On 2022-05-05 15:23, olcott wrote:
> On 5/5/2022 1:06 PM, André G. Isaak wrote:
>> On 2022-05-05 11:57, olcott wrote:
>>> On 5/4/2022 9:46 PM, Richard Damon wrote:
>>
>>>> But if G is true in the Theory, it is BY DEFINITION not provable in
>>>> the Theory, so the space of the Theory is shown to have a True
>>>> Statement which is not provable, thus the system of the Theory in
>>>> Incomplete.
>>>
>>> G is self-contradictory on the theory and non self-contradictory in
>>> the meta-theory.
>>
>> G is not self-contradictory in either the theory or the meta-theory.
>>
>> The Liar Paradox and G are not the same sentence. You keep treating
>> them as if they were based solely on Gödel's claim that there is a
>> close relationship between them. But saying two things are closely
>> related does not mean they are the same.
>>
>> G asserts a claim about arithmetic. It asserts nothing about itself.
>>
>> André
>>
>
> From the quote below:
> We are therefore confronted with a proposition which asserts its own
> unprovability.

Even if G did assert its own unprovability (it doesn't -- not directly),
there's nothing self-contradictory about that. You can't treat it like
The Liar which *is* self-contradictory.

> Gödel says:
> The analogy between this result and Richard’s antinomy leaps to the eye;
> there is also a close relationship with the “liar” antinomy,14 since the
> undecidable proposition [R(q); q] states precisely that q belongs to K,
> i.e. according to (1), that [R(q); q] is not provable. We are therefore
> confronted with a proposition which asserts its own unprovability.

There is an analogy. That does not mean they are the same. No one denies
that Gödel acknowledges an analogy/close relationship between his G and
The Liar. But they are not the same, and none of these quotes from Gödel
suggest they are the same.

André

--
To email remove 'invalid' & replace 'gm' with well known Google mail
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On 5/5/2022 8:41 PM, André G. Isaak wrote:
> On 2022-05-05 15:23, olcott wrote:
>> On 5/5/2022 1:06 PM, André G. Isaak wrote:
>>> On 2022-05-05 11:57, olcott wrote:
>>>> On 5/4/2022 9:46 PM, Richard Damon wrote:
>>>
>>>>> But if G is true in the Theory, it is BY DEFINITION not provable in
>>>>> the Theory, so the space of the Theory is shown to have a True
>>>>> Statement which is not provable, thus the system of the Theory in
>>>>> Incomplete.
>>>>
>>>> G is self-contradictory on the theory and non self-contradictory in
>>>> the meta-theory.
>>>
>>> G is not self-contradictory in either the theory or the meta-theory.
>>>
>>> The Liar Paradox and G are not the same sentence. You keep treating
>>> them as if they were based solely on Gödel's claim that there is a
>>> close relationship between them. But saying two things are closely
>>> related does not mean they are the same.
>>>
>>> G asserts a claim about arithmetic. It asserts nothing about itself.
>>>
>>> André
>>>
>>
>>  From the quote below:
>> We are therefore confronted with a proposition which asserts its own
>> unprovability.
>
> Even if G did assert its own unprovability (it doesn't -- not directly),
> there's nothing self-contradictory about that. You can't treat it like
> The Liar which *is* self-contradictory.
>
>> Gödel says:
>> The analogy between this result and Richard’s antinomy leaps to the
>> eye; there is also a close relationship with the “liar” antinomy,14
>> since the undecidable proposition [R(q); q] states precisely that q
>> belongs to K, i.e. according to (1), that [R(q); q] is not provable.
>> We are therefore confronted with a proposition which asserts its own
>> unprovability.
>
> There is an analogy. That does not mean they are the same. No one denies
> that Gödel acknowledges an analogy/close relationship between his G and
> The Liar. But they are not the same, and none of these quotes from Gödel
> suggest they are the same.
You might add that Godel's breakthrough was figuring out how something
that looked paradoxical could do real mathematical work. Similar use was
made of the dilemma of the set of all sets; around the 1900s a number of
mathematicians rather than fear it as a paradox, made it do work.
The fact that Godel could find inspiration from an apparent
inconsistency and make it do work guarantees that he will be remembered
by man longer than Christ. It's also the opposite reason - utter lack of
insight - that dooms PO and sock puppet to only be remembered for a
short time as clowns.
--
Jeff Barnett

On 5/5/2022 9:41 PM, André G. Isaak wrote:
> On 2022-05-05 15:23, olcott wrote:
>> On 5/5/2022 1:06 PM, André G. Isaak wrote:
>>> On 2022-05-05 11:57, olcott wrote:
>>>> On 5/4/2022 9:46 PM, Richard Damon wrote:
>>>
>>>>> But if G is true in the Theory, it is BY DEFINITION not provable in
>>>>> the Theory, so the space of the Theory is shown to have a True
>>>>> Statement which is not provable, thus the system of the Theory in
>>>>> Incomplete.
>>>>
>>>> G is self-contradictory on the theory and non self-contradictory in
>>>> the meta-theory.
>>>
>>> G is not self-contradictory in either the theory or the meta-theory.
>>>
>>> The Liar Paradox and G are not the same sentence. You keep treating
>>> them as if they were based solely on Gödel's claim that there is a
>>> close relationship between them. But saying two things are closely
>>> related does not mean they are the same.
>>>
>>> G asserts a claim about arithmetic. It asserts nothing about itself.
>>>
>>> André
>>>
>>
>>  From the quote below:
>> We are therefore confronted with a proposition which asserts its own
>> unprovability.
>
> Even if G did assert its own unprovability (it doesn't -- not directly),
> there's nothing self-contradictory about that. You can't treat it like
> The Liar which *is* self-contradictory.
>
>> Gödel says:
>> The analogy between this result and Richard’s antinomy leaps to the
>> eye; there is also a close relationship with the “liar” antinomy,14
>> since the undecidable proposition [R(q); q] states precisely that q
>> belongs to K, i.e. according to (1), that [R(q); q] is not provable.
>> We are therefore confronted with a proposition which asserts its own
>> unprovability.
>
> There is an analogy. That does not mean they are the same.
>

(from the above quote)
*We are therefore confronted with a proposition which asserts its own
unprovability*

*We are therefore confronted with a proposition which asserts its own
unprovability*

*We are therefore confronted with a proposition which asserts its own
unprovability*

> No one denies
> that Gödel acknowledges an analogy/close relationship between his G and
> The Liar. But they are not the same, and none of these quotes from Gödel
> suggest they are the same.
>
> André
>
>

--
Copyright 2022 Pete Olcott "Talent hits a target no one else can hit;
Genius hits a target no one else can see." Arthur Schopenhauer

On 5/5/22 10:37 PM, olcott wrote:
> On 5/5/2022 9:24 PM, Richard Damon wrote:
>> On 5/5/22 1:57 PM, olcott wrote:

>>> G is self-contradictory on the theory and non self-contradictory in
>>> the meta-theory.
>>
>> No, because in the theory, G doesn't even reference itself, so it
>> can't be self-contradictory.
>>
>
> Gödel says:
> ...We are therefore confronted with a proposition which asserts its own
> unprovability.
>

Which is Godel making a comment about G, and not a statement in G itself.

