...14 Every epistemological antinomy can likewise be
used for a similar undecidability proof...(Gödel 1931:43-44)
An epistemological antinomy is a self-contradictory expression.
∀L ∈ Formal_System
(Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x)))
The only possible place to insert Gödel's above reference
to an epistemological antinomy in the above definition of
Incompleteness is x.
So Gödel's quote is saying that a formal system <is> incomplete
when it cannot prove or refute a self-contradictory expression.
He says this even though the actual reason that self-contradictory expressions cannot be proven or refuted is that there is something
wrong with them.
...14 Every epistemological antinomy can likewise be
used for a similar undecidability proof...(Gödel 1931:43-44)
An epistemological antinomy is a self-contradictory expression.
∀L ∈ Formal_System
(Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x)))
The only possible place to insert Gödel's above reference
to an epistemological antinomy in the above definition of
Incompleteness is x.
So Gödel's quote is saying that a formal system <is> incomplete
when it cannot prove or refute a self-contradictory expression.
On 11/20/2023 3:39 PM, olcott wrote:
...14 Every epistemological antinomy can likewise be
used for a similar undecidability proof...(Gödel 1931:43-44)
An epistemological antinomy is a self-contradictory expression.
∀L ∈ Formal_System
(Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x)))
The only possible place to insert Gödel's above reference
to an epistemological antinomy in the above definition of
Incompleteness is x.
When I do this then we can see that the definition of incompleteness determines that a formal system is incomplete when it cannot prove self-contradictory expressions.
Since self-contradictory expressions are defective then this proves
that the definition of incompleteness is terribly incorrect.
So Gödel's quote is saying that a formal system <is> incomplete
when it cannot prove or refute a self-contradictory expression.
When I insert the term {epistemological antinomy} from his above quote
then we see that Gödel is affirming that defective expressions do prove
that a formal system is incomplete, never noticing that the real issue
is that these expressions are defective.
On 11/20/2023 3:39 PM, olcott wrote:
...14 Every epistemological antinomy can likewise be
used for a similar undecidability proof...(Gödel 1931:43-44)
An epistemological antinomy is a self-contradictory expression.
∀L ∈ Formal_System
(Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x)))
The first incompleteness theorem states that in any consistent formal
system F within which a certain amount of arithmetic can be carried out, there are statements of the language of F which can neither be proved
nor disproved in F.
https://plato.stanford.edu/entries/goedel-incompleteness/
The only possible place to insert Gödel's above reference
to an epistemological antinomy in the above definition of
Incompleteness is x.
So Gödel's quote is saying that a formal system <is> incomplete
when it cannot prove or refute a self-contradictory expression.
He says this even though the actual reason that self-contradictory
expressions cannot be proven or refuted is that there is something
wrong with them.
On 11/20/2023 3:39 PM, olcott wrote:
...14 Every epistemological antinomy can likewise be
used for a similar undecidability proof...(Gödel 1931:43-44)
An epistemological antinomy is a self-contradictory expression.
∀L ∈ Formal_System
(Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x)))
The only possible place to insert Gödel's above reference
to an epistemological antinomy in the above definition of
Incompleteness is x.
When I do this then we can see that the definition of incompleteness determines that a formal system is incomplete when it cannot prove self-contradictory expressions.
Since self-contradictory expressions are defective then this proves
that the definition of incompleteness is terribly incorrect.
So Gödel's quote is saying that a formal system <is> incomplete
when it cannot prove or refute a self-contradictory expression.
When I insert the term {epistemological antinomy} from his above quote
then we see that Gödel is affirming that defective expressions do prove
that a formal system is incomplete, never noticing that the real issue
is that these expressions are defective.
...14 Every epistemological antinomy can likewise be
used for a similar undecidability proof...(Gödel 1931:43-44)
An epistemological antinomy is a self-contradictory expression.
∀L ∈ Formal_System
(Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x)))
The only possible place to insert Gödel's above reference
to an epistemological antinomy in the above definition of
Incompleteness is x.
