On 4/20/2024 2:41 AM, Mikko wrote:
On 2024-04-19 02:25:48 +0000, olcott said:
On 4/18/2024 8:58 PM, Richard Damon wrote:
Godel's proof you are quoting from had NOTHING to do with undecidability, >>>*Mendelson (and everyone that knows these things) disagrees*
https://sistemas.fciencias.unam.mx/~lokylog/images/Notas/la_aldea_de_la_logica/Libros_notas_varios/L_02_MENDELSON,%20E%20-%20Introduction%20to%20Mathematical%20Logic,%206th%20Ed%20-%20CRC%20Press%20(2015).pdf
On questions whether Gödel said something or not the sumpreme authority
is not Mendelson but Gödel.
When some authors affirm that undecidability and incompleteness
are the exact same thing then whenever Gödel uses the term
incompleteness then he is also referring to the term undecidability.
On 4/22/2024 3:26 AM, Mikko wrote:
On 2024-04-21 14:34:44 +0000, olcott said:
On 4/21/2024 2:50 AM, Mikko wrote:
On 2024-04-20 16:37:27 +0000, olcott said:
On 4/20/2024 2:41 AM, Mikko wrote:
On 2024-04-19 02:25:48 +0000, olcott said:
On 4/18/2024 8:58 PM, Richard Damon wrote:
Godel's proof you are quoting from had NOTHING to do with undecidability,
*Mendelson (and everyone that knows these things) disagrees*
https://sistemas.fciencias.unam.mx/~lokylog/images/Notas/la_aldea_de_la_logica/Libros_notas_varios/L_02_MENDELSON,%20E%20-%20Introduction%20to%20Mathematical%20Logic,%206th%20Ed%20-%20CRC%20Press%20(2015).pdf
On questions whether Gödel said something or not the sumpreme authority >>>>>> is not Mendelson but Gödel.
When some authors affirm that undecidability and incompleteness
are the exact same thing then whenever Gödel uses the term
incompleteness then he is also referring to the term undecidability.
That does not follow. Besides, a reference to the term "undecidability" >>>> is not a reference to the concept 'undecidability'.
In other words you deny the identity principle thus X=X is false.
It is not a good idea to lie where the truth can be seen.
That is the best that I could make about the above quote. There is no standard practice of using different kind of quotes that I am aware of."undecidability" is not a reference to the concept 'undecidability'.
On 4/22/2024 10:14 AM, Mikko wrote:
On 2024-04-22 14:03:05 +0000, olcott said:
On 4/22/2024 3:26 AM, Mikko wrote:
On 2024-04-21 14:34:44 +0000, olcott said:standard practice of using different kind of quotes that I am aware of.
On 4/21/2024 2:50 AM, Mikko wrote:
On 2024-04-20 16:37:27 +0000, olcott said:
On 4/20/2024 2:41 AM, Mikko wrote:That does not follow. Besides, a reference to the term "undecidability" >>>>>> is not a reference to the concept 'undecidability'.
On 2024-04-19 02:25:48 +0000, olcott said:
On 4/18/2024 8:58 PM, Richard Damon wrote:
Godel's proof you are quoting from had NOTHING to do with undecidability,
*Mendelson (and everyone that knows these things) disagrees* >>>>>>>>>
https://sistemas.fciencias.unam.mx/~lokylog/images/Notas/la_aldea_de_la_logica/Libros_notas_varios/L_02_MENDELSON,%20E%20-%20Introduction%20to%20Mathematical%20Logic,%206th%20Ed%20-%20CRC%20Press%20(2015).pdf
On questions whether Gödel said something or not the sumpreme authority
is not Mendelson but Gödel.
When some authors affirm that undecidability and incompleteness
are the exact same thing then whenever Gödel uses the term
incompleteness then he is also referring to the term undecidability. >>>>>>
In other words you deny the identity principle thus X=X is false.
It is not a good idea to lie where the truth can be seen.