G does not directly mention itself in the Theory.

>> I think you don't even know what G is, but have only read the clift
>> notes edition that actually explain it in the meta-theory.
>
>

So this comment of mine is now proven.

And you have prooved to be a Liar and an idiot, as you abolutely don't
know what you are talking about.

On 2022-05-04 17:55:11 +0000, olcott said:

> On 5/4/2022 1:49 AM, Mikko wrote:
>> On 2022-05-04 04:06:43 +0000, André G. Isaak said:
>>
>>> That's very different from G, where no proof exists of either G or of ¬G.
>>
>> However, (G ∨ ¬G) is provable in PA.
>>
>> Mikko
>>
>
> Incomplete(PA) ↔ ∃G ((PA ⊬ G) ∧ (PA ⊬ ¬G)).

Irrelevant but yes, one could say it that way.

> Then Gödel is wrong.

Too vague to mean anything but probably irrelevant and false.

Mikko

On 2022-05-05 23:16, olcott wrote:
> On 5/5/2022 9:41 PM, André G. Isaak wrote:
>> On 2022-05-05 15:23, olcott wrote:
>>> On 5/5/2022 1:06 PM, André G. Isaak wrote:
>>>> On 2022-05-05 11:57, olcott wrote:
>>>>> On 5/4/2022 9:46 PM, Richard Damon wrote:
>>>>
>>>>>> But if G is true in the Theory, it is BY DEFINITION not provable
>>>>>> in the Theory, so the space of the Theory is shown to have a True
>>>>>> Statement which is not provable, thus the system of the Theory in
>>>>>> Incomplete.
>>>>>
>>>>> G is self-contradictory on the theory and non self-contradictory in
>>>>> the meta-theory.
>>>>
>>>> G is not self-contradictory in either the theory or the meta-theory.
>>>>
>>>> The Liar Paradox and G are not the same sentence. You keep treating
>>>> them as if they were based solely on Gödel's claim that there is a
>>>> close relationship between them. But saying two things are closely
>>>> related does not mean they are the same.
>>>>
>>>> G asserts a claim about arithmetic. It asserts nothing about itself.
>>>>
>>>> André
>>>>
>>>
>>>  From the quote below:
>>> We are therefore confronted with a proposition which asserts its own
>>> unprovability.
>>
>> Even if G did assert its own unprovability (it doesn't -- not
>> directly), there's nothing self-contradictory about that. You can't
>> treat it like The Liar which *is* self-contradictory.
>>
>>> Gödel says:
>>> The analogy between this result and Richard’s antinomy leaps to the
>>> eye; there is also a close relationship with the “liar” antinomy,14
>>> since the undecidable proposition [R(q); q] states precisely that q
>>> belongs to K, i.e. according to (1), that [R(q); q] is not provable.
>>> We are therefore confronted with a proposition which asserts its own
>>> unprovability.
>>
>> There is an analogy. That does not mean they are the same.
> >
>
> (from the above quote)
> *We are therefore confronted with a proposition which asserts its own
> unprovability*

Which is Gödel's commentary on G, not what G actually states. And it is
also irrelevant to my central point which was that even if G did state
this it would not be self-contradictory.

If a statement asserts its own falsity like the Liar, it is
self-contradictory.

If a statement asserts its own unprovability, it is not self-contradictory.

André

--
To email remove 'invalid' & replace 'gm' with well known Google mail
service.

On 5/6/2022 8:39 AM, André G. Isaak wrote:
> On 2022-05-05 23:16, olcott wrote:
>> On 5/5/2022 9:41 PM, André G. Isaak wrote:
>>> On 2022-05-05 15:23, olcott wrote:
>>>> On 5/5/2022 1:06 PM, André G. Isaak wrote:
>>>>> On 2022-05-05 11:57, olcott wrote:
>>>>>> On 5/4/2022 9:46 PM, Richard Damon wrote:
>>>>>
>>>>>>> But if G is true in the Theory, it is BY DEFINITION not provable
>>>>>>> in the Theory, so the space of the Theory is shown to have a True
>>>>>>> Statement which is not provable, thus the system of the Theory in
>>>>>>> Incomplete.
>>>>>>
>>>>>> G is self-contradictory on the theory and non self-contradictory
>>>>>> in the meta-theory.
>>>>>
>>>>> G is not self-contradictory in either the theory or the meta-theory.
>>>>>
>>>>> The Liar Paradox and G are not the same sentence. You keep treating
>>>>> them as if they were based solely on Gödel's claim that there is a
>>>>> close relationship between them. But saying two things are closely
>>>>> related does not mean they are the same.
>>>>>
>>>>> G asserts a claim about arithmetic. It asserts nothing about itself.
>>>>>
>>>>> André
>>>>>
>>>>
>>>>  From the quote below:
>>>> We are therefore confronted with a proposition which asserts its own
>>>> unprovability.
>>>
>>> Even if G did assert its own unprovability (it doesn't -- not
>>> directly), there's nothing self-contradictory about that. You can't
>>> treat it like The Liar which *is* self-contradictory.
>>>
>>>> Gödel says:
>>>> The analogy between this result and Richard’s antinomy leaps to the
>>>> eye; there is also a close relationship with the “liar” antinomy,14
>>>> since the undecidable proposition [R(q); q] states precisely that q
>>>> belongs to K, i.e. according to (1), that [R(q); q] is not provable.
>>>> We are therefore confronted with a proposition which asserts its own
>>>> unprovability.
>>>
>>> There is an analogy. That does not mean they are the same.
>>  >
>>
>> (from the above quote)
>> *We are therefore confronted with a proposition which asserts its own
>> unprovability*
>
> Which is Gödel's commentary on G, not what G actually states. And it is
> also irrelevant to my central point which was that even if G did state
> this it would not be self-contradictory.
>
> If a statement asserts its own falsity like the Liar, it is
> self-contradictory.
>
> If a statement asserts its own unprovability, it is not self-contradictory.
>
> André
>

The problem is that this formula simply assumes that G is a logic
sentence, thus assumes that semantically incorrect expressions of the
language F are semantically correct.

Incomplete(F) ↔ ∃G ((F ⊬ G) ∧ (F ⊬ ¬G)).

When G is not a logic sentence it is incorrect to conclude
Incomplete(F). G is only unprovable because it is semantically incoherent.

