway).
There are more reasons than those detailed in the summary, but if you'd like to correctly, technically counter argue the summary as it is, you could of course
make yourself be heard.
Comprehensive reply from Alex Eustis, Ph.D. Mathematics, University of
California, San Diego (2013)
No, that's not possible. Even without fully understanding the substance
of your complaint, I can surmise that you're wrong, and with as much
certainty as it is ever possible to have about anything.
If all of Gödel's results happen to be complete bunk because he wrote <
instead of >
or something like that, then there is absolutely no chance that nearly a
century's worth of scrutiny by the whole worldwide Mathematical
community somehow missed this — and yet you, some random dude with a
B.A. in Mathematics and an inflated ego, are the first person to notice
this error. I mean, how arrogant is that?
This paper was a breakthrough — it more or less kicked off the entire
study of Incompleteness, Independence, provability logic, and so forth.
But just as we can do Galois Theory without having to refer back to
Galois’ original notebooks that he frantically scribbled the night
before his fatal duel, we can today do Incompleteness Theory and prove
the independence of various statements from certain axiomatic systems
without directly citing Gödel's 1931 paper.
For instance, according to Wikipedia,
Stephen Cole Kleene (1943) presented a proof of Gödel's
incompleteness theorem using basic results of computability theory.
Franzén (2005, p. 73) explains how Matiyasevich's solution to
Hilbert's 10th problem can be used to obtain a proof to Gödel's first
incompleteness theorem.
Smorynski (1977, p. 842) shows how the existence of inseparable
sets can be used to prove the first incompleteness theorem.
Chaitin's incompleteness theorem gives a different method of
producing independent sentences, based on Kolmogorov complexity.
So even if you happen to be right (which you aren't), and there is some
fundamental, uncorrectable error with Gödel numbering (which there
isn't) — can you find an ingenius rebuttal for all these proofs, and
more? Can you single-handedly overturn a century of Mathematics?
As George Greene would say: _that_ is NOT an argument (or counter argument)
at all - to say the least. Let me quote my response to his inquisition innuendo
below:

<quote>

Your "Even without fully understanding the substance of your complaint, I can surmise that you're wrong, and with as much certainty as it is ever possible to have about anything" is nothing but innuendo and is of character-assassination nature and has zero technical content (i.e. not a counter argument at all).

On your "and yet you, some random dude with a B.A. in Mathematics and an inflated ego, are the first person to notice this error. I mean, how arrogant is that?" is pure Inquisition in term of knowledge exchange.

On your "if you keep asking questions like these you'll start to look like the mathematical equivalent of a Flat-Earther", suppose you had a time machine, went back to 1878 (a year before Einstein was born) and explained to hardcore Newtonian PhD's, Professors, that if one is on a moving train one would see the length of the train-station be contracted, you would be treated not just as Flat-Earther: you might be sent to a mental institution!

Would you be able to *technically* respond better than what you’ve done?

</quote>
Khong Dong
2020-03-28 19:40:42 UTC
Permalink
Post by Khong Dong
https://qr.ae/pNvAdF
which should be a *summary* of why Gödel's 1931 incompleteness paper is
an invalid argument (1st Incompleteness statement anyway).
There are more reasons than those detailed in the summary, but if you'd like to correctly, technically counter argue the summary as it is, you could of course
make yourself be heard.
Btw, Me, Rupert, Goerge Greene, et al., in R. B. BRAITHWAITE's introduction to
a translated version of Goedel's 1931 paper one would see (reformatted slightly
for ease of reading):

<quote>

For the part of his argument which establishes the 'unprovability' of
Neg (v Gen r) requires at one point considering a statement about all numbers,
whether or not they are Gödel numbers; and this statement cannot be construed
without change as a statement about all strings, since a number which is not
a Gödel number does not correspond to any string.

But it is easy to close the gap in the recasting by considering the numbers
which are Gödel numbers as arranged in a sequence of increasing magnitude,
and then using, instead of a Gödel number itself, the number which gives the
place of this Gödel number in the sequence. To be precise, if n is the
(m + 1)-th Gödel number in increasing order, call m the G-number of the string
of which n is the Gödel number, and use the G-number m wherever Gödel in his
argument uses the Gödel number n. Then every natural number 0, 1, 2, etc. will
be the G-number of some string; and there will be a recursive one-to-one
correspondence between natural numbers and strings.