;"undecidability" is not a reference to the concept 'undecidability'. >>> That is the best that I could make about the above quote. There is no
Dishonest partial quoting is not a good idea, either.
It is like you are saying "cats" are not 'cats'
On 4/23/2024 3:10 AM, Mikko wrote:
On 2024-04-22 15:49:56 +0000, olcott said:
On 4/22/2024 10:14 AM, Mikko wrote:
On 2024-04-22 14:03:05 +0000, olcott said:
On 4/22/2024 3:26 AM, Mikko wrote:Dishonest partial quoting is not a good idea, either.
On 2024-04-21 14:34:44 +0000, olcott said:
On 4/21/2024 2:50 AM, Mikko wrote:It is not a good idea to lie where the truth can be seen.
On 2024-04-20 16:37:27 +0000, olcott said:
On 4/20/2024 2:41 AM, Mikko wrote:That does not follow. Besides, a reference to the term "undecidability"
On 2024-04-19 02:25:48 +0000, olcott said:
On 4/18/2024 8:58 PM, Richard Damon wrote:
Godel's proof you are quoting from had NOTHING to do with undecidability,
*Mendelson (and everyone that knows these things) disagrees* >>>>>>>>>>>
https://sistemas.fciencias.unam.mx/~lokylog/images/Notas/la_aldea_de_la_logica/Libros_notas_varios/L_02_MENDELSON,%20E%20-%20Introduction%20to%20Mathematical%20Logic,%206th%20Ed%20-%20CRC%20Press%20(2015).pdf
On questions whether Gödel said something or not the sumpreme authority
is not Mendelson but Gödel.
When some authors affirm that undecidability and incompleteness >>>>>>>>> are the exact same thing then whenever Gödel uses the term
incompleteness then he is also referring to the term undecidability. >>>>>>>>
is not a reference to the concept 'undecidability'.
In other words you deny the identity principle thus X=X is false. >>>>>>
;"undecidability" is not a reference to the concept 'undecidability'. >>>>> That is the best that I could make about the above quote. There is no >>>>> standard practice of using different kind of quotes that I am aware of. >>>>
It is like you are saying "cats" are not 'cats'
There is nothing like "cats" or 'cats' in the part of sentence "Besides,
a reference to the term" that you deceptively omitted.
Gibberish nonsense:
"undecidability" is not a reference to the concept 'undecidability'.
On 4/22/2024 5:54 PM, Richard Damon wrote:
On 4/22/24 10:03 AM, olcott wrote:
On 4/22/2024 3:26 AM, Mikko wrote:
On 2024-04-21 14:34:44 +0000, olcott said:standard practice of using different kind of quotes that I am aware of.
On 4/21/2024 2:50 AM, Mikko wrote:
On 2024-04-20 16:37:27 +0000, olcott said:
On 4/20/2024 2:41 AM, Mikko wrote:That does not follow. Besides, a reference to the term "undecidability" >>>>>> is not a reference to the concept 'undecidability'.
On 2024-04-19 02:25:48 +0000, olcott said:
On 4/18/2024 8:58 PM, Richard Damon wrote:
Godel's proof you are quoting from had NOTHING to do with undecidability,
*Mendelson (and everyone that knows these things) disagrees* >>>>>>>>>
https://sistemas.fciencias.unam.mx/~lokylog/images/Notas/la_aldea_de_la_logica/Libros_notas_varios/L_02_MENDELSON,%20E%20-%20Introduction%20to%20Mathematical%20Logic,%206th%20Ed%20-%20CRC%20Press%20(2015).pdf
On questions whether Gödel said something or not the sumpreme authority
is not Mendelson but Gödel.
When some authors affirm that undecidability and incompleteness
are the exact same thing then whenever Gödel uses the term
incompleteness then he is also referring to the term undecidability. >>>>>>
In other words you deny the identity principle thus X=X is false.
It is not a good idea to lie where the truth can be seen.