So Gödel's quote is saying that a formal system <is> incomplete
when it cannot prove or refute a self-contradictory expression.
He says this even though the actual reason that self-contradictory expressions cannot be proven or refuted is that there is something
wrong with them.
...14 Every epistemological antinomy can likewise be
used for a similar undecidability proof...(Gödel 1931:43-44)
An epistemological antinomy is a self-contradictory expression.
∀L ∈ Formal_System
(Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x)))
The only possible place to insert Gödel's above reference
to an epistemological antinomy in the above definition of
Incompleteness is x.
So Gödel's quote is saying that a formal system <is> incomplete
when it cannot prove or refute a self-contradictory expression.
He says this even though the actual reason that self-contradictory expressions cannot be proven or refuted is that there is something
wrong with them.
On 11/20/2023 3:39 PM, olcott wrote:
...14 Every epistemological antinomy can likewise be
used for a similar undecidability proof...(Gödel 1931:43-44)
"x can NOT be an epistemological antinomy in a normal logic
system, as those are not members of the Language(L)"
In other words you are saying that Gödel was referring
to a situation that he knew can't possibly ever occur.
*I think that it may be time for you to give up rebuttal mode*
An epistemological antinomy is a self-contradictory expression.
∀L ∈ Formal_System
(Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x)))
The only possible place to insert Gödel's above reference
to an epistemological antinomy in the above definition of
Incompleteness is x.
So Gödel's quote is saying that a formal system <is> incomplete
when it cannot prove or refute a self-contradictory expression.
He says this even though the actual reason that self-contradictory
expressions cannot be proven or refuted is that there is something
wrong with them.
On 11/20/2023 3:39 PM, olcott wrote:
...14 Every epistemological antinomy can likewise be
used for a similar undecidability proof...(Gödel 1931:43-44)
"x can NOT be an epistemological antinomy in a normal logic
system, as those are not members of the Language(L)"
In other words you are saying that Gödel was referring
to a situation that he knew can't possibly ever occur.
*I think that it may be time for you to give up rebuttal mode*
On 11/20/2023 5:58 PM, olcott wrote:
On 11/20/2023 3:39 PM, olcott wrote:
...14 Every epistemological antinomy can likewise be
used for a similar undecidability proof...(Gödel 1931:43-44)
"x can NOT be an epistemological antinomy in a normal logic
system, as those are not members of the Language(L)"
In other words you are saying that Gödel was referring
to a situation that he knew can't possibly ever occur.
*I think that it may be time for you to give up rebuttal mode*
"His G is most certainly an element of the Language(F)"
I have only been referring to the single quote above
for the last fifty posts and you know this.
I have only been referring to the single quote above
for the last fifty posts and you know this.
I have only been referring to the single quote above
for the last fifty posts and you know this.
I have only been referring to the single quote above
for the last fifty posts and you know this.
I have only been referring to the single quote above
for the last fifty posts and you know this.
I have only been referring to the single quote above
for the last fifty posts and you know this.
I have only been referring to the single quote above
for the last fifty posts and you know this.
I have only been referring to the single quote above
for the last fifty posts and you know this.
I have only been referring to the single quote above
for the last fifty posts and you know this.
I have only been referring to the single quote above
for the last fifty posts and you know this.
I have only been referring to the single quote above
for the last fifty posts and you know this.
I have only been referring to the single quote above
for the last fifty posts and you know this.
I have only been referring to the single quote above
for the last fifty posts and you know this.
I have only been referring to the single quote above
for the last fifty posts and you know this.
I have only been referring to the single quote above
for the last fifty posts and you know this.
...14 Every epistemological antinomy can likewise be
used for a similar undecidability proof...(Gödel 1931:43-44)
An epistemological antinomy is a self-contradictory expression.
∀L ∈ Formal_System
(Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x)))
The only possible place to insert Gödel's above reference
to an epistemological antinomy in the above definition of
Incompleteness is x.
So Gödel's quote is saying that a formal system <is> incomplete
when it cannot prove or refute a self-contradictory expression.