</quote>

Would you be able to see from the above passage why Goedel's encoding is
invalid as I've pointed out in one way or another (about Goedel having not
formalized the unary "magnitude" function)?
Khong Dong
2020-03-28 19:44:36 UTC
Permalink
Post by Khong Dong
Post by Khong Dong
https://qr.ae/pNvAdF
which should be a *summary* of why Gödel's 1931 incompleteness paper is
an invalid argument (1st Incompleteness statement anyway).
There are more reasons than those detailed in the summary, but if you'd like to correctly, technically counter argue the summary as it is, you could of course
make yourself be heard.
Btw, Me, Rupert, Goerge Greene, et al., in R. B. BRAITHWAITE's introduction to
a translated version of Goedel's 1931 paper one would see (reformatted slightly
"George Greene" I meant. (My apology).
Post by Khong Dong
<quote>
For the part of his argument which establishes the 'unprovability' of
Neg (v Gen r) requires at one point considering a statement about all numbers,
whether or not they are Gödel numbers; and this statement cannot be construed
without change as a statement about all strings, since a number which is not
a Gödel number does not correspond to any string.
But it is easy to close the gap in the recasting by considering the numbers
which are Gödel numbers as arranged in a sequence of increasing magnitude,
and then using, instead of a Gödel number itself, the number which gives the
place of this Gödel number in the sequence. To be precise, if n is the
(m + 1)-th Gödel number in increasing order, call m the G-number of the string
of which n is the Gödel number, and use the G-number m wherever Gödel in his
argument uses the Gödel number n. Then every natural number 0, 1, 2, etc. will
be the G-number of some string; and there will be a recursive one-to-one
correspondence between natural numbers and strings.
</quote>
Would you be able to see from the above passage why Goedel's encoding is
invalid as I've pointed out in one way or another (about Goedel having not
formalized the unary "magnitude" function)?
Rupert
2020-03-28 21:14:53 UTC
Permalink
Post by Khong Dong
Post by Khong Dong
https://qr.ae/pNvAdF
which should be a *summary* of why Gödel's 1931 incompleteness paper is
an invalid argument (1st Incompleteness statement anyway).
There are more reasons than those detailed in the summary, but if you'd like to correctly, technically counter argue the summary as it is, you could of course
make yourself be heard.
Btw, Me, Rupert, Goerge Greene, et al., in R. B. BRAITHWAITE's introduction to
a translated version of Goedel's 1931 paper one would see (reformatted slightly
<quote>
For the part of his argument which establishes the 'unprovability' of
Neg (v Gen r) requires at one point considering a statement about all numbers,
whether or not they are Gödel numbers; and this statement cannot be construed
without change as a statement about all strings, since a number which is not
a Gödel number does not correspond to any string.
But it is easy to close the gap in the recasting by considering the numbers
which are Gödel numbers as arranged in a sequence of increasing magnitude,
and then using, instead of a Gödel number itself, the number which gives the
place of this Gödel number in the sequence. To be precise, if n is the
(m + 1)-th Gödel number in increasing order, call m the G-number of the string
of which n is the Gödel number, and use the G-number m wherever Gödel in his
argument uses the Gödel number n. Then every natural number 0, 1, 2, etc. will
be the G-number of some string; and there will be a recursive one-to-one
correspondence between natural numbers and strings.
</quote>
Would you be able to see from the above passage why Goedel's encoding is
invalid as I've pointed out in one way or another (about Goedel having not
formalized the unary "magnitude" function)?
Goedel does not at any stage invoke a unary function called "magnitude". He does make use of the phrase "in order of magnitude", at least in the English translation. The meaning of this is crystal clear. This quotation from Braithwaite doesn't help you any.
Khong Dong
2020-03-28 23:57:50 UTC
Permalink
Post by Rupert
Post by Khong Dong
Post by Khong Dong
https://qr.ae/pNvAdF
which should be a *summary* of why Gödel's 1931 incompleteness paper is
an invalid argument (1st Incompleteness statement anyway).
There are more reasons than those detailed in the summary, but if you'd like to correctly, technically counter argue the summary as it is, you could of course
make yourself be heard.
Btw, Me, Rupert, Goerge Greene, et al., in R. B. BRAITHWAITE's introduction to
a translated version of Goedel's 1931 paper one would see (reformatted slightly
<quote>
For the part of his argument which establishes the 'unprovability' of
Neg (v Gen r) requires at one point considering a statement about all numbers,
whether or not they are Gödel numbers; and this statement cannot be construed
without change as a statement about all strings, since a number which is not
a Gödel number does not correspond to any string.
But it is easy to close the gap in the recasting by considering the numbers
which are Gödel numbers as arranged in a sequence of increasing magnitude,
and then using, instead of a Gödel number itself, the number which gives the
place of this Gödel number in the sequence. To be precise, if n is the
(m + 1)-th Gödel number in increasing order, call m the G-number of the string
of which n is the Gödel number, and use the G-number m wherever Gödel in his
argument uses the Gödel number n. Then every natural number 0, 1, 2, etc. will
be the G-number of some string; and there will be a recursive one-to-one
correspondence between natural numbers and strings.
</quote>
Would you be able to see from the above passage why Goedel's encoding is
invalid as I've pointed out in one way or another (about Goedel having not
formalized the unary "magnitude" function)?
Goedel does not at any stage invoke a unary function called "magnitude".
Like I said in the Quora post:

<quote>

It's submitted here that the above phrase would be completely meaningless, nonsensical unless all the following are true:

"prime number" connotes the existence of a unary predicate denoted as "prime" which is a definable symbol of the language of arithmetic L(0,S,+,*,<).
"order" connotes the existence of a binary predicate denoted as "<" which is a formal symbol of the language of arithmetic L(0,S,+,*,<).
"magnitude" connotes the existence of a unary function denoted here as "magnitude" which is supposed to be a definable symbol - but yet hasn't been defined - of the language of arithmetic L(0,S,+,*,<).

</quote>
Post by Rupert
He does make use of the phrase "in order of magnitude", at least in the
English translation. The meaning of this is crystal clear.
Again, (for the nth time ?), it's trivial "in order of magnitude" of a _number_
doesn't connote a the existence of unary function then the word "mathematics"
is quite meaningless.

"Magnitude" of a complex number z = (a + bi) connotes the formalized definition:

magnitude(z) = sqrt(a^2 + b^2).