--
Copyright 2022 Pete Olcott "Talent hits a target no one else can hit;
Genius hits a target no one else can see." Arthur Schopenhauer

On 5/6/2022 7:46 AM, Mikko wrote:
> On 2022-05-04 17:55:11 +0000, olcott said:
>
>> On 5/4/2022 1:49 AM, Mikko wrote:
>>> On 2022-05-04 04:06:43 +0000, André G. Isaak said:
>>>
>>>> That's very different from G, where no proof exists of either G or
>>>> of ¬G.
>>>
>>> However, (G ∨ ¬G) is provable in PA.

No this is incorrect.
∃G ((PA ⊬ G) ∧ (PA ⊬ ¬G)).

>>>
>>> Mikko
>>>
>>
>> Incomplete(PA) ↔ ∃G ((PA ⊬ G) ∧ (PA ⊬ ¬G)).
>
> Irrelevant but yes, one could say it that way.
>
>> Then Gödel is wrong.
>
> Too vague to mean anything but probably irrelevant and false.
>
> Mikko
>

Incomplete(F) ↔ ∃G ((F ⊬ G) ∧ (F ⊬ ¬G)).

The above formula mischaracterizes Incomplete(F) on the basis of
Semantically_Incoherent(G) of the language of F.

F has its own provability operator, unlike PA.

--
Copyright 2022 Pete Olcott "Talent hits a target no one else can hit;
Genius hits a target no one else can see." Arthur Schopenhauer

On 2022-05-06 11:17, olcott wrote:
> On 5/6/2022 8:39 AM, André G. Isaak wrote:
>> On 2022-05-05 23:16, olcott wrote:

>>> *We are therefore confronted with a proposition which asserts its own
>>> unprovability*
>>
>> Which is Gödel's commentary on G, not what G actually states. And it
>> is also irrelevant to my central point which was that even if G did
>> state this it would not be self-contradictory.
>>
>> If a statement asserts its own falsity like the Liar, it is
>> self-contradictory.
>>
>> If a statement asserts its own unprovability, it is not
>> self-contradictory.
>>
>> André
>>
>
> The problem is that this formula simply assumes that G is a logic
> sentence, thus assumes that semantically incorrect expressions of the
> language F are semantically correct.
>
> Incomplete(F) ↔ ∃G ((F ⊬ G) ∧ (F ⊬ ¬G)).
>
> When G is not a logic sentence it is incorrect to conclude
> Incomplete(F). G is only unprovable because it is semantically incoherent.

Gödel's G asserts that a particular polynomial equation has at least one
integer solution. In what possible sense is that 'semantically
incoherent'? Every polynomial equation either has at least one integer
solution or it does not.

André

--
To email remove 'invalid' & replace 'gm' with well known Google mail
service.

On 5/6/2022 12:31 PM, André G. Isaak wrote:
> On 2022-05-06 11:17, olcott wrote:
>> On 5/6/2022 8:39 AM, André G. Isaak wrote:
>>> On 2022-05-05 23:16, olcott wrote:
>
>>>> *We are therefore confronted with a proposition which asserts its
>>>> own unprovability*
>>>
>>> Which is Gödel's commentary on G, not what G actually states. And it
>>> is also irrelevant to my central point which was that even if G did
>>> state this it would not be self-contradictory.
>>>
>>> If a statement asserts its own falsity like the Liar, it is
>>> self-contradictory.
>>>
>>> If a statement asserts its own unprovability, it is not
>>> self-contradictory.
>>>
>>> André
>>>
>>
>> The problem is that this formula simply assumes that G is a logic
>> sentence, thus assumes that semantically incorrect expressions of the
>> language F are semantically correct.
>>
>> Incomplete(F) ↔ ∃G ((F ⊬ G) ∧ (F ⊬ ¬G)).
>>
>> When G is not a logic sentence it is incorrect to conclude
>> Incomplete(F). G is only unprovable because it is semantically
>> incoherent.
>
> Gödel's G asserts that a particular polynomial equation has at least one
> integer solution. In what possible sense is that 'semantically
> incoherent'? Every polynomial equation either has at least one integer
> solution or it does not.
>
> André
>

In the sense the G is implemented in a formal system having its own
Provable() predicate so that it can directly implement ⊬ and not dance
all around these semantics to artificially contrive an approximation of
the concept of provability in a language that is woefully insufficiently
expressive.

Only when we do that and boil incompleteness down to its barest essence
can we see the underlying relationships clearly enough to see that they
are erroneous.

Tarski actually used the Liar Paradox itself as his basis in his
undefinability proof. He then converted the formalized LP to the barest
essence of the semantics of of Gödel's G.

x ⋶ Pr if and only if p // Tarski's barest essence of Gödel' G
where the symbol 'p' represents the whole sentence x

x ⋶ Tr if and only if p // Tarski's formalized Liar Paradox
where the symbol 'p' represents the whole sentence x

THEOREM I. (α) In whatever way the symbol 'Tr', denoting a class of
expressions, is defined in the metatheory, it will be possible to derive
from it the negation of one of the sentences which were described in the
condition (α) of the convention T;

It would then be possible to reconstruct the antinomy of the liar in the
metalanguage, by forming in the language itself a sentence x such that
the sentence of the metalanguage which is correlated with x asserts that
x is not a true sentence. https://liarparadox.org/Tarski_247_248.pdf

In accordance with the first part of Th. I we can obtain the negation of
one of the sentences in condition (α) of convention T of § 3 as a
consequence of the definition of the symbol 'Pr' (provided we replace
'Tr' in this convention by 'Pr').
https://liarparadox.org/Tarski_275_276.pdf

x ⋶ Tr if and only if p
where the symbol 'p' represents the whole sentence x

Converting the above formalized Liar Paradox into a very simplified
version of Gödel's G:

x ⋶ Pr if and only if p
where the symbol 'p' represents the whole sentence x

--
Copyright 2022 Pete Olcott "Talent hits a target no one else can hit;
Genius hits a target no one else can see." Arthur Schopenhauer

On 2022-05-06 12:36, olcott wrote:
> On 5/6/2022 12:31 PM, André G. Isaak wrote:
>> On 2022-05-06 11:17, olcott wrote:
>>> On 5/6/2022 8:39 AM, André G. Isaak wrote:
>>>> On 2022-05-05 23:16, olcott wrote:
>>
>>>>> *We are therefore confronted with a proposition which asserts its
>>>>> own unprovability*
>>>>
>>>> Which is Gödel's commentary on G, not what G actually states. And it
>>>> is also irrelevant to my central point which was that even if G did
>>>> state this it would not be self-contradictory.
>>>>
>>>> If a statement asserts its own falsity like the Liar, it is
>>>> self-contradictory.
>>>>
>>>> If a statement asserts its own unprovability, it is not
>>>> self-contradictory.
>>>>
>>>> André
>>>>
>>>
>>> The problem is that this formula simply assumes that G is a logic
>>> sentence, thus assumes that semantically incorrect expressions of the
>>> language F are semantically correct.
>>>
>>> Incomplete(F) ↔ ∃G ((F ⊬ G) ∧ (F ⊬ ¬G)).
>>>
>>> When G is not a logic sentence it is incorrect to conclude
>>> Incomplete(F). G is only unprovable because it is semantically
>>> incoherent.
>>
>> Gödel's G asserts that a particular polynomial equation has at least
>> one integer solution. In what possible sense is that 'semantically
>> incoherent'? Every polynomial equation either has at least one integer
>> solution or it does not.
>>
>> André
>>
>
> In the sense the G is implemented in a formal system having its own
> Provable() predicate so that it can directly implement ⊬ and not dance
> all around these semantics to artificially contrive an approximation of
> the concept of provability in a language that is woefully insufficiently
> expressive.