;"undecidability" is not a reference to the concept 'undecidability'. >>> That is the best that I could make about the above quote. There is no
Except that undeciability and incompleteness are not the EXACT same thing. >>
So you were paying attention?
He said that undecidability is not the same thing as undecidability.
Somehow he felt that two different kinds of quotes mean something.
On 4/22/2024 3:26 AM, Mikko wrote:
On 2024-04-21 14:34:44 +0000, olcott said:
On 4/21/2024 2:50 AM, Mikko wrote:
On 2024-04-20 16:37:27 +0000, olcott said:
On 4/20/2024 2:41 AM, Mikko wrote:
On 2024-04-19 02:25:48 +0000, olcott said:
On 4/18/2024 8:58 PM, Richard Damon wrote:
Godel's proof you are quoting from had NOTHING to do with undecidability,
*Mendelson (and everyone that knows these things) disagrees*
https://sistemas.fciencias.unam.mx/~lokylog/images/Notas/la_aldea_de_la_logica/Libros_notas_varios/L_02_MENDELSON,%20E%20-%20Introduction%20to%20Mathematical%20Logic,%206th%20Ed%20-%20CRC%20Press%20(2015).pdf
On questions whether Gödel said something or not the sumpreme authority >>>>>> is not Mendelson but Gödel.
When some authors affirm that undecidability and incompleteness
are the exact same thing then whenever Gödel uses the term
incompleteness then he is also referring to the term undecidability.
That does not follow. Besides, a reference to the term "undecidability" >>>> is not a reference to the concept 'undecidability'.
In other words you deny the identity principle thus X=X is false.
It is not a good idea to lie where the truth can be seen.
It is not a good idea to say gibberish nonsense and
expect it to be understood.
a reference to the term "undecidability"
is not a reference to the concept 'undecidability'.
An undecidable sentence of a theory K is a closed wf ℬ of K such that
neither ℬ nor ¬ℬ is a theorem of K, that is, such that not-⊢K ℬ and
not-⊢K ¬ℬ. (Mendelson: 2015:208)
So that is what "undecideble" means in Mendelson: 2015. Elsewhere it may
mean something else.
It usually means one cannot make up one's mind.
In math it means an epistemological antinomy expression
is not a proposition thus a type mismatch error for every
bivalent system of logic.
not-⊢K ℬ and not-⊢K ¬ℬ. (Mendelson: 2015:208)
K ⊬ ℬ and K ⊬ ¬ℬ. // switching notational conventions
Incomplete(F) ≡ ∃x ∈ L ((L ⊬ x) ∧ (L ⊬ ¬x))
So not the same.
When an expression cannot be proved or refuted is a formal system
this is exactly the same as an expression cannot be proved or refuted
in a formal system.
On 4/24/2024 4:49 AM, Mikko wrote:
On 2024-04-23 14:54:09 +0000, olcott said:
On 4/22/2024 3:26 AM, Mikko wrote:
On 2024-04-21 14:34:44 +0000, olcott said:
On 4/21/2024 2:50 AM, Mikko wrote:
On 2024-04-20 16:37:27 +0000, olcott said:
On 4/20/2024 2:41 AM, Mikko wrote:That does not follow. Besides, a reference to the term "undecidability" >>>>>> is not a reference to the concept 'undecidability'.
On 2024-04-19 02:25:48 +0000, olcott said:
On 4/18/2024 8:58 PM, Richard Damon wrote:
Godel's proof you are quoting from had NOTHING to do with undecidability,
*Mendelson (and everyone that knows these things) disagrees* >>>>>>>>>
https://sistemas.fciencias.unam.mx/~lokylog/images/Notas/la_aldea_de_la_logica/Libros_notas_varios/L_02_MENDELSON,%20E%20-%20Introduction%20to%20Mathematical%20Logic,%206th%20Ed%20-%20CRC%20Press%20(2015).pdf
On questions whether Gödel said something or not the sumpreme authority
is not Mendelson but Gödel.