Are you implying you forget such a trivial definition of Complex number?
Post by Rupert
This quotation from Braithwaite doesn't help you any.
Let's have an agreement on understanding the necessity of the "magnitude"
function in GIT1 first. (To be frank, your kept saying "this is crystal clear"
isn't an argument, and doesn't sound like it's clear to you what "this is crystal clear" phrase is really about).
Khong Dong
2020-03-29 19:14:36 UTC
Permalink
Post by Khong Dong
Post by Rupert
Post by Khong Dong
Post by Khong Dong
https://qr.ae/pNvAdF
which should be a *summary* of why Gödel's 1931 incompleteness paper is
an invalid argument (1st Incompleteness statement anyway).
There are more reasons than those detailed in the summary, but if you'd like to correctly, technically counter argue the summary as it is, you could of course
make yourself be heard.
Btw, Me, Rupert, Goerge Greene, et al., in R. B. BRAITHWAITE's introduction to
a translated version of Goedel's 1931 paper one would see (reformatted slightly
<quote>
For the part of his argument which establishes the 'unprovability' of
Neg (v Gen r) requires at one point considering a statement about all numbers,
whether or not they are Gödel numbers; and this statement cannot be construed
without change as a statement about all strings, since a number which is not
a Gödel number does not correspond to any string.
But it is easy to close the gap in the recasting by considering the numbers
which are Gödel numbers as arranged in a sequence of increasing magnitude,
and then using, instead of a Gödel number itself, the number which gives the
place of this Gödel number in the sequence. To be precise, if n is the
(m + 1)-th Gödel number in increasing order, call m the G-number of the string
of which n is the Gödel number, and use the G-number m wherever Gödel in his
argument uses the Gödel number n. Then every natural number 0, 1, 2, etc. will
be the G-number of some string; and there will be a recursive one-to-one
correspondence between natural numbers and strings.
</quote>
Would you be able to see from the above passage why Goedel's encoding is
invalid as I've pointed out in one way or another (about Goedel having not
formalized the unary "magnitude" function)?
Goedel does not at any stage invoke a unary function called "magnitude".
<quote>
"prime number" connotes the existence of a unary predicate denoted as "prime" which is a definable symbol of the language of arithmetic L(0,S,+,*,<).
"order" connotes the existence of a binary predicate denoted as "<" which is a formal symbol of the language of arithmetic L(0,S,+,*,<).
"magnitude" connotes the existence of a unary function denoted here as "magnitude" which is supposed to be a definable symbol - but yet hasn't been defined - of the language of arithmetic L(0,S,+,*,<).
</quote>
Post by Rupert
He does make use of the phrase "in order of magnitude", at least in the
English translation. The meaning of this is crystal clear.
Again, (for the nth time ?), it's trivial "in order of magnitude" of a _number_
doesn't connote a the existence of unary function then the word "mathematics"
is quite meaningless.
magnitude(z) = sqrt(a^2 + b^2).
Are you implying you forget such a trivial definition of Complex number?
Post by Rupert
This quotation from Braithwaite doesn't help you any.
Let's have an agreement on understanding the necessity of the "magnitude"
function in GIT1 first. (To be frank, your kept saying "this is crystal clear"
isn't an argument, and doesn't sound like it's clear to you what "this is crystal clear" phrase is really about).
In fact, Rupert, the invalidity of Gödel's encoding is even more basic
(more "raw") than you and many mathematicians might be to willing to admit.

Few pages down from where he wrote "wo pk die k-te Primzhal (Der Größe nach) bedeutet", you'd see his definition of prime numbers - "Prim(x)" - being a
function of the familiar complete order binary relation denoted by '<'!
_That_ in itself is an invalid definition: since natural prime numbers should
be _defined_ in the context of a _multiplicative monoid_ that has only _one_
_NON_ logical symbol denoting the multiplication operation itself.

Even if the Fundamental Theorem of Arithmetic would allude to something like
"(by [order of] magnitude)", that still doesn't logically entail that the definition (of primes) - itself - be a function of the familiar symbols '<',
'S' at all. The correct definition of primes should look something like the
below - free of occurrences of '<' and 'S':

prime(p) <-> ((∃x∀y[x*y=y]) /\ (~p=0) /\ ~∀z[p*z=z] /\ ((p=x*y) -> (∀z[x*z=z] \/ ∀z[y*z=z]))).

Certainly all these aren't the only reasons why Gödel's Incompleteness (1)
is invalid (there are also problems about HP as a law of thought, object vs
semantic quantification, if nothing else). But I can confidently say: I rest
my case.
Ben Bacarisse
2020-03-29 20:12:01 UTC
Permalink
Khong Dong <***@gmail.com> writes:
<cut>
... the invalidity of Gödel's encoding is even more basic
(more "raw") than you and many mathematicians might be to willing to admit.
Few pages down from where he wrote "wo pk die k-te Primzhal (Der Größe
nach) bedeutet",
It's very musch a detail, but where possible you should cut and paste
when copying from a language you don't know because Gödel did not write
that.
you'd see his definition of prime numbers - "Prim(x)"
- being a function of the familiar complete order binary relation
denoted by '<'!
Your surprise is revealing.
The correct definition of primes should look something like the below
prime(p) <-> ((∃x∀y[x*y=y]) /\ (~p=0) /\ ~∀z[p*z=z] /\ ((p=x*y) ->
(∀z[x*z=z] \/ ∀z[y*z=z]))).
For students who are serious about learning this theorem your mistake
here is interesting. Here's a question for any students here. Why must
Gödel /not/ use this definition? Why is his the right one and this one
wrong?