The above is simply word-salad.

What follows is completely irrelevant as it fails to address my
question: How can an assertion that a polynomial has an integer solution
be 'incoherent'?

André

> Only when we do that and boil incompleteness down to its barest essence
> can we see the underlying relationships clearly enough to see that they
> are erroneous.
>
> Tarski actually used the Liar Paradox itself as his basis in his
> undefinability proof. He then converted the formalized LP to the barest
> essence of the semantics of of Gödel's G.
>
> x ⋶ Pr if and only if p // Tarski's barest essence of Gödel' G
> where the symbol 'p' represents the whole sentence x
>
> x ⋶ Tr if and only if p // Tarski's formalized Liar Paradox
> where the symbol 'p' represents the whole sentence x
>
> THEOREM I. (α) In whatever way the symbol 'Tr', denoting a class of
> expressions, is defined in the metatheory, it will be possible to derive
> from it the negation of one of the sentences which were described in the
> condition (α) of the convention T;
>
> It would then be possible to reconstruct the antinomy of the liar in the
> metalanguage, by forming in the language itself a sentence x such that
> the sentence of the metalanguage which is correlated with x asserts that
> x is not a true sentence. https://liarparadox.org/Tarski_247_248.pdf
>
> In accordance with the first part of Th. I we can obtain the negation of
> one of the sentences in condition (α) of convention T of § 3 as a
> consequence of the definition of the symbol 'Pr' (provided we replace
> 'Tr' in this convention by 'Pr').
> https://liarparadox.org/Tarski_275_276.pdf
>
> x ⋶ Tr if and only if p
> where the symbol 'p' represents the whole sentence x
>
> Converting the above formalized Liar Paradox into a very simplified
> version of Gödel's G:
>
> x ⋶ Pr if and only if p
> where the symbol 'p' represents the whole sentence x
>

--
To email remove 'invalid' & replace 'gm' with well known Google mail
service.

On 5/6/2022 1:41 PM, André G. Isaak wrote:
> On 2022-05-06 12:36, olcott wrote:
>> On 5/6/2022 12:31 PM, André G. Isaak wrote:
>>> On 2022-05-06 11:17, olcott wrote:
>>>> On 5/6/2022 8:39 AM, André G. Isaak wrote:
>>>>> On 2022-05-05 23:16, olcott wrote:
>>>
>>>>>> *We are therefore confronted with a proposition which asserts its
>>>>>> own unprovability*
>>>>>
>>>>> Which is Gödel's commentary on G, not what G actually states. And
>>>>> it is also irrelevant to my central point which was that even if G
>>>>> did state this it would not be self-contradictory.
>>>>>
>>>>> If a statement asserts its own falsity like the Liar, it is
>>>>> self-contradictory.
>>>>>
>>>>> If a statement asserts its own unprovability, it is not
>>>>> self-contradictory.
>>>>>
>>>>> André
>>>>>
>>>>
>>>> The problem is that this formula simply assumes that G is a logic
>>>> sentence, thus assumes that semantically incorrect expressions of
>>>> the language F are semantically correct.
>>>>
>>>> Incomplete(F) ↔ ∃G ((F ⊬ G) ∧ (F ⊬ ¬G)).
>>>>
>>>> When G is not a logic sentence it is incorrect to conclude
>>>> Incomplete(F). G is only unprovable because it is semantically
>>>> incoherent.
>>>
>>> Gödel's G asserts that a particular polynomial equation has at least
>>> one integer solution. In what possible sense is that 'semantically
>>> incoherent'? Every polynomial equation either has at least one
>>> integer solution or it does not.
>>>
>>> André
>>>
>>
>> In the sense the G is implemented in a formal system having its own
>> Provable() predicate so that it can directly implement ⊬ and not dance
>> all around these semantics to artificially contrive an approximation
>> of the concept of provability in a language that is woefully
>> insufficiently expressive.
>
> The above is simply word-salad.
>

Tarski anchors all of the details of this

x ⋶ Pr if and only if p // Tarski's barest essence of Gödel' G
where the symbol 'p' represents the whole sentence x
and 'Pr' means provable

The above is derived from the following by simply swapping "Tr" for "Pr"
"we replace 'Tr' in this convention by 'Pr'".

x ⋶ Tr if and only if p // Tarski's formalized Liar Paradox
where the symbol 'p' represents the whole sentence x
and 'Tr' means true.

> What follows is completely irrelevant as it fails to address my
> question: How can an assertion that a polynomial has an integer solution
> be 'incoherent'?
>

When we look at the same thing that is being proved at its highest level
of abstraction then we find that p is unprovable because it is
semantically incoherent. In the dozens of pages of hundreds of formulas
of the Gödel G form of this the details obscure the essential structure.

> André
>
>> Only when we do that and boil incompleteness down to its barest
>> essence can we see the underlying relationships clearly enough to see
>> that they are erroneous.
>>
>> Tarski actually used the Liar Paradox itself as his basis in his
>> undefinability proof. He then converted the formalized LP to the
>> barest essence of the semantics of of Gödel's G.
>>
>> x ⋶ Pr if and only if p // Tarski's barest essence of Gödel' G
>> where the symbol 'p' represents the whole sentence x
>>
>> x ⋶ Tr if and only if p // Tarski's formalized Liar Paradox
>> where the symbol 'p' represents the whole sentence x
>>
>> THEOREM I. (α) In whatever way the symbol 'Tr', denoting a class of
>> expressions, is defined in the metatheory, it will be possible to
>> derive from it the negation of one of the sentences which were
>> described in the condition (α) of the convention T;
>>
>> It would then be possible to reconstruct the antinomy of the liar in
>> the metalanguage, by forming in the language itself a sentence x such
>> that the sentence of the metalanguage which is correlated with x
>> asserts that x is not a true sentence.
>> https://liarparadox.org/Tarski_247_248.pdf
>>
>> In accordance with the first part of Th. I we can obtain the negation
>> of one of the sentences in condition (α) of convention T of § 3 as a
>> consequence of the definition of the symbol 'Pr' (provided we replace
>> 'Tr' in this convention by 'Pr').
>> https://liarparadox.org/Tarski_275_276.pdf
>>
>> x ⋶ Tr if and only if p
>> where the symbol 'p' represents the whole sentence x
>>
>> Converting the above formalized Liar Paradox into a very simplified
>> version of Gödel's G:
>>
>> x ⋶ Pr if and only if p
>> where the symbol 'p' represents the whole sentence x
>>
>
>