When some authors affirm that undecidability and incompleteness
are the exact same thing then whenever Gödel uses the term
incompleteness then he is also referring to the term undecidability. >>>>>>
In other words you deny the identity principle thus X=X is false.
It is not a good idea to lie where the truth can be seen.
It is not a good idea to say gibberish nonsense and
expect it to be understood.
a reference to the term "undecidability"
is not a reference to the concept 'undecidability'.
That is how a sentence must be quoted. The proof that the quoted
sentence can be understood is that Richard Damon undesstood it.
An undecidable sentence of a theory K is a closed wf ℬ of K such that >>>>> neither ℬ nor ¬ℬ is a theorem of K, that is, such that not-⊢K ℬ and
not-⊢K ¬ℬ. (Mendelson: 2015:208)
So that is what "undecideble" means in Mendelson: 2015. Elsewhere it may >>>> mean something else.
It usually means one cannot make up one's mind.
In math it means an epistemological antinomy expression
is not a proposition thus a type mismatch error for every
bivalent system of logic.
No, it doesn't. There is no reference to an epistemological
anitnomy in "undecidable".
...14 Every epistemological antinomy can likewise be used for a similar undecidability proof...(Gödel 1931:43-44)
On 4/25/2024 3:26 AM, Mikko wrote:
epistemological antinomy
It <is> part of the current (thus incorrect) definition
of undecidability because expressions of language that
are neither true nor false (epistemological antinomies)
do prove undecidability even though these expressions
are not truth bearers thus not propositions.
On 4/26/2024 3:42 AM, Mikko wrote:
On 2024-04-25 14:27:23 +0000, olcott said:
On 4/25/2024 3:26 AM, Mikko wrote:
epistemological antinomy
It <is> part of the current (thus incorrect) definition
of undecidability because expressions of language that
are neither true nor false (epistemological antinomies)
do prove undecidability even though these expressions
are not truth bearers thus not propositions.
That a definition is current does not mean that is incorrect.
...14 Every epistemological antinomy can likewise be used for a similar undecidability proof...(Gödel 1931:43-44)
An epistemological antinomy can only be an undecidable sentence
if it can be a sentence. What epistemological antinomies you
can find that can be expressed in, say, first order goup theory
or first order arithmetic or first order set tehory?
It only matters that they can be expressed in some formal system.
If they cannot be expressed in any formal system then Gödel is
wrong for a different reason.
On 4/27/2024 3:18 AM, Mikko wrote:
On 2024-04-26 15:28:08 +0000, olcott said:
On 4/26/2024 3:42 AM, Mikko wrote:
On 2024-04-25 14:27:23 +0000, olcott said:
On 4/25/2024 3:26 AM, Mikko wrote:
epistemological antinomy
It <is> part of the current (thus incorrect) definition
of undecidability because expressions of language that
are neither true nor false (epistemological antinomies)
do prove undecidability even though these expressions
are not truth bearers thus not propositions.
That a definition is current does not mean that is incorrect.
...14 Every epistemological antinomy can likewise be used for a similar
undecidability proof...(Gödel 1931:43-44)
An epistemological antinomy can only be an undecidable sentence
if it can be a sentence. What epistemological antinomies you
can find that can be expressed in, say, first order goup theory
or first order arithmetic or first order set tehory?
It only matters that they can be expressed in some formal system.
If they cannot be expressed in any formal system then Gödel is
wrong for a different reason.
How is it relevant to the incompleteness of a theory whether an
epistemological antińomy can be expressed in some other formal
system?
When an expression of language cannot be proved in a formal system only because it is contradictory in this formal system then the inability to
prove this expression does not place any actual limit on what can be
proven because formal system are not supposed to prove contradictions.