(I doubt any serious students are reading this, but it struck me as an
interesting study question.)
--
Ben.
Khong Dong
2020-03-29 20:42:08 UTC
Permalink
Post by Ben Bacarisse
<cut>
... the invalidity of Gödel's encoding is even more basic
(more "raw") than you and many mathematicians might be to willing to admit.
Few pages down from where he wrote "wo pk die k-te Primzhal (Der Größe
nach) bedeutet",
It's very musch a detail, but where possible you should cut and paste
when copying from a language you don't know because Gödel did not write
that.
Perhaps you ask Rupert if I had misquoted _that_ , or you've had absolutely
no clue as to what Gödel wrote - in all these posts (and related) you've
frantically tried to "technically" (counter) argued ...
Post by Ben Bacarisse
you'd see his definition of prime numbers - "Prim(x)"
- being a function of the familiar complete order binary relation
denoted by '<'!
Your surprise is revealing.
Sounds very much an Inquisition dishonesty. I'm not too shocked.

Can you present to sci.logic (which might include respected Professors who
might have had read this conversations) a _proof_ that I had "surprise" on
this at all, Ben Bacarisse?
Khong Dong
2020-03-30 03:22:29 UTC
Permalink
Post by Ben Bacarisse
<cut>
... the invalidity of Gödel's encoding is even more basic
(more "raw") than you and many mathematicians might be to willing to admit.
Few pages down from where he wrote "wo pk die k-te Primzhal (Der Größe
nach) bedeutet",
It's very musch a detail, but where possible you should cut and paste
when copying from a language you don't know because Gödel did not write
that.
you'd see his definition of prime numbers - "Prim(x)"
- being a function of the familiar complete order binary relation
denoted by '<'!
The correct definition of primes should look something like the below
prime(p) <-> ((∃x∀y[x*y=y]) /\ (~p=0) /\ ~∀z[p*z=z] /\ ((p=x*y) ->
(∀z[x*z=z] \/ ∀z[y*z=z]))).
For students who are serious about learning this theorem your mistake
here is interesting.
That, coming from you as an educator who supervised, as well "supervised", PhD
students but who failed to notice the short but a key phrase of Gödel's
encoding scheme ("wo pk die k-te Primzhal (Der Größe nach) bedeutet"), is so
tragically comical!

I've never believed the entire post Gödel education system is bankrupted, but
the part you're representing as an educator is definitely so. I feel so sad
for your students having been brainwashed by you in much a worse way than the
way you've alluded/accused WM done to his students.
Post by Ben Bacarisse
Here's a question for any students here. Why must
Gödel /not/ use this definition? Why is his the right one and this one
wrong?
(I doubt any serious students are reading this, but it struck me as an
interesting study question.)
Let me correct your method of "educating" your PhD students: Have them ask
experts in mathematics - who you're _NOT_ one - to explain what the below quote means:

"one must look at whole numbers in a different light — leaving addition
aside and seeing the multiplication structure as something malleable and
deformable".

I'm certain that you have zero clue what "seeing the multiplication structure
as something malleable and deformable" would connote, because you've wrongly
insisted the multiplicative monoid structure where primes would come from must
necessarily conform to Gödel's ill-fated (invalid) definition of primes.

In fact, as a mathematics educator, you've failed to understand the basic
mathematical concepts of prime numbers, multiplicative monoid, ...

Note: Just in case you wonder who said what is quoted above, the below source
seems to allude that that's what Fesenko might have said about the natural
numbers. In ZERO way can Gödel's invalid definition of primes would
entail that!

https://www.scientificamerican.com/article/math-mystery-shinichi-mochizuki-and-the-impenetrable-proof/
Peter Percival
2020-03-30 06:32:23 UTC
Permalink
Post by Khong Dong
Post by Ben Bacarisse
<cut>
... the invalidity of Gödel's encoding is even more basic
(more "raw") than you and many mathematicians might be to willing to admit.
Few pages down from where he wrote "wo pk die k-te Primzhal (Der Größe
nach) bedeutet",
It's very musch a detail, but where possible you should cut and paste
when copying from a language you don't know because Gödel did not write
that.
you'd see his definition of prime numbers - "Prim(x)"
- being a function of the familiar complete order binary relation
denoted by '<'!
The correct definition of primes should look something like the below
prime(p) <-> ((∃x∀y[x*y=y]) /\ (~p=0) /\ ~∀z[p*z=z] /\ ((p=x*y) ->
(∀z[x*z=z] \/ ∀z[y*z=z]))).
For students who are serious about learning this theorem your mistake
here is interesting.
That, coming from you as an educator who supervised, as well "supervised", PhD
students but who failed to notice the short but a key phrase of Gödel's
encoding scheme ("wo pk die k-te Primzhal (Der Größe nach) bedeutet"), is so
tragically comical!
I've never believed the entire post Gödel education system is bankrupted, but
the part you're representing as an educator is definitely so. I feel so sad
for your students having been brainwashed by you in much a worse way than the
way you've alluded/accused WM done to his students.
Post by Ben Bacarisse
Here's a question for any students here. Why must
Gödel /not/ use this definition? Why is his the right one and this one
wrong?
(I doubt any serious students are reading this, but it struck me as an
interesting study question.)
Let me correct your method of "educating" your PhD students: Have them ask
"one must look at whole numbers in a different light — leaving addition
aside and seeing the multiplication structure as something malleable and
deformable".
I'm certain that you have zero clue what "seeing the multiplication structure
as something malleable and deformable" would connote, because you've wrongly
insisted the multiplicative monoid structure where primes would come from must
necessarily conform to Gödel's ill-fated (invalid) definition of primes.
In fact, as a mathematics educator, you've failed to understand the basic
mathematical concepts of prime numbers, multiplicative monoid, ...
Note: Just in case you wonder who said what is quoted above, the below source
seems to allude that that's what Fesenko might have said about the natural
numbers. In ZERO way can Gödel's invalid definition of primes would
entail that!
https://www.scientificamerican.com/article/math-mystery-shinichi-mochizuki-and-the-impenetrable-proof/
Don't change the subject! You claimed