--
Copyright 2022 Pete Olcott "Talent hits a target no one else can hit;
Genius hits a target no one else can see." Arthur Schopenhauer

On 2022-05-06 13:03, olcott wrote:
> On 5/6/2022 1:41 PM, André G. Isaak wrote:
>> On 2022-05-06 12:36, olcott wrote:
>>> On 5/6/2022 12:31 PM, André G. Isaak wrote:
>>>> On 2022-05-06 11:17, olcott wrote:
>>>>> On 5/6/2022 8:39 AM, André G. Isaak wrote:
>>>>>> On 2022-05-05 23:16, olcott wrote:
>>>>
>>>>>>> *We are therefore confronted with a proposition which asserts its
>>>>>>> own unprovability*
>>>>>>
>>>>>> Which is Gödel's commentary on G, not what G actually states. And
>>>>>> it is also irrelevant to my central point which was that even if G
>>>>>> did state this it would not be self-contradictory.
>>>>>>
>>>>>> If a statement asserts its own falsity like the Liar, it is
>>>>>> self-contradictory.
>>>>>>
>>>>>> If a statement asserts its own unprovability, it is not
>>>>>> self-contradictory.
>>>>>>
>>>>>> André
>>>>>>
>>>>>
>>>>> The problem is that this formula simply assumes that G is a logic
>>>>> sentence, thus assumes that semantically incorrect expressions of
>>>>> the language F are semantically correct.
>>>>>
>>>>> Incomplete(F) ↔ ∃G ((F ⊬ G) ∧ (F ⊬ ¬G)).
>>>>>
>>>>> When G is not a logic sentence it is incorrect to conclude
>>>>> Incomplete(F). G is only unprovable because it is semantically
>>>>> incoherent.
>>>>
>>>> Gödel's G asserts that a particular polynomial equation has at least
>>>> one integer solution. In what possible sense is that 'semantically
>>>> incoherent'? Every polynomial equation either has at least one
>>>> integer solution or it does not.
>>>>
>>>> André
>>>>
>>>
>>> In the sense the G is implemented in a formal system having its own
>>> Provable() predicate so that it can directly implement ⊬ and not
>>> dance all around these semantics to artificially contrive an
>>> approximation of the concept of provability in a language that is
>>> woefully insufficiently expressive.
>>
>> The above is simply word-salad.
>>
>
> Tarski anchors all of the details of this
>
> x ⋶ Pr if and only if p // Tarski's barest essence of Gödel' G
> where the symbol 'p' represents the whole sentence x
> and 'Pr' means provable
>
> The above is derived from the following by simply swapping "Tr" for "Pr"
> "we replace 'Tr' in this convention by 'Pr'".
>
> x ⋶ Tr if and only if p // Tarski's formalized Liar Paradox
> where the symbol 'p' represents the whole sentence x
> and 'Tr' means true.
>
>
>> What follows is completely irrelevant as it fails to address my
>> question: How can an assertion that a polynomial has an integer
>> solution be 'incoherent'?
>>
>
> When we look at the same thing that is being proved at its highest level
> of abstraction then we find that p is unprovable because it is
> semantically incoherent. In the dozens of pages of hundreds of formulas
> of the Gödel G form of this the details obscure the essential structure.

Again, you are not answering the question. G asserts that a specific
polynomial equation has an integer solution. That's *all* it asserts.
How can such an assertion be 'semantically incoherent'?

How do you define 'semantic coherence'? How does G violate this definition?

Is the assertion that "a² + b² = c² has at least one integer solution"
semantically incoherent? If not, how is G different from this?

André

--
To email remove 'invalid' & replace 'gm' with well known Google mail
service.

On 5/6/2022 2:11 PM, André G. Isaak wrote:
> On 2022-05-06 13:03, olcott wrote:
>> On 5/6/2022 1:41 PM, André G. Isaak wrote:
>>> On 2022-05-06 12:36, olcott wrote:
>>>> On 5/6/2022 12:31 PM, André G. Isaak wrote:
>>>>> On 2022-05-06 11:17, olcott wrote:
>>>>>> On 5/6/2022 8:39 AM, André G. Isaak wrote:
>>>>>>> On 2022-05-05 23:16, olcott wrote:
>>>>>
>>>>>>>> *We are therefore confronted with a proposition which asserts
>>>>>>>> its own unprovability*
>>>>>>>
>>>>>>> Which is Gödel's commentary on G, not what G actually states. And
>>>>>>> it is also irrelevant to my central point which was that even if
>>>>>>> G did state this it would not be self-contradictory.
>>>>>>>
>>>>>>> If a statement asserts its own falsity like the Liar, it is
>>>>>>> self-contradictory.
>>>>>>>
>>>>>>> If a statement asserts its own unprovability, it is not
>>>>>>> self-contradictory.
>>>>>>>
>>>>>>> André
>>>>>>>
>>>>>>
>>>>>> The problem is that this formula simply assumes that G is a logic
>>>>>> sentence, thus assumes that semantically incorrect expressions of
>>>>>> the language F are semantically correct.
>>>>>>
>>>>>> Incomplete(F) ↔ ∃G ((F ⊬ G) ∧ (F ⊬ ¬G)).
>>>>>>
>>>>>> When G is not a logic sentence it is incorrect to conclude
>>>>>> Incomplete(F). G is only unprovable because it is semantically
>>>>>> incoherent.
>>>>>
>>>>> Gödel's G asserts that a particular polynomial equation has at
>>>>> least one integer solution. In what possible sense is that
>>>>> 'semantically incoherent'? Every polynomial equation either has at
>>>>> least one integer solution or it does not.
>>>>>
>>>>> André
>>>>>
>>>>
>>>> In the sense the G is implemented in a formal system having its own
>>>> Provable() predicate so that it can directly implement ⊬ and not
>>>> dance all around these semantics to artificially contrive an
>>>> approximation of the concept of provability in a language that is
>>>> woefully insufficiently expressive.
>>>
>>> The above is simply word-salad.
>>>
>>
>> Tarski anchors all of the details of this
>>
>> x ⋶ Pr if and only if p // Tarski's barest essence of Gödel' G
>> where the symbol 'p' represents the whole sentence x
>> and 'Pr' means provable
>>
>> The above is derived from the following by simply swapping "Tr" for "Pr"
>> "we replace 'Tr' in this convention by 'Pr'".
>>
>> x ⋶ Tr if and only if p // Tarski's formalized Liar Paradox
>> where the symbol 'p' represents the whole sentence x
>> and 'Tr' means true.
>>
>>
>>> What follows is completely irrelevant as it fails to address my
>>> question: How can an assertion that a polynomial has an integer
>>> solution be 'incoherent'?
>>>
>>
>> When we look at the same thing that is being proved at its highest
>> level of abstraction then we find that p is unprovable because it is
>> semantically incoherent. In the dozens of pages of hundreds of
>> formulas of the Gödel G form of this the details obscure the essential
>> structure.
>
> Again, you are not answering the question. G asserts that a specific
> polynomial equation has an integer solution. That's *all* it asserts.
> How can such an assertion be 'semantically incoherent'?
>

x ⋶ Pr if and only if p
is the same G at a much higher level of abstraction so that

"We are therefore confronted with a proposition which asserts its own
unprovability." without any extranelous details.