On 4/28/2024 4:34 AM, Mikko wrote:
On 2024-04-27 13:36:56 +0000, olcott said:
On 4/27/2024 3:18 AM, Mikko wrote:
On 2024-04-26 15:28:08 +0000, olcott said:
On 4/26/2024 3:42 AM, Mikko wrote:
On 2024-04-25 14:27:23 +0000, olcott said:
On 4/25/2024 3:26 AM, Mikko wrote:
epistemological antinomy
It <is> part of the current (thus incorrect) definition
of undecidability because expressions of language that
are neither true nor false (epistemological antinomies)
do prove undecidability even though these expressions
are not truth bearers thus not propositions.
That a definition is current does not mean that is incorrect.
...14 Every epistemological antinomy can likewise be used for a similar >>>>> undecidability proof...(Gödel 1931:43-44)
An epistemological antinomy can only be an undecidable sentence
if it can be a sentence. What epistemological antinomies you
can find that can be expressed in, say, first order goup theory
or first order arithmetic or first order set tehory?
It only matters that they can be expressed in some formal system.
If they cannot be expressed in any formal system then Gödel is
wrong for a different reason.
How is it relevant to the incompleteness of a theory whether an
epistemological antińomy can be expressed in some other formal
system?
When an expression of language cannot be proved in a formal system only
because it is contradictory in this formal system then the inability to
prove this expression does not place any actual limit on what can be
proven because formal system are not supposed to prove contradictions.
The first order theories of Peano arithmetic, ZFC set theory, and
group theroy are said to be incomplete but you have not shown any
fromula of any of them that could be called an epistemoloigcal
antinomy.
The details of the semantics of the inference steps are hidden behind arithmetization and diagonalization in Gödel's actual proof.
($) ⊢k G ⇔ (∀x2) ¬𝒫𝑓 (x2, ⌜G⌝)
Observe that, in terms of the standard interpretation (∀x2) ¬𝒫𝑓 (x2, ⌜G⌝) says that there is no natural number that is the Gödel number of a proof in K of the wf G, which is equivalent to asserting that there is
no proof in K of G.
On 4/29/2024 4:09 AM, Mikko wrote:
On 2024-04-28 13:41:50 +0000, olcott said:
On 4/28/2024 4:34 AM, Mikko wrote:
On 2024-04-27 13:36:56 +0000, olcott said:
On 4/27/2024 3:18 AM, Mikko wrote:The first order theories of Peano arithmetic, ZFC set theory, and
On 2024-04-26 15:28:08 +0000, olcott said:
On 4/26/2024 3:42 AM, Mikko wrote:
On 2024-04-25 14:27:23 +0000, olcott said:
On 4/25/2024 3:26 AM, Mikko wrote:
epistemological antinomy
It <is> part of the current (thus incorrect) definition
of undecidability because expressions of language that
are neither true nor false (epistemological antinomies)
do prove undecidability even though these expressions
are not truth bearers thus not propositions.
That a definition is current does not mean that is incorrect.
...14 Every epistemological antinomy can likewise be used for a similar >>>>>>> undecidability proof...(Gödel 1931:43-44)
An epistemological antinomy can only be an undecidable sentence >>>>>>>> if it can be a sentence. What epistemological antinomies you
can find that can be expressed in, say, first order goup theory >>>>>>>> or first order arithmetic or first order set tehory?
It only matters that they can be expressed in some formal system. >>>>>>> If they cannot be expressed in any formal system then Gödel is
wrong for a different reason.
How is it relevant to the incompleteness of a theory whether an
epistemological antińomy can be expressed in some other formal
system?
When an expression of language cannot be proved in a formal system only >>>>> because it is contradictory in this formal system then the inability to >>>>> prove this expression does not place any actual limit on what can be >>>>> proven because formal system are not supposed to prove contradictions. >>>>
group theroy are said to be incomplete but you have not shown any
fromula of any of them that could be called an epistemoloigcal
antinomy.
The details of the semantics of the inference steps are hidden behind
arithmetization and diagonalization in Gödel's actual proof.