He says this even though the actual reason that self-contradictory expressions cannot be proven or refuted is that there is something
wrong with them.
...14 Every epistemological antinomy can likewise be
used for a similar undecidability proof...(Gödel 1931:43-44)
An epistemological antinomy is a self-contradictory expression.
∀L ∈ Formal_System
(Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x)))
The only possible place to insert Gödel's above reference
to an epistemological antinomy in the above definition of
Incompleteness is x.
So Gödel's quote is saying that a formal system <is> incomplete
when it cannot prove or refute a self-contradictory expression.
He says this even though the actual reason that self-contradictory expressions cannot be proven or refuted is that there is something
wrong with them.
On 11/20/2023 3:39 PM, olcott wrote:
...14 Every epistemological antinomy can likewise be
used for a similar undecidability proof...(Gödel 1931:43-44)
An epistemological antinomy is a self-contradictory expression.
∀L ∈ Formal_System
(Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x)))
The only possible place to insert Gödel's above reference
to an epistemological antinomy in the above definition of
Incompleteness is x.
So Gödel's quote is saying that a formal system <is> incomplete
when it cannot prove or refute a self-contradictory expression.
He says this even though the actual reason that self-contradictory
expressions cannot be proven or refuted is that there is something
wrong with them.
I am only referring to the above Gödel quote and stipulating that
anything else that he ever said or did is an dishonest dodge
way from the point.
I am also only referring to the above definition of incompleteness
thus not any naive paraphrase.
The first incompleteness theorem states that in any consistent formal
system F within which a certain amount of arithmetic can be carried out, there are statements of the language of F which can neither be proved
nor disproved in F.
https://plato.stanford.edu/entries/goedel-incompleteness/
On 11/20/2023 3:39 PM, olcott wrote:
...14 Every epistemological antinomy can likewise be
used for a similar undecidability proof...(Gödel 1931:43-44)
An epistemological antinomy is a self-contradictory expression.
∀L ∈ Formal_System
(Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x)))
The only possible place to insert Gödel's above reference
to an epistemological antinomy in the above definition of
Incompleteness is x.
So Gödel's quote is saying that a formal system <is> incomplete
when it cannot prove or refute a self-contradictory expression.
He says this even though the actual reason that self-contradictory
expressions cannot be proven or refuted is that there is something
wrong with them.
I am only referring to the above Gödel quote and stipulating that
[referring to] anything else that he ever said or did is an dishonest
dodge way from the point.
I am also only referring to the above definition of incompleteness
thus not any naive paraphrase.
The first incompleteness theorem states that in any consistent formal
system F within which a certain amount of arithmetic can be carried out, there are statements of the language of F which can neither be proved
nor disproved in F.
https://plato.stanford.edu/entries/goedel-incompleteness/
On 11/20/2023 6:36 PM, olcott wrote:
On 11/20/2023 3:39 PM, olcott wrote:
...14 Every epistemological antinomy can likewise be
used for a similar undecidability proof...(Gödel 1931:43-44)
An epistemological antinomy is a self-contradictory expression.
∀L ∈ Formal_System
(Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x)))
The only possible place to insert Gödel's above reference
to an epistemological antinomy in the above definition of
Incompleteness is x.
So Gödel's quote is saying that a formal system <is> incomplete
when it cannot prove or refute a self-contradictory expression.
He says this even though the actual reason that self-contradictory
expressions cannot be proven or refuted is that there is something
wrong with them.
I am only referring to the above Gödel quote and stipulating that
[referring to] anything else that he ever said or did is an dishonest
dodge way from the point.
I am also only referring to the above definition of incompleteness
thus not any naive paraphrase.
The first incompleteness theorem states that in any consistent formal
system F within which a certain amount of arithmetic can be carried out,
there are statements of the language of F which can neither be proved
nor disproved in F.
https://plato.stanford.edu/entries/goedel-incompleteness/
∀L ∈ Formal_System
(Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x)))
When we do plug "epistemological antinomy" into x we prove that
the notion of mathematical incompleteness is incorrect because
it determines that L is incomplete on the basis that x is self- contradictory.