"The correct definition of primes should look something like the
below - free of occurrences of '<' and 'S':

prime(p) <-> ((∃x∀y[x*y=y]) /\ (~p=0) /\ ~∀z[p*z=z] /\ ((p=x*y) ->
(∀z[x*z=z] \/ ∀z[y*z=z])))."

That may generally be ok as a definition of prime, but it won't do for
Gödel. Why not? If you don't know, why not have the courage to admit it?

Here's a big hint. The answer may be found on page 602 and page 603 of
van Heijenoort. I bet that even with that big helping hand, you don't
know the answer.

Reference's to the /Collected papers/ supplied if requested.
Khong Dong
2020-03-30 17:17:41 UTC
Permalink
Post by Peter Percival
Post by Khong Dong
Post by Ben Bacarisse
<cut>
... the invalidity of Gödel's encoding is even more basic
(more "raw") than you and many mathematicians might be to willing to admit.
Few pages down from where he wrote "wo pk die k-te Primzhal (Der Größe
nach) bedeutet",
It's very musch a detail, but where possible you should cut and paste
when copying from a language you don't know because Gödel did not write
that.
you'd see his definition of prime numbers - "Prim(x)"
- being a function of the familiar complete order binary relation
denoted by '<'!
The correct definition of primes should look something like the below
prime(p) <-> ((∃x∀y[x*y=y]) /\ (~p=0) /\ ~∀z[p*z=z] /\ ((p=x*y) ->
(∀z[x*z=z] \/ ∀z[y*z=z]))).
For students who are serious about learning this theorem your mistake
here is interesting.
That, coming from you as an educator who supervised, as well "supervised", PhD
students but who failed to notice the short but a key phrase of Gödel's
encoding scheme ("wo pk die k-te Primzhal (Der Größe nach) bedeutet"), is so
tragically comical!
I've never believed the entire post Gödel education system is bankrupted, but
the part you're representing as an educator is definitely so. I feel so sad
for your students having been brainwashed by you in much a worse way than the
way you've alluded/accused WM done to his students.
Post by Ben Bacarisse
Here's a question for any students here. Why must
Gödel /not/ use this definition? Why is his the right one and this one
wrong?
(I doubt any serious students are reading this, but it struck me as an
interesting study question.)
Let me correct your method of "educating" your PhD students: Have them ask
"one must look at whole numbers in a different light — leaving addition
aside and seeing the multiplication structure as something malleable and
deformable".
I'm certain that you have zero clue what "seeing the multiplication structure
as something malleable and deformable" would connote, because you've wrongly
insisted the multiplicative monoid structure where primes would come from must
necessarily conform to Gödel's ill-fated (invalid) definition of primes.
In fact, as a mathematics educator, you've failed to understand the basic
mathematical concepts of prime numbers, multiplicative monoid, ...
Note: Just in case you wonder who said what is quoted above, the below source
seems to allude that that's what Fesenko might have said about the natural
numbers. In ZERO way can Gödel's invalid definition of primes would
entail that!
https://www.scientificamerican.com/article/math-mystery-shinichi-mochizuki-and-the-impenetrable-proof/
Don't change the subject! You claimed
"The correct definition of primes should look something like the
prime(p) <-> ((∃x∀y[x*y=y]) /\ (~p=0) /\ ~∀z[p*z=z] /\ ((p=x*y) ->
(∀z[x*z=z] \/ ∀z[y*z=z])))."
That may generally be ok as a definition of prime, but it won't do for
Gödel. Why not?
Yeah. The currently ok definition of Power set won't do for some cranks
who fail to regocnize the validity of Cantor theorem, but so what?

Suppose Goldbach sunddenly appeared among us the currently
ok definition of prime which excludes 1 wouldn't do for him, but so what?