> How do you define 'semantic coherence'? How does G violate this definition?
>
> Is the assertion that "a² + b² = c² has at least one integer solution"
> semantically incoherent? If not, how is G different from this?
>
> André
>

--
Copyright 2022 Pete Olcott "Talent hits a target no one else can hit;
Genius hits a target no one else can see." Arthur Schopenhauer

On 2022-05-06 13:23, olcott wrote:
> On 5/6/2022 2:11 PM, André G. Isaak wrote:
>> On 2022-05-06 13:03, olcott wrote:
>>> On 5/6/2022 1:41 PM, André G. Isaak wrote:
>>>> On 2022-05-06 12:36, olcott wrote:
>>>>> On 5/6/2022 12:31 PM, André G. Isaak wrote:
>>>>>> On 2022-05-06 11:17, olcott wrote:
>>>>>>> On 5/6/2022 8:39 AM, André G. Isaak wrote:
>>>>>>>> On 2022-05-05 23:16, olcott wrote:
>>>>>>
>>>>>>>>> *We are therefore confronted with a proposition which asserts
>>>>>>>>> its own unprovability*
>>>>>>>>
>>>>>>>> Which is Gödel's commentary on G, not what G actually states.
>>>>>>>> And it is also irrelevant to my central point which was that
>>>>>>>> even if G did state this it would not be self-contradictory.
>>>>>>>>
>>>>>>>> If a statement asserts its own falsity like the Liar, it is
>>>>>>>> self-contradictory.
>>>>>>>>
>>>>>>>> If a statement asserts its own unprovability, it is not
>>>>>>>> self-contradictory.
>>>>>>>>
>>>>>>>> André
>>>>>>>>
>>>>>>>
>>>>>>> The problem is that this formula simply assumes that G is a logic
>>>>>>> sentence, thus assumes that semantically incorrect expressions of
>>>>>>> the language F are semantically correct.
>>>>>>>
>>>>>>> Incomplete(F) ↔ ∃G ((F ⊬ G) ∧ (F ⊬ ¬G)).
>>>>>>>
>>>>>>> When G is not a logic sentence it is incorrect to conclude
>>>>>>> Incomplete(F). G is only unprovable because it is semantically
>>>>>>> incoherent.
>>>>>>
>>>>>> Gödel's G asserts that a particular polynomial equation has at
>>>>>> least one integer solution. In what possible sense is that
>>>>>> 'semantically incoherent'? Every polynomial equation either has at
>>>>>> least one integer solution or it does not.
>>>>>>
>>>>>> André
>>>>>>
>>>>>
>>>>> In the sense the G is implemented in a formal system having its own
>>>>> Provable() predicate so that it can directly implement ⊬ and not
>>>>> dance all around these semantics to artificially contrive an
>>>>> approximation of the concept of provability in a language that is
>>>>> woefully insufficiently expressive.
>>>>
>>>> The above is simply word-salad.
>>>>
>>>
>>> Tarski anchors all of the details of this
>>>
>>> x ⋶ Pr if and only if p // Tarski's barest essence of Gödel' G
>>> where the symbol 'p' represents the whole sentence x
>>> and 'Pr' means provable
>>>
>>> The above is derived from the following by simply swapping "Tr" for "Pr"
>>> "we replace 'Tr' in this convention by 'Pr'".
>>>
>>> x ⋶ Tr if and only if p // Tarski's formalized Liar Paradox
>>> where the symbol 'p' represents the whole sentence x
>>> and 'Tr' means true.
>>>
>>>
>>>> What follows is completely irrelevant as it fails to address my
>>>> question: How can an assertion that a polynomial has an integer
>>>> solution be 'incoherent'?
>>>>
>>>
>>> When we look at the same thing that is being proved at its highest
>>> level of abstraction then we find that p is unprovable because it is
>>> semantically incoherent. In the dozens of pages of hundreds of
>>> formulas of the Gödel G form of this the details obscure the
>>> essential structure.
>>
>> Again, you are not answering the question. G asserts that a specific
>> polynomial equation has an integer solution. That's *all* it asserts.
>> How can such an assertion be 'semantically incoherent'?
>>
>
> x ⋶ Pr if and only if p
> is the same G at a much higher level of abstraction so that

No. It isn't. Why don't you learn what G actually is.

And I didn't ask anything about some higher level; I asked how the
assertion that a specific polynomial has an integer solution can be
'semantically incoherent'.

> "We are therefore confronted with a proposition which asserts its own
> unprovability." without any extranelous details.

Repeatedly quoting this serves no purpose. That's Gödel's commentary,
*not* G. Gödel assumes that people are actually following along with the
proof and therefore will grasp his meaning. You are not, and therefore
do not.

>> How do you define 'semantic coherence'? How does G violate this
>> definition?

No answer?

>> Is the assertion that "a² + b² = c² has at least one integer solution"
>> semantically incoherent? If not, how is G different from this?

No answer?

André

--
To email remove 'invalid' & replace 'gm' with well known Google mail
service.