On 11/20/2023 6:36 PM, olcott wrote:
On 11/20/2023 3:39 PM, olcott wrote:
...14 Every epistemological antinomy can likewise be
used for a similar undecidability proof...(Gödel 1931:43-44)
An epistemological antinomy is a self-contradictory expression.
∀L ∈ Formal_System
(Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x)))
The only possible place to insert Gödel's above reference
to an epistemological antinomy in the above definition of
Incompleteness is x.
So Gödel's quote is saying that a formal system <is> incomplete
when it cannot prove or refute a self-contradictory expression.
He says this even though the actual reason that self-contradictory
expressions cannot be proven or refuted is that there is something
wrong with them.
I am only referring to the above Gödel quote and stipulating that
[referring to] anything else that he ever said or did is an dishonest
dodge way from the point.
I am also only referring to the above definition of incompleteness
thus not any naive paraphrase.
The first incompleteness theorem states that in any consistent formal
system F within which a certain amount of arithmetic can be carried out,
there are statements of the language of F which can neither be proved
nor disproved in F.
https://plato.stanford.edu/entries/goedel-incompleteness/
∀L ∈ Formal_System
(Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x)))
When we do plug "epistemological antinomy" into x we prove that
the notion of mathematical incompleteness is incorrect because
it determines that L is incomplete on the basis that x is self- contradictory.
On 11/20/2023 8:54 PM, olcott wrote:
On 11/20/2023 6:36 PM, olcott wrote:
On 11/20/2023 3:39 PM, olcott wrote:
...14 Every epistemological antinomy can likewise be
used for a similar undecidability proof...(Gödel 1931:43-44)
An epistemological antinomy is a self-contradictory expression.
∀L ∈ Formal_System
(Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x)))
The only possible place to insert Gödel's above reference
to an epistemological antinomy in the above definition of
Incompleteness is x.
So Gödel's quote is saying that a formal system <is> incomplete
when it cannot prove or refute a self-contradictory expression.
He says this even though the actual reason that self-contradictory
expressions cannot be proven or refuted is that there is something
wrong with them.
I am only referring to the above Gödel quote and stipulating that
[referring to] anything else that he ever said or did is an dishonest
dodge way from the point.
I am also only referring to the above definition of incompleteness
thus not any naive paraphrase.
The first incompleteness theorem states that in any consistent formal
system F within which a certain amount of arithmetic can be carried out, >>> there are statements of the language of F which can neither be proved
nor disproved in F.
https://plato.stanford.edu/entries/goedel-incompleteness/
∀L ∈ Formal_System
(Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x)))
When we do plug "epistemological antinomy" into x we prove that
the notion of mathematical incompleteness is incorrect because
it determines that L is incomplete on the basis that x is self-
contradictory.
And by what justification do you plug a
epistemological antinomy into "x?"
...14 Every epistemological antinomy can likewise be
used for a similar undecidability proof...(Gödel 1931:43-44)
*Paraphrasing that*
No epistemological antinomy can be prohibited from being
used for a similar undecidability proof
On 11/20/2023 8:54 PM, olcott wrote:
On 11/20/2023 6:36 PM, olcott wrote:
On 11/20/2023 3:39 PM, olcott wrote:
...14 Every epistemological antinomy can likewise be
used for a similar undecidability proof...(Gödel 1931:43-44)
An epistemological antinomy is a self-contradictory expression.
∀L ∈ Formal_System
(Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x)))
The only possible place to insert Gödel's above reference
to an epistemological antinomy in the above definition of
Incompleteness is x.
So Gödel's quote is saying that a formal system <is> incomplete
when it cannot prove or refute a self-contradictory expression.
He says this even though the actual reason that self-contradictory
expressions cannot be proven or refuted is that there is something
wrong with them.
I am only referring to the above Gödel quote and stipulating that
[referring to] anything else that he ever said or did is an dishonest
dodge way from the point.
I am also only referring to the above definition of incompleteness
thus not any naive paraphrase.