Crank.
Post by Peter Percival
If you don't know, why not have the courage to admit it?
Usual idiotic barking from the crank PP.
Peter Percival
2020-03-30 18:47:30 UTC
Permalink
Post by Khong Dong
Post by Peter Percival
Post by Khong Dong
Post by Ben Bacarisse
<cut>
... the invalidity of Gödel's encoding is even more basic
(more "raw") than you and many mathematicians might be to willing to admit.
Few pages down from where he wrote "wo pk die k-te Primzhal (Der Größe
nach) bedeutet",
It's very musch a detail, but where possible you should cut and paste
when copying from a language you don't know because Gödel did not write
that.
you'd see his definition of prime numbers - "Prim(x)"
- being a function of the familiar complete order binary relation
denoted by '<'!
The correct definition of primes should look something like the below
prime(p) <-> ((∃x∀y[x*y=y]) /\ (~p=0) /\ ~∀z[p*z=z] /\ ((p=x*y) ->
(∀z[x*z=z] \/ ∀z[y*z=z]))).
For students who are serious about learning this theorem your mistake
here is interesting.
That, coming from you as an educator who supervised, as well "supervised", PhD
students but who failed to notice the short but a key phrase of Gödel's
encoding scheme ("wo pk die k-te Primzhal (Der Größe nach) bedeutet"), is so
tragically comical!
I've never believed the entire post Gödel education system is bankrupted, but
the part you're representing as an educator is definitely so. I feel so sad
for your students having been brainwashed by you in much a worse way than the
way you've alluded/accused WM done to his students.
Post by Ben Bacarisse
Here's a question for any students here. Why must
Gödel /not/ use this definition? Why is his the right one and this one
wrong?
(I doubt any serious students are reading this, but it struck me as an
interesting study question.)
Let me correct your method of "educating" your PhD students: Have them ask
"one must look at whole numbers in a different light — leaving addition
aside and seeing the multiplication structure as something malleable and
deformable".
I'm certain that you have zero clue what "seeing the multiplication structure
as something malleable and deformable" would connote, because you've wrongly
insisted the multiplicative monoid structure where primes would come from must
necessarily conform to Gödel's ill-fated (invalid) definition of primes.
In fact, as a mathematics educator, you've failed to understand the basic
mathematical concepts of prime numbers, multiplicative monoid, ...
Note: Just in case you wonder who said what is quoted above, the below source
seems to allude that that's what Fesenko might have said about the natural
numbers. In ZERO way can Gödel's invalid definition of primes would
entail that!
https://www.scientificamerican.com/article/math-mystery-shinichi-mochizuki-and-the-impenetrable-proof/
Don't change the subject! You claimed
"The correct definition of primes should look something like the
prime(p) <-> ((∃x∀y[x*y=y]) /\ (~p=0) /\ ~∀z[p*z=z] /\ ((p=x*y) ->
(∀z[x*z=z] \/ ∀z[y*z=z])))."
That may generally be ok as a definition of prime, but it won't do for
Gödel. Why not?
Yeah. The currently ok definition of Power set won't do for some cranks
who fail to regocnize the validity of Cantor theorem, but so what?
Suppose Goldbach sunddenly appeared among us the currently
ok definition of prime which excludes 1 wouldn't do for him, but so what?
Crank.
Post by Peter Percival
If you don't know, why not have the courage to admit it?
Usual idiotic barking from the crank PP.
What is wrong with your definition of prime(p)? You don't know, do you?
Peter Percival
2020-03-29 20:38:19 UTC
Permalink
Post by Khong Dong
Post by Khong Dong
Post by Rupert
Post by Khong Dong
Post by Khong Dong
https://qr.ae/pNvAdF
which should be a *summary* of why Gödel's 1931 incompleteness paper is
an invalid argument (1st Incompleteness statement anyway).
There are more reasons than those detailed in the summary, but if you'd like to correctly, technically counter argue the summary as it is, you could of course
make yourself be heard.
Btw, Me, Rupert, Goerge Greene, et al., in R. B. BRAITHWAITE's introduction to
a translated version of Goedel's 1931 paper one would see (reformatted slightly
<quote>
For the part of his argument which establishes the 'unprovability' of
Neg (v Gen r) requires at one point considering a statement about all numbers,
whether or not they are Gödel numbers; and this statement cannot be construed
without change as a statement about all strings, since a number which is not
a Gödel number does not correspond to any string.
But it is easy to close the gap in the recasting by considering the numbers
which are Gödel numbers as arranged in a sequence of increasing magnitude,
and then using, instead of a Gödel number itself, the number which gives the
place of this Gödel number in the sequence. To be precise, if n is the
(m + 1)-th Gödel number in increasing order, call m the G-number of the string
of which n is the Gödel number, and use the G-number m wherever Gödel in his
argument uses the Gödel number n. Then every natural number 0, 1, 2, etc. will
be the G-number of some string; and there will be a recursive one-to-one
correspondence between natural numbers and strings.
</quote>
Would you be able to see from the above passage why Goedel's encoding is
invalid as I've pointed out in one way or another (about Goedel having not
formalized the unary "magnitude" function)?
Goedel does not at any stage invoke a unary function called "magnitude".
<quote>
"prime number" connotes the existence of a unary predicate denoted as "prime" which is a definable symbol of the language of arithmetic L(0,S,+,*,<).
"order" connotes the existence of a binary predicate denoted as "<" which is a formal symbol of the language of arithmetic L(0,S,+,*,<).
"magnitude" connotes the existence of a unary function denoted here as "magnitude" which is supposed to be a definable symbol - but yet hasn't been defined - of the language of arithmetic L(0,S,+,*,<).
</quote>
Post by Rupert
He does make use of the phrase "in order of magnitude", at least in the