On 5/6/2022 2:27 PM, André G. Isaak wrote:
> On 2022-05-06 13:23, olcott wrote:
>> On 5/6/2022 2:11 PM, André G. Isaak wrote:
>>> On 2022-05-06 13:03, olcott wrote:
>>>> On 5/6/2022 1:41 PM, André G. Isaak wrote:
>>>>> On 2022-05-06 12:36, olcott wrote:
>>>>>> On 5/6/2022 12:31 PM, André G. Isaak wrote:
>>>>>>> On 2022-05-06 11:17, olcott wrote:
>>>>>>>> On 5/6/2022 8:39 AM, André G. Isaak wrote:
>>>>>>>>> On 2022-05-05 23:16, olcott wrote:
>>>>>>>
>>>>>>>>>> *We are therefore confronted with a proposition which asserts
>>>>>>>>>> its own unprovability*
>>>>>>>>>
>>>>>>>>> Which is Gödel's commentary on G, not what G actually states.
>>>>>>>>> And it is also irrelevant to my central point which was that
>>>>>>>>> even if G did state this it would not be self-contradictory.
>>>>>>>>>
>>>>>>>>> If a statement asserts its own falsity like the Liar, it is
>>>>>>>>> self-contradictory.
>>>>>>>>>
>>>>>>>>> If a statement asserts its own unprovability, it is not
>>>>>>>>> self-contradictory.
>>>>>>>>>
>>>>>>>>> André
>>>>>>>>>
>>>>>>>>
>>>>>>>> The problem is that this formula simply assumes that G is a
>>>>>>>> logic sentence, thus assumes that semantically incorrect
>>>>>>>> expressions of the language F are semantically correct.
>>>>>>>>
>>>>>>>> Incomplete(F) ↔ ∃G ((F ⊬ G) ∧ (F ⊬ ¬G)).
>>>>>>>>
>>>>>>>> When G is not a logic sentence it is incorrect to conclude
>>>>>>>> Incomplete(F). G is only unprovable because it is semantically
>>>>>>>> incoherent.
>>>>>>>
>>>>>>> Gödel's G asserts that a particular polynomial equation has at
>>>>>>> least one integer solution. In what possible sense is that
>>>>>>> 'semantically incoherent'? Every polynomial equation either has
>>>>>>> at least one integer solution or it does not.
>>>>>>>
>>>>>>> André
>>>>>>>
>>>>>>
>>>>>> In the sense the G is implemented in a formal system having its
>>>>>> own Provable() predicate so that it can directly implement ⊬ and
>>>>>> not dance all around these semantics to artificially contrive an
>>>>>> approximation of the concept of provability in a language that is
>>>>>> woefully insufficiently expressive.
>>>>>
>>>>> The above is simply word-salad.
>>>>>
>>>>
>>>> Tarski anchors all of the details of this
>>>>
>>>> x ⋶ Pr if and only if p // Tarski's barest essence of Gödel' G
>>>> where the symbol 'p' represents the whole sentence x
>>>> and 'Pr' means provable
>>>>
>>>> The above is derived from the following by simply swapping "Tr" for
>>>> "Pr"
>>>> "we replace 'Tr' in this convention by 'Pr'".
>>>>
>>>> x ⋶ Tr if and only if p // Tarski's formalized Liar Paradox
>>>> where the symbol 'p' represents the whole sentence x
>>>> and 'Tr' means true.
>>>>
>>>>
>>>>> What follows is completely irrelevant as it fails to address my
>>>>> question: How can an assertion that a polynomial has an integer
>>>>> solution be 'incoherent'?
>>>>>
>>>>
>>>> When we look at the same thing that is being proved at its highest
>>>> level of abstraction then we find that p is unprovable because it is
>>>> semantically incoherent. In the dozens of pages of hundreds of
>>>> formulas of the Gödel G form of this the details obscure the
>>>> essential structure.
>>>
>>> Again, you are not answering the question. G asserts that a specific
>>> polynomial equation has an integer solution. That's *all* it asserts.
>>> How can such an assertion be 'semantically incoherent'?
>>>
>>
>> x ⋶ Pr if and only if p
>> is the same G at a much higher level of abstraction so that
>
> No. It isn't. Why don't you learn what G actually is.
>
> And I didn't ask anything about some higher level; I asked how the
> assertion that a specific polynomial has an integer solution can be
> 'semantically incoherent'.
>
>> "We are therefore confronted with a proposition which asserts its own
>> unprovability." without any extranelous details.
>
> Repeatedly quoting this serves no purpose. That's Gödel's commentary,
> *not* G. Gödel assumes that people are actually following along with the
> proof and therefore will grasp his meaning. You are not, and therefore
> do not.
>
>>> How do you define 'semantic coherence'? How does G violate this
>>> definition?
>
> No answer?

∃G ∈ language of F (Incoherent(G) ↔ ((F ⊬ G) ∧ (F ⊬ ¬G)))

>
>>> Is the assertion that "a² + b² = c² has at least one integer
>>> solution" semantically incoherent? If not, how is G different from this?
>
> No answer?
>
> André
>

--
Copyright 2022 Pete Olcott "Talent hits a target no one else can hit;
Genius hits a target no one else can see." Arthur Schopenhauer

On 2022-05-06 13:39, olcott wrote:
> On 5/6/2022 2:27 PM, André G. Isaak wrote:
>> On 2022-05-06 13:23, olcott wrote:
>>> On 5/6/2022 2:11 PM, André G. Isaak wrote:

>>>> How do you define 'semantic coherence'? How does G violate this
>>>> definition?
>>
>> No answer?
>
> ∃G ∈ language of F (Incoherent(G) ↔ ((F ⊬ G) ∧ (F ⊬ ¬G)))

That's not an answer. That just a poorly formalized definition of
'incomplete' with the word 'incomplete' replaced with 'incoherent'. If
'incoherent' is just your synonym for 'incomplete' then it serves no
purpose whatsoever, and certainly doesn't provide any justification for
claiming that G is in any way ill-formed.

A definition of semantic coherence will begin as follows:

An expression E is semantically coherent iff ...

Please provide such a definition and then show how G fails to meet this
definition. Note that this will require you to learn what G actually is.

>>
>>>> Is the assertion that "a² + b² = c² has at least one integer
>>>> solution" semantically incoherent? If not, how is G different from
>>>> this?
>>
>> No answer?

Still no answer?

André

--
To email remove 'invalid' & replace 'gm' with well known Google mail
service.

On 5/6/2022 2:46 PM, André G. Isaak wrote:
> On 2022-05-06 13:39, olcott wrote:
>> On 5/6/2022 2:27 PM, André G. Isaak wrote:
>>> On 2022-05-06 13:23, olcott wrote:
>>>> On 5/6/2022 2:11 PM, André G. Isaak wrote:
>
>>>>> How do you define 'semantic coherence'? How does G violate this
>>>>> definition?
>>>
>>> No answer?
>>
>> ∃G ∈ language of F (Incoherent(G) ↔ ((F ⊬ G) ∧ (F ⊬ ¬G)))
>
> That's not an answer. That just a poorly formalized definition of
> 'incomplete' with the word 'incomplete' replaced with 'incoherent'. If
> 'incoherent' is just your synonym for 'incomplete' then it serves no
> purpose whatsoever, and certainly doesn't provide any justification for
> claiming that G is in any way ill-formed.
>

If a cat cannot lay eggs does that make the cat "incomplete" ?

Everything that is referred to in math as incomplete(F) is actually
incorrect(G).