The first incompleteness theorem states that in any consistent formal
system F within which a certain amount of arithmetic can be carried out, >>> there are statements of the language of F which can neither be proved
nor disproved in F.
https://plato.stanford.edu/entries/goedel-incompleteness/
∀L ∈ Formal_System
(Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x)))
When we do plug "epistemological antinomy" into x we prove that
the notion of mathematical incompleteness is incorrect because
it determines that L is incomplete on the basis that x is self-
contradictory.
And by what justification do you plug a
epistemological antinomy into "x?"
...14 Every epistemological antinomy can likewise be
used for a similar undecidability proof...(Gödel 1931:43-44) >
*Paraphrasing that*
No epistemological antinomy can be prohibited from being
used for a similar undecidability proof
On 11/20/2023 8:54 PM, olcott wrote:
On 11/20/2023 6:36 PM, olcott wrote:
On 11/20/2023 3:39 PM, olcott wrote:
...14 Every epistemological antinomy can likewise be
used for a similar undecidability proof...(Gödel 1931:43-44)
An epistemological antinomy is a self-contradictory expression.
∀L ∈ Formal_System
(Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x)))
The only possible place to insert Gödel's above reference
to an epistemological antinomy in the above definition of
Incompleteness is x.
So Gödel's quote is saying that a formal system <is> incomplete
when it cannot prove or refute a self-contradictory expression.
He says this even though the actual reason that self-contradictory
expressions cannot be proven or refuted is that there is something
wrong with them.
I am only referring to the above Gödel quote and stipulating that
[referring to] anything else that he ever said or did is an dishonest
dodge way from the point.
I am also only referring to the above definition of incompleteness
thus not any naive paraphrase.
The first incompleteness theorem states that in any consistent formal
system F within which a certain amount of arithmetic can be carried out, >>> there are statements of the language of F which can neither be proved
nor disproved in F.
https://plato.stanford.edu/entries/goedel-incompleteness/
∀L ∈ Formal_System
(Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x)))
When we do plug "epistemological antinomy" into x we prove that
the notion of mathematical incompleteness is incorrect because
it determines that L is incomplete on the basis that x is self-
contradictory.
And by what justification do you plug a
epistemological antinomy into "x?"
...14 Every epistemological antinomy can likewise be
used for a similar undecidability proof...(Gödel 1931:43-44)
*Paraphrasing that*
No epistemological antinomy can be prohibited from being
used for a similar undecidability proof
On 11/20/2023 9:13 PM, olcott wrote:
On 11/20/2023 8:54 PM, olcott wrote:
On 11/20/2023 6:36 PM, olcott wrote:
On 11/20/2023 3:39 PM, olcott wrote:
...14 Every epistemological antinomy can likewise be
used for a similar undecidability proof...(Gödel 1931:43-44)
An epistemological antinomy is a self-contradictory expression.
∀L ∈ Formal_System
(Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x))) >>>>>
The only possible place to insert Gödel's above reference
to an epistemological antinomy in the above definition of
Incompleteness is x.
So Gödel's quote is saying that a formal system <is> incomplete
when it cannot prove or refute a self-contradictory expression.
He says this even though the actual reason that self-contradictory
expressions cannot be proven or refuted is that there is something
wrong with them.
I am only referring to the above Gödel quote and stipulating that
[referring to] anything else that he ever said or did is an dishonest
dodge way from the point.
I am also only referring to the above definition of incompleteness
thus not any naive paraphrase.
The first incompleteness theorem states that in any consistent formal
system F within which a certain amount of arithmetic can be carried
out,
there are statements of the language of F which can neither be proved
nor disproved in F.
https://plato.stanford.edu/entries/goedel-incompleteness/
∀L ∈ Formal_System
(Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x)))
When we do plug "epistemological antinomy" into x we prove that
the notion of mathematical incompleteness is incorrect because
it determines that L is incomplete on the basis that x is self-
contradictory.
And by what justification do you plug a
epistemological antinomy into "x?"