The correctness of a proof can be checked without any consideration of
semantics. If the proof is fully formal there is an algorithm to check
the correctness.
($) ⊢k G ⇔ (∀x2) ¬𝒫𝑓 (x2, ⌜G⌝)
Observe that, in terms of the standard interpretation (∀x2) ¬𝒫𝑓 (x2,
⌜G⌝) says that there is no natural number that is the Gödel number of a
proof in K of the wf G, which is equivalent to asserting that there is
no proof in K of G.
The standard interpretation of artihmetic does not say anything about
proofs and Gödel numbers.
That was a direct quote from a math textbook, here it is again:
On 4/29/2024 10:13 AM, Mikko wrote:
On 2024-04-29 14:22:36 +0000, olcott said:
On 4/29/2024 4:09 AM, Mikko wrote:
On 2024-04-28 13:41:50 +0000, olcott said:
On 4/28/2024 4:34 AM, Mikko wrote:
On 2024-04-27 13:36:56 +0000, olcott said:
On 4/27/2024 3:18 AM, Mikko wrote:The first order theories of Peano arithmetic, ZFC set theory, and
On 2024-04-26 15:28:08 +0000, olcott said:
On 4/26/2024 3:42 AM, Mikko wrote:
On 2024-04-25 14:27:23 +0000, olcott said:
On 4/25/2024 3:26 AM, Mikko wrote:
epistemological antinomy
It <is> part of the current (thus incorrect) definition
of undecidability because expressions of language that
are neither true nor false (epistemological antinomies)
do prove undecidability even though these expressions
are not truth bearers thus not propositions.
That a definition is current does not mean that is incorrect. >>>>>>>>>>
...14 Every epistemological antinomy can likewise be used for a similar
undecidability proof...(Gödel 1931:43-44)
An epistemological antinomy can only be an undecidable sentence >>>>>>>>>> if it can be a sentence. What epistemological antinomies you >>>>>>>>>> can find that can be expressed in, say, first order goup theory >>>>>>>>>> or first order arithmetic or first order set tehory?
It only matters that they can be expressed in some formal system. >>>>>>>>> If they cannot be expressed in any formal system then Gödel is >>>>>>>>> wrong for a different reason.
How is it relevant to the incompleteness of a theory whether an >>>>>>>> epistemological antińomy can be expressed in some other formal >>>>>>>> system?
When an expression of language cannot be proved in a formal system only >>>>>>> because it is contradictory in this formal system then the inability to >>>>>>> prove this expression does not place any actual limit on what can be >>>>>>> proven because formal system are not supposed to prove contradictions. >>>>>>
group theroy are said to be incomplete but you have not shown any
fromula of any of them that could be called an epistemoloigcal
antinomy.
The details of the semantics of the inference steps are hidden behind >>>>> arithmetization and diagonalization in Gödel's actual proof.
The correctness of a proof can be checked without any consideration of >>>> semantics. If the proof is fully formal there is an algorithm to check >>>> the correctness.
($) ⊢k G ⇔ (∀x2) ¬𝒫𝑓 (x2, ⌜G⌝)
Observe that, in terms of the standard interpretation (∀x2) ¬𝒫𝑓 (x2,
⌜G⌝) says that there is no natural number that is the Gödel number of a
proof in K of the wf G, which is equivalent to asserting that there is >>>>> no proof in K of G.
The standard interpretation of artihmetic does not say anything about
proofs and Gödel numbers.
That was a direct quote from a math textbook, here it is again:
That quote didn't define "standard semantics".
It need not define standard semantics once is has summed of the essence
of that whole proof as: G says “I am not provable in K”.