> A definition of semantic coherence will begin as follows:
>
> An expression E is semantically coherent iff ...
>
> Please provide such a definition and then show how G fails to meet this
> definition. Note that this will require you to learn what G actually is.
>
>>>
>>>>> Is the assertion that "a² + b² = c² has at least one integer
>>>>> solution" semantically incoherent? If not, how is G different from
>>>>> this?
>>>
>>> No answer?
>
> Still no answer?
>
> André
>
>

--
Copyright 2022 Pete Olcott "Talent hits a target no one else can hit;
Genius hits a target no one else can see." Arthur Schopenhauer

On 2022-05-06 13:51, olcott wrote:
> On 5/6/2022 2:46 PM, André G. Isaak wrote:
>> On 2022-05-06 13:39, olcott wrote:
>>> On 5/6/2022 2:27 PM, André G. Isaak wrote:
>>>> On 2022-05-06 13:23, olcott wrote:
>>>>> On 5/6/2022 2:11 PM, André G. Isaak wrote:
>>
>>>>>> How do you define 'semantic coherence'? How does G violate this
>>>>>> definition?
>>>>
>>>> No answer?
>>>
>>> ∃G ∈ language of F (Incoherent(G) ↔ ((F ⊬ G) ∧ (F ⊬ ¬G)))
>>
>> That's not an answer. That just a poorly formalized definition of
>> 'incomplete' with the word 'incomplete' replaced with 'incoherent'. If
>> 'incoherent' is just your synonym for 'incomplete' then it serves no
>> purpose whatsoever, and certainly doesn't provide any justification
>> for claiming that G is in any way ill-formed.
>>
>
> If a cat cannot lay eggs does that make the cat "incomplete" ?

If an analogy has absolutely no connection to the thing which it
purports to be a response to is there any reason I should take it
seriously ?

> Everything that is referred to in math as incomplete(F) is actually
> incorrect(G).

A completely groundless assertion. You can't just declare something to
be 'incorrect' because you don't like it. You need to show exactly what
accepted principles it violates.

>> A definition of semantic coherence will begin as follows:
>>
>> An expression E is semantically coherent iff ...

Still no answer?

>> Please provide such a definition and then show how G fails to meet
>> this definition. Note that this will require you to learn what G
>> actually is.
>>
>>>>
>>>>>> Is the assertion that "a² + b² = c² has at least one integer
>>>>>> solution" semantically incoherent? If not, how is G different from
>>>>>> this?
>>>>
>>>> No answer?
>>
>> Still no answer?

Still no answer?

André

--
To email remove 'invalid' & replace 'gm' with well known Google mail
service.

On 5/6/2022 6:43 AM, Richard Damon wrote:
>
> On 5/5/22 10:37 PM, olcott wrote:
>> On 5/5/2022 9:24 PM, Richard Damon wrote:
>>> On 5/5/22 1:57 PM, olcott wrote:
>
>>>> G is self-contradictory on the theory and non self-contradictory in
>>>> the meta-theory.
>>>
>>> No, because in the theory, G doesn't even reference itself, so it
>>> can't be self-contradictory.
>>>
>>
>> Gödel says:
>> ...We are therefore confronted with a proposition which asserts its
>> own unprovability.
>>
>
> Which is Godel making a comment about G, and not a statement in G itself.
>
> G does not directly mention itself in the Theory.

Gödel says that it does with dodgy words that also says that it does not.

15 In spite of appearances, there is nothing circular about such a
proposition, since it begins by asserting the unprovability of a wholly
determinate formula (namely the q-th in the alphabetical arrangement
with a definite substitution), and only subsequently (and in some way by
accident)does it emerge that this formula is precisely that by which the
proposition was itself expressed.END:(Gödel 1931:39-41)

Gödel's footnote 15 is dodgy in that although it denies the circularity
of his proposition he affirms its circularity in the same paragraph that
he denies it:

Removing the dodgy words from the above.
a proposition...begins by asserting the unprovability of a wholly
determinate formula...this formula is precisely that by which the
proposition was itself expressed.

Paraphrasing the above using less clumsy words:
a proposition asserts the unprovability of a formula that expresses this
same proposition

https://www.researchgate.net/publication/350789898_Prolog_detects_and_rejects_pathological_self_reference_in_the_Godel_sentence

>
>>> I think you don't even know what G is, but have only read the clift
>>> notes edition that actually explain it in the meta-theory.
>>
>>
>
> So this comment of mine is now proven.
>
> And you have prooved to be a Liar and an idiot, as you abolutely don't
> know what you are talking about.

--
Copyright 2022 Pete Olcott "Talent hits a target no one else can hit;
Genius hits a target no one else can see." Arthur Schopenhauer

On 5/6/2022 2:55 PM, André G. Isaak wrote:
> On 2022-05-06 13:51, olcott wrote:
>> On 5/6/2022 2:46 PM, André G. Isaak wrote:
>>> On 2022-05-06 13:39, olcott wrote:
>>>> On 5/6/2022 2:27 PM, André G. Isaak wrote:
>>>>> On 2022-05-06 13:23, olcott wrote:
>>>>>> On 5/6/2022 2:11 PM, André G. Isaak wrote:
>>>
>>>>>>> How do you define 'semantic coherence'? How does G violate this
>>>>>>> definition?
>>>>>
>>>>> No answer?
>>>>
>>>> ∃G ∈ language of F (Incoherent(G) ↔ ((F ⊬ G) ∧ (F ⊬ ¬G)))
>>>
>>> That's not an answer. That just a poorly formalized definition of
>>> 'incomplete' with the word 'incomplete' replaced with 'incoherent'.
>>> If 'incoherent' is just your synonym for 'incomplete' then it serves
>>> no purpose whatsoever, and certainly doesn't provide any
>>> justification for claiming that G is in any way ill-formed.
>>>
>>
>> If a cat cannot lay eggs does that make the cat "incomplete" ?
>
> If an analogy has absolutely no connection to the thing which it
> purports to be a response to is there any reason I should take it
> seriously ?
>
>> Everything that is referred to in math as incomplete(F) is actually
>> incorrect(G).
>
> A completely groundless assertion. You can't just declare something to
> be 'incorrect' because you don't like it. You need to show exactly what
> accepted principles it violates.
>
>>> A definition of semantic coherence will begin as follows:
>>>
>>> An expression E is semantically coherent iff ...
>
> Still no answer?

Should I repeat it a few hundred more times before you notice that I
ever said it?

∃G ∈ language of F (Incoherent(G) ↔ ((F ⊬ G) ∧ (F ⊬ ¬G)))

Wittgenstein says the same thing in my quote of him on page 6
https://www.researchgate.net/publication/333907915_Proof_that_Wittgenstein_is_correct_about_Godel

>
>>> Please provide such a definition and then show how G fails to meet
>>> this definition. Note that this will require you to learn what G
>>> actually is.
>>>
>>>>>
>>>>>>> Is the assertion that "a² + b² = c² has at least one integer
>>>>>>> solution" semantically incoherent? If not, how is G different
>>>>>>> from this?
>>>>>
>>>>> No answer?
>>>
>>> Still no answer?
>
> Still no answer?
>
> André
>

--
Copyright 2022 Pete Olcott "Talent hits a target no one else can hit;
Genius hits a target no one else can see." Arthur Schopenhauer