...14 Every epistemological antinomy can likewise be
used for a similar undecidability proof...(Gödel 1931:43-44)
*Paraphrasing that*
No epistemological antinomy can be prohibited from being
used for a similar undecidability proof
It <is> the case that an epistemological antinomy inserted in
x does cause the incompleteness criteria to determine that
formal system L <is> incomplete even though the actual issue
is that x is self-contradictory.
The only last step of this is whether or not it is possible
or logically impossible to define a formal system that can
express actual epistemological antinomies in its language.
Another way of saying this is whether or not it is logically
impossible to precisely correctly formalize the Liar Paradox.
If we can precisely formalize the Liar Paradox then we know
that it is not logically impossible to formalize epistemological
antinomies.
On 11/20/2023 9:13 PM, olcott wrote:
On 11/20/2023 8:54 PM, olcott wrote:
On 11/20/2023 6:36 PM, olcott wrote:
On 11/20/2023 3:39 PM, olcott wrote:
...14 Every epistemological antinomy can likewise be
used for a similar undecidability proof...(Gödel 1931:43-44)
An epistemological antinomy is a self-contradictory expression.
∀L ∈ Formal_System
(Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x))) >>>>>
The only possible place to insert Gödel's above reference
to an epistemological antinomy in the above definition of
Incompleteness is x.
So Gödel's quote is saying that a formal system <is> incomplete
when it cannot prove or refute a self-contradictory expression.
He says this even though the actual reason that self-contradictory
expressions cannot be proven or refuted is that there is something
wrong with them.
I am only referring to the above Gödel quote and stipulating that
[referring to] anything else that he ever said or did is an dishonest
dodge way from the point.
I am also only referring to the above definition of incompleteness
thus not any naive paraphrase.
The first incompleteness theorem states that in any consistent formal
system F within which a certain amount of arithmetic can be carried
out,
there are statements of the language of F which can neither be proved
nor disproved in F.
https://plato.stanford.edu/entries/goedel-incompleteness/
∀L ∈ Formal_System
(Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x)))
When we do plug "epistemological antinomy" into x we prove that
the notion of mathematical incompleteness is incorrect because
it determines that L is incomplete on the basis that x is self-
contradictory.
And by what justification do you plug a
epistemological antinomy into "x?"
...14 Every epistemological antinomy can likewise be
used for a similar undecidability proof...(Gödel 1931:43-44)
*Paraphrasing that*
No epistemological antinomy can be prohibited from being
used for a similar undecidability proof
It <is> the case that an epistemological antinomy inserted in
x does cause the incompleteness criteria to determine that
formal system L <is> incomplete even though the actual issue
is that x is self-contradictory.
The only last step of this is whether or not it is possible
or logically impossible to define a formal system that can
express actual epistemological antinomies in its language.
Another way of saying this is whether or not it is logically
impossible to precisely correctly formalize the Liar Paradox.
If we can precisely formalize the Liar Paradox then we know
that it is not logically impossible to formalize epistemological
antinomies.
On 11/20/2023 9:43 PM, olcott wrote:
On 11/20/2023 9:13 PM, olcott wrote:
On 11/20/2023 8:54 PM, olcott wrote:
On 11/20/2023 6:36 PM, olcott wrote:
On 11/20/2023 3:39 PM, olcott wrote:
...14 Every epistemological antinomy can likewise be
used for a similar undecidability proof...(Gödel 1931:43-44)
An epistemological antinomy is a self-contradictory expression.
∀L ∈ Formal_System
(Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x))) >>>>>>
The only possible place to insert Gödel's above reference
to an epistemological antinomy in the above definition of
Incompleteness is x.
So Gödel's quote is saying that a formal system <is> incomplete
when it cannot prove or refute a self-contradictory expression.
He says this even though the actual reason that self-contradictory >>>>>> expressions cannot be proven or refuted is that there is something >>>>>> wrong with them.
I am only referring to the above Gödel quote and stipulating that
[referring to] anything else that he ever said or did is an dishonest >>>>> dodge way from the point.
I am also only referring to the above definition of incompleteness
thus not any naive paraphrase.