English translation. The meaning of this is crystal clear.
Again, (for the nth time ?), it's trivial "in order of magnitude" of a _number_
doesn't connote a the existence of unary function then the word "mathematics"
is quite meaningless.
magnitude(z) = sqrt(a^2 + b^2).
Are you implying you forget such a trivial definition of Complex number?
Post by Rupert
This quotation from Braithwaite doesn't help you any.
Let's have an agreement on understanding the necessity of the "magnitude"
function in GIT1 first. (To be frank, your kept saying "this is crystal clear"
isn't an argument, and doesn't sound like it's clear to you what "this is crystal clear" phrase is really about).
In fact, Rupert, the invalidity of Gödel's encoding is even more basic
(more "raw") than you and many mathematicians might be to willing to admit.
Few pages down from where he wrote "wo pk die k-te Primzhal (Der Größe nach) bedeutet", you'd see his definition of prime numbers - "Prim(x)" - being a
function of the familiar complete order binary relation denoted by '<'!
_That_ in itself is an invalid definition: since natural prime numbers should
be _defined_ in the context of a _multiplicative monoid_ that has only _one_
_NON_ logical symbol denoting the multiplication operation itself.
Even if the Fundamental Theorem of Arithmetic would allude to something like
"(by [order of] magnitude)", that still doesn't logically entail that the definition (of primes) - itself - be a function of the familiar symbols '<',
'S' at all. The correct definition of primes should look something like the
prime(p) <-> ((∃x∀y
Two errors have already occurred. It is *not* sufficient (given Gödel's
need) to have on the RHS of that <-> a formula that is true of all and
only primes p. The formula has to have another very specific property:
it has to be ... which requires that the quantifiers have to be ... .
Now, go on Nam grab the opportunity to show that you know something
about your subject matter (GIT1) by filling in the two blanks. Here's
betting that you can't and that you will try to hide your ignorance with
a smokescreen about Nazis or inquisitors or something.
Post by Khong Dong
[x*y=y]) /\ (~p=0) /\ ~∀z[p*z=z] /\ ((p=x*y) -> (∀z[x*z=z] \/ ∀z[y*z=z]))).
Certainly all these aren't the only reasons why Gödel's Incompleteness (1)
is invalid (there are also problems about HP as a law of thought, object vs
semantic quantification, if nothing else). But I can confidently say: I rest
my case.
Rupert
2020-03-30 18:29:45 UTC
Permalink
Post by Khong Dong
Post by Rupert
Post by Khong Dong
Post by Khong Dong
https://qr.ae/pNvAdF
which should be a *summary* of why Gödel's 1931 incompleteness paper is
an invalid argument (1st Incompleteness statement anyway).
There are more reasons than those detailed in the summary, but if you'd like to correctly, technically counter argue the summary as it is, you could of course
make yourself be heard.
Btw, Me, Rupert, Goerge Greene, et al., in R. B. BRAITHWAITE's introduction to
a translated version of Goedel's 1931 paper one would see (reformatted slightly
<quote>
For the part of his argument which establishes the 'unprovability' of
Neg (v Gen r) requires at one point considering a statement about all numbers,
whether or not they are Gödel numbers; and this statement cannot be construed
without change as a statement about all strings, since a number which is not
a Gödel number does not correspond to any string.
But it is easy to close the gap in the recasting by considering the numbers
which are Gödel numbers as arranged in a sequence of increasing magnitude,
and then using, instead of a Gödel number itself, the number which gives the
place of this Gödel number in the sequence. To be precise, if n is the
(m + 1)-th Gödel number in increasing order, call m the G-number of the string
of which n is the Gödel number, and use the G-number m wherever Gödel in his
argument uses the Gödel number n. Then every natural number 0, 1, 2, etc. will
be the G-number of some string; and there will be a recursive one-to-one
correspondence between natural numbers and strings.
</quote>
Would you be able to see from the above passage why Goedel's encoding is
invalid as I've pointed out in one way or another (about Goedel having not
formalized the unary "magnitude" function)?
Goedel does not at any stage invoke a unary function called "magnitude".
<quote>
"prime number" connotes the existence of a unary predicate denoted as "prime" which is a definable symbol of the language of arithmetic L(0,S,+,*,<).
"order" connotes the existence of a binary predicate denoted as "<" which is a formal symbol of the language of arithmetic L(0,S,+,*,<).
"magnitude" connotes the existence of a unary function denoted here as "magnitude" which is supposed to be a definable symbol - but yet hasn't been defined - of the language of arithmetic L(0,S,+,*,<).
</quote>
Yeah, but that's wrong. "The k-th prime number in order of magnitude" simply means, well, you know, what it obviously means. If you can't understand what it means, then you're kind of out of the running for understanding advanced mathematics.
Post by Khong Dong
Post by Rupert
He does make use of the phrase "in order of magnitude", at least in the
English translation. The meaning of this is crystal clear.
Again, (for the nth time ?), it's trivial "in order of magnitude" of a _number_
doesn't connote a the existence of unary function then the word "mathematics"
is quite meaningless.
magnitude(z) = sqrt(a^2 + b^2).
Are you implying you forget such a trivial definition of Complex number?
Wow.
Peter Percival
2020-03-30 19:12:46 UTC
Permalink
Post by Khong Dong
magnitude(z) = sqrt(a^2 + b^2).
The function 'magnitude' that you have defined will serve just as well
for natural numbers z. (As indeed it must since every natural number is
a complex number.) So it turns out that you *do* know what the
magnitude of a natural number is. It's only taken you 24 months to get it.
Post by Khong Dong
Are you implying you forget such a trivial definition of Complex number?
You haven't defined complex number, and I doubt that you could.