On 4/30/2024 7:04 AM, Mikko wrote:
On 2024-04-29 15:26:23 +0000, olcott said:
On 4/29/2024 10:13 AM, Mikko wrote:
On 2024-04-29 14:22:36 +0000, olcott said:
On 4/29/2024 4:09 AM, Mikko wrote:
On 2024-04-28 13:41:50 +0000, olcott said:
On 4/28/2024 4:34 AM, Mikko wrote:
On 2024-04-27 13:36:56 +0000, olcott said:
On 4/27/2024 3:18 AM, Mikko wrote:
On 2024-04-26 15:28:08 +0000, olcott said:
On 4/26/2024 3:42 AM, Mikko wrote:
On 2024-04-25 14:27:23 +0000, olcott said:
On 4/25/2024 3:26 AM, Mikko wrote:
epistemological antinomy
It <is> part of the current (thus incorrect) definition >>>>>>>>>>>>> of undecidability because expressions of language that >>>>>>>>>>>>> are neither true nor false (epistemological antinomies) >>>>>>>>>>>>> do prove undecidability even though these expressions >>>>>>>>>>>>> are not truth bearers thus not propositions.
That a definition is current does not mean that is incorrect. >>>>>>>>>>>>
...14 Every epistemological antinomy can likewise be used for a similar
undecidability proof...(Gödel 1931:43-44)
An epistemological antinomy can only be an undecidable sentence >>>>>>>>>>>> if it can be a sentence. What epistemological antinomies you >>>>>>>>>>>> can find that can be expressed in, say, first order goup theory >>>>>>>>>>>> or first order arithmetic or first order set tehory?
It only matters that they can be expressed in some formal system. >>>>>>>>>>> If they cannot be expressed in any formal system then Gödel is >>>>>>>>>>> wrong for a different reason.
How is it relevant to the incompleteness of a theory whether an >>>>>>>>>> epistemological antińomy can be expressed in some other formal >>>>>>>>>> system?
When an expression of language cannot be proved in a formal system only
because it is contradictory in this formal system then the inability to
prove this expression does not place any actual limit on what can be >>>>>>>>> proven because formal system are not supposed to prove contradictions.
The first order theories of Peano arithmetic, ZFC set theory, and >>>>>>>> group theroy are said to be incomplete but you have not shown any >>>>>>>> fromula of any of them that could be called an epistemoloigcal >>>>>>>> antinomy.
The details of the semantics of the inference steps are hidden behind >>>>>>> arithmetization and diagonalization in Gödel's actual proof.
The correctness of a proof can be checked without any consideration of >>>>>> semantics. If the proof is fully formal there is an algorithm to check >>>>>> the correctness.
($) ⊢k G ⇔ (∀x2) ¬𝒫𝑓 (x2, ⌜G⌝)
Observe that, in terms of the standard interpretation (∀x2) ¬𝒫𝑓 (x2,
⌜G⌝) says that there is no natural number that is the Gödel number of a
proof in K of the wf G, which is equivalent to asserting that there is >>>>>>> no proof in K of G.
The standard interpretation of artihmetic does not say anything about >>>>>> proofs and Gödel numbers.
That was a direct quote from a math textbook, here it is again:
That quote didn't define "standard semantics".
It need not define standard semantics once is has summed of the essence
of that whole proof as: G says “I am not provable in K”.
In the standard semantics of arithmetic nothing means "I am not provable
in K". That simply is not an arithmetic statement about numbers.
Mendelson says that Gödel's actual proof is equivalent to:
G says “I am not provable in K”.
On 5/1/2024 4:01 AM, Mikko wrote:
On 2024-04-30 16:08:08 +0000, olcott said:
On 4/30/2024 7:04 AM, Mikko wrote:
On 2024-04-29 15:26:23 +0000, olcott said:
On 4/29/2024 10:13 AM, Mikko wrote:
On 2024-04-29 14:22:36 +0000, olcott said:
On 4/29/2024 4:09 AM, Mikko wrote:
On 2024-04-28 13:41:50 +0000, olcott said:
On 4/28/2024 4:34 AM, Mikko wrote:The correctness of a proof can be checked without any consideration of >>>>>>>> semantics. If the proof is fully formal there is an algorithm to check >>>>>>>> the correctness.