The first incompleteness theorem states that in any consistent formal >>>>> system F within which a certain amount of arithmetic can be carried
out,
there are statements of the language of F which can neither be proved >>>>> nor disproved in F.
https://plato.stanford.edu/entries/goedel-incompleteness/
∀L ∈ Formal_System
(Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x)))
When we do plug "epistemological antinomy" into x we prove that
the notion of mathematical incompleteness is incorrect because
it determines that L is incomplete on the basis that x is self-
contradictory.
And by what justification do you plug a
epistemological antinomy into "x?"
...14 Every epistemological antinomy can likewise be
used for a similar undecidability proof...(Gödel 1931:43-44)
*Paraphrasing that*
No epistemological antinomy can be prohibited from being
used for a similar undecidability proof
It <is> the case that an epistemological antinomy inserted in
x does cause the incompleteness criteria to determine that
formal system L <is> incomplete even though the actual issue
is that x is self-contradictory.
The only last step of this is whether or not it is possible
or logically impossible to define a formal system that can
express actual epistemological antinomies in its language.
Another way of saying this is whether or not it is logically
impossible to precisely correctly formalize the Liar Paradox.
If we can precisely formalize the Liar Paradox then we know
that it is not logically impossible to formalize epistemological
antinomies.
Finally a decent review that does not use dishonest dodge as the
rebuttal tactic.
"And yes, a logical system that admits self-contradictory statements
as elements of its language is incomplete, but that doesn't invaldate
the concept of Incompleteness, just shows that it may not mean a lot
in edge cases that have other problems."
It is incorrect to determine that a formal system is incomplete on the
basis of its inability to prove or refute self-contradictory statements.
Formal systems cannot be required to prove or refute self-contradictory expressions.
We will have to get back to your other points after we finish this one.
On 11/20/2023 6:36 PM, olcott wrote:
On 11/20/2023 3:39 PM, olcott wrote:
...14 Every epistemological antinomy can likewise be
used for a similar undecidability proof...(Gödel 1931:43-44)
An epistemological antinomy is a self-contradictory expression.
∀L ∈ Formal_System
(Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x)))
The only possible place to insert Gödel's above reference
to an epistemological antinomy in the above definition of
Incompleteness is x.
So Gödel's quote is saying that a formal system <is> incomplete
when it cannot prove or refute a self-contradictory expression.
He says this even though the actual reason that self-contradictory
expressions cannot be proven or refuted is that there is something
wrong with them.
I am only referring to the above Gödel quote and stipulating that
[referring to] anything else that he ever said or did is an dishonest
dodge way from the point.
I am also only referring to the above definition of incompleteness
thus not any naive paraphrase.
The first incompleteness theorem states that in any consistent formal
system F within which a certain amount of arithmetic can be carried out,
there are statements of the language of F which can neither be proved
nor disproved in F.
https://plato.stanford.edu/entries/goedel-incompleteness/
∀L ∈ Formal_System
(Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x)))
When we do plug "epistemological antinomy" into x we prove that
the notion of mathematical incompleteness is incorrect because
it determines that L is incomplete on the basis that x is self- contradictory.
On 11/20/2023 8:54 PM, olcott wrote:
On 11/20/2023 6:36 PM, olcott wrote:
On 11/20/2023 3:39 PM, olcott wrote:
...14 Every epistemological antinomy can likewise be
used for a similar undecidability proof...(Gödel 1931:43-44)
An epistemological antinomy is a self-contradictory expression.
∀L ∈ Formal_System
(Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x)))
The only possible place to insert Gödel's above reference
to an epistemological antinomy in the above definition of
Incompleteness is x.
So Gödel's quote is saying that a formal system <is> incomplete
when it cannot prove or refute a self-contradictory expression.
He says this even though the actual reason that self-contradictory
expressions cannot be proven or refuted is that there is something
wrong with them.
I am only referring to the above Gödel quote and stipulating that
[referring to] anything else that he ever said or did is an dishonest
dodge way from the point.