g***@gmail.com
2020-03-29 04:00:36 UTC
Permalink
Post by Khong Dong
https://qr.ae/pNvAdF
which should be a *summary* of why Gödel's 1931 incompleteness paper is
an invalid argument (1st Incompleteness statement anyway).
No its not an invalid argument. There are ommissions and flaws such as:


isProof( <x1,x2,...xk> )

An
IsAxiom( xn )
V
DirectlyFollows( <xi,xj> xn )


Now-a-days this is a recursive modus ponens routine.

Godel omitted how to decode a godel statement.

Godel numbers aren't used, formulas are alpha-numeric orderable expressions already

Godel numbers can also be substituted with index-augmented logic which is FOL -
e.g. directly referring to the own formula by number

Godel uses weak logic to DESCRIBE EXIST(x) instead of formulating the rule algorithmically

Godels 44 statements are not a proof, just a set of definitions

Godel failed his own definition of formal by not formally inferring each next rule

Godel stopped at the set of definitions without a final proven theorem

Godel wrongly defines NEXT_PRIME and gives no algorithm

Godel wrongly infers an infinite succession of larger sets but the godel statement is identical

---------------------


All these and more flaws aside its still a valid argument

Somewhere in most FOL this statement EXISTS

not(exist(p) proof(p,*THIS*))


And the CONCLUSION is misleading, ternary logic (stratification) easily solves this paradox of Boolean logic.

GIVEN ENOUGH LOGICAL EXPRESSIVE POWER + SELF REFERENCE + ALL-WFF-TorF
=
PARADOX
g***@gmail.com
2020-03-29 04:19:34 UTC
Permalink
Post by g***@gmail.com
GIVEN ENOUGH LOGICAL EXPRESSIVE POWER + SELF REFERENCE + ALL-WFF-TorF
=
PARADOX
also:

Godel defeated his own argument with IsProof(<x1,x2,...xk>) sizeX>0

This easily resolves to

~EXIST(p) IsProof(p)
_____________________
FALSE

IsProof({}) returns FALSE
*************************


also Godel misinterpreted the concept of PROOF(X,Y) - it CHECKS if X is a proof


isProof( <x1,x2,...xk> )

An
IsAxiom( xn )
V
DirectlyFollows( <xi,xj> xn )


merely by checking EXISTS(xi,xj) both GIVEN
but the definition of PROVE(Y)=X
will automatically check AXIOMS to guess xi & xj, constructing the proof in reverse

so it doesn't prove anything - PROOF(X,Y)!
Peter Percival
2020-03-29 21:22:48 UTC
Permalink
Post by g***@gmail.com
Post by Khong Dong
https://qr.ae/pNvAdF
which should be a *summary* of why Gödel's 1931 incompleteness paper is
an invalid argument (1st Incompleteness statement anyway).
isProof( <x1,x2,...xk> )
An
IsAxiom( xn )
V
DirectlyFollows( <xi,xj> xn )
Now-a-days this is a recursive modus ponens routine.
Godel omitted how to decode a godel statement.
Godel numbers aren't used, formulas are alpha-numeric orderable expressions already
Godel numbers can also be substituted with index-augmented logic which is FOL -
An irrelevance (whatever it means): Gödel works with type theory and his
work can formalized in PRA.
Post by g***@gmail.com
e.g. directly referring to the own formula by number
Godel uses weak logic to DESCRIBE EXIST(x) instead of formulating the rule algorithmically
The existential quantifier is *defined* in Gödel. See page 600 in van
Heijenoort. He does not state the definition because he expects the
reader to know that (Ex) is xPi for x of all types.
Post by g***@gmail.com
Godels 44 statements are not a proof, just a set of definitions
Do you mean the 46 statements on pp 603-6 in van Heijenoort. If so,
you're right, they are not a proof they are a set of definitions. So what?
Post by g***@gmail.com
Godel failed his own definition of formal by not formally inferring each next rule
Godel stopped at the set of definitions without a final proven theorem
Generally, G­ödel proves things but rather telegraphically. According
to the last sentence of the original paper, he intended to publish more
on systems other than P, but in the end he did not. (It is claimed that
the ready acceptance of his results indicated to him that there was no
need for a second paper. See note 68a on page 616 of van Heijenoort.)
Post by g***@gmail.com
Godel wrongly defines NEXT_PRIME and gives no algorithm
What are you referring to? His dfn (no. 5.) of Pr(n) - nth prime number?
Post by g***@gmail.com
Godel wrongly infers an infinite succession of larger sets but the godel statement is identical
I don't understand that.
Post by g***@gmail.com
---------------------
If you, or anyone, wants references to the collected works rather than
to van Heijenoort, just ask.
Post by g***@gmail.com
All these and more flaws aside its still a valid argument
Somewhere in most FOL this statement EXISTS
not(exist(p) proof(p,*THIS*))
And the CONCLUSION is misleading, ternary logic (stratification) easily solves this paradox of Boolean logic.
GIVEN ENOUGH LOGICAL EXPRESSIVE POWER + SELF REFERENCE + ALL-WFF-TorF
=
PARADOX
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