On 2024-04-27 13:36:56 +0000, olcott said:
On 4/27/2024 3:18 AM, Mikko wrote:
On 2024-04-26 15:28:08 +0000, olcott said:
On 4/26/2024 3:42 AM, Mikko wrote:
On 2024-04-25 14:27:23 +0000, olcott said:
On 4/25/2024 3:26 AM, Mikko wrote:
epistemological antinomy
It <is> part of the current (thus incorrect) definition >>>>>>>>>>>>>>> of undecidability because expressions of language that >>>>>>>>>>>>>>> are neither true nor false (epistemological antinomies) >>>>>>>>>>>>>>> do prove undecidability even though these expressions >>>>>>>>>>>>>>> are not truth bearers thus not propositions.
That a definition is current does not mean that is incorrect. >>>>>>>>>>>>>>
...14 Every epistemological antinomy can likewise be used for a similar
undecidability proof...(Gödel 1931:43-44)
An epistemological antinomy can only be an undecidable sentence >>>>>>>>>>>>>> if it can be a sentence. What epistemological antinomies you >>>>>>>>>>>>>> can find that can be expressed in, say, first order goup theory >>>>>>>>>>>>>> or first order arithmetic or first order set tehory? >>>>>>>>>>>>>>
It only matters that they can be expressed in some formal system. >>>>>>>>>>>>> If they cannot be expressed in any formal system then Gödel is >>>>>>>>>>>>> wrong for a different reason.
How is it relevant to the incompleteness of a theory whether an >>>>>>>>>>>> epistemological antińomy can be expressed in some other formal >>>>>>>>>>>> system?
When an expression of language cannot be proved in a formal system only
because it is contradictory in this formal system then the inability to
prove this expression does not place any actual limit on what can be
proven because formal system are not supposed to prove contradictions.
The first order theories of Peano arithmetic, ZFC set theory, and >>>>>>>>>> group theroy are said to be incomplete but you have not shown any >>>>>>>>>> fromula of any of them that could be called an epistemoloigcal >>>>>>>>>> antinomy.
The details of the semantics of the inference steps are hidden behind >>>>>>>>> arithmetization and diagonalization in Gödel's actual proof. >>>>>>>>
($) ⊢k G ⇔ (∀x2) ¬𝒫𝑓 (x2, ⌜G⌝)
Observe that, in terms of the standard interpretation (∀x2) ¬𝒫𝑓 (x2,
⌜G⌝) says that there is no natural number that is the Gödel number of a
proof in K of the wf G, which is equivalent to asserting that there is
no proof in K of G.
The standard interpretation of artihmetic does not say anything about >>>>>>>> proofs and Gödel numbers.
That was a direct quote from a math textbook, here it is again:
That quote didn't define "standard semantics".
It need not define standard semantics once is has summed of the essence >>>>> of that whole proof as: G says “I am not provable in K”.
In the standard semantics of arithmetic nothing means "I am not provable >>>> in K". That simply is not an arithmetic statement about numbers.
Mendelson says that Gödel's actual proof is equivalent to:
G says “I am not provable in K”.
For a certain kind of equivalence. But they are not semantically equivalent: >> G is an arithmetic sentence and in the standard smeantics it is interpreted >> as an arithmetic statement about numbers. The statement "I am not provable >> in K" is an English sentence and in the standard semantics of English it
does not refer to numbers.
You can regard the two sentences equivalent if you use some non-stadard
semantics for arithmetic or English or both. Or you may try to find a
mapping from one system to another that has the necessary properties to
prove someting.
G says “I am not provable in K”. // Mendelson
...We are therefore confronted with a proposition which asserts its own unprovability. 15 ...(Gödel 1931:43-44)
This is the correct way to encode that:
∃G ∈ K (G := (K ⊬ G))
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