I am also only referring to the above definition of incompleteness
thus not any naive paraphrase.
The first incompleteness theorem states that in any consistent formal
system F within which a certain amount of arithmetic can be carried out, >>> there are statements of the language of F which can neither be proved
nor disproved in F.
https://plato.stanford.edu/entries/goedel-incompleteness/
∀L ∈ Formal_System
(Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x)))
When we do plug "epistemological antinomy" into x we prove that
the notion of mathematical incompleteness is incorrect because
it determines that L is incomplete on the basis that x is self-
contradictory.
"Problem, if x is an epistemological antinony, what is ¬x?"
*Was a particularly good point*
On 11/20/2023 8:54 PM, olcott wrote:
On 11/20/2023 6:36 PM, olcott wrote:
On 11/20/2023 3:39 PM, olcott wrote:
...14 Every epistemological antinomy can likewise be
used for a similar undecidability proof...(Gödel 1931:43-44)
An epistemological antinomy is a self-contradictory expression.
∀L ∈ Formal_System
(Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x)))
The only possible place to insert Gödel's above reference
to an epistemological antinomy in the above definition of
Incompleteness is x.
So Gödel's quote is saying that a formal system <is> incomplete
when it cannot prove or refute a self-contradictory expression.
He says this even though the actual reason that self-contradictory
expressions cannot be proven or refuted is that there is something
wrong with them.
I am only referring to the above Gödel quote and stipulating that
[referring to] anything else that he ever said or did is an dishonest
dodge way from the point.
I am also only referring to the above definition of incompleteness
thus not any naive paraphrase.
The first incompleteness theorem states that in any consistent formal
system F within which a certain amount of arithmetic can be carried out, >>> there are statements of the language of F which can neither be proved
nor disproved in F.
https://plato.stanford.edu/entries/goedel-incompleteness/
∀L ∈ Formal_System
(Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x)))
When we do plug "epistemological antinomy" into x we prove that
the notion of mathematical incompleteness is incorrect because
it determines that L is incomplete on the basis that x is self-
contradictory.
"Problem, if x is an epistemological antinony, what is ¬x?"
*Was a particularly good point*
On 11/21/2023 10:52 AM, olcott wrote:
On 11/20/2023 8:54 PM, olcott wrote:
On 11/20/2023 6:36 PM, olcott wrote:
On 11/20/2023 3:39 PM, olcott wrote:
...14 Every epistemological antinomy can likewise be
used for a similar undecidability proof...(Gödel 1931:43-44)
An epistemological antinomy is a self-contradictory expression.
∀L ∈ Formal_System
(Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x))) >>>>>
The only possible place to insert Gödel's above reference
to an epistemological antinomy in the above definition of
Incompleteness is x.
So Gödel's quote is saying that a formal system <is> incomplete
when it cannot prove or refute a self-contradictory expression.
He says this even though the actual reason that self-contradictory
expressions cannot be proven or refuted is that there is something
wrong with them.
I am only referring to the above Gödel quote and stipulating that
[referring to] anything else that he ever said or did is an dishonest
dodge way from the point.
I am also only referring to the above definition of incompleteness
thus not any naive paraphrase.
The first incompleteness theorem states that in any consistent formal
system F within which a certain amount of arithmetic can be carried
out,
there are statements of the language of F which can neither be proved
nor disproved in F.
https://plato.stanford.edu/entries/goedel-incompleteness/
∀L ∈ Formal_System
(Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x)))
When we do plug "epistemological antinomy" into x we prove that
the notion of mathematical incompleteness is incorrect because
it determines that L is incomplete on the basis that x is self-
contradictory.
"Problem, if x is an epistemological antinony, what is ¬x?"
*Was a particularly good point*
"One reason why useful logic system exclude logical self-contradiction
from them!"
*That is my whole point, modern logic systems do not do that*
When we take the set of all human knowledge expressed as HOL
actually incompleteness is only unknown truths yet the
incompleteness criteria incorrectly determines that these systems
are also incomplete on the basis that they cannot prove self-
contradictory expressions.
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