We are therefore confronted with a proposition which asserts its own unprovability. 15
14 Every epistemological antinomy can likewise be used for a similar undecidability proof.
(Gödel 1931:40)
Antinomy
...term often used in logic and epistemology, when describing a paradox
or unresolvable contradiction. https://www.newworldencyclopedia.org/entry/Antinomy
Gödel, Kurt 1931.
On Formally Undecidable Propositions of Principia Mathematica And
Related Systems
https://mavdisk.mnsu.edu/pj2943kt/Fall%202015/Promotion%20Application/Previous%20Years%20Article%2022%20Materials/godel-1931.pdf
On this basis we define a much more powerful F in a formal system having
its own unprovability operator: ⊬
The eliminates the need for the complexity of arithmetization and diagonalization.
G := (F ⊬ G) means G is defined to be another name for (F ⊬ G) https://en.wikipedia.org/wiki/List_of_logic_symbols
∃G ∈ F (G := (F ⊬ G))
There exists a G in F that proves its own unprovability in F
Within this much more powerful F a proof of G in F requires a sequence
of inference steps in F that prove that they themselves do not exist.
On 4/25/23 12:47 PM, olcott wrote:
We are therefore confronted with a proposition which asserts its own
unprovability. 15
Right, a statement in Meta-F proved from G.
14 Every epistemological antinomy can likewise be used for a similar
undecidability proof.
(Gödel 1931:40)
Right "Used" as in, establish a form that gets TRANSFORMED into th proof.
Antinomy
...term often used in logic and epistemology, when describing a
paradox or unresolvable contradiction.
https://www.newworldencyclopedia.org/entry/Antinomy
Gödel, Kurt 1931.
On Formally Undecidable Propositions of Principia Mathematica And
Related Systems
https://mavdisk.mnsu.edu/pj2943kt/Fall%202015/Promotion%20Application/Previous%20Years%20Article%2022%20Materials/godel-1931.pdf
On this basis we define a much more powerful F in a formal system
having its own unprovability operator: ⊬
Is such an operator actually computable? or possible to know the answer
of in general?
YOu are just showing you lack of understanding of how things work.
YOU JUST DON'T UNDERSTAND HOW LOGIC WORKS.
You can't just postulate that something exists and then use its
existance to prove something.
The eliminates the need for the complexity of arithmetization and
diagonalization.
So?
G := (F ⊬ G) means G is defined to be another name for (F ⊬ G)
https://en.wikipedia.org/wiki/List_of_logic_symbols
∃G ∈ F (G := (F ⊬ G))
There exists a G in F that proves its own unprovability in F
Within this much more powerful F a proof of G in F requires a sequence
of inference steps in F that prove that they themselves do not exist.
But since this is a DIFFERENT G, it doesn't disprove that Godel's G is actually True but Unprovable.
Again, you fall into the trap of your own strawman.
You can't argue that a statement can't be correct if you have replaced
the statement with something it isn't.
You are just proving your stupiditiy.
On 4/25/2023 5:42 PM, Richard Damon wrote:
On 4/25/23 12:47 PM, olcott wrote:
We are therefore confronted with a proposition which asserts its own
unprovability. 15
Right, a statement in Meta-F proved from G.
Not at all look at page 40 of the link.
14 Every epistemological antinomy can likewise be used for a similar
undecidability proof.
(Gödel 1931:40)
Right "Used" as in, establish a form that gets TRANSFORMED into th proof.
That the liar paradox cannot be proved or refuted because it is self- contradictory derives an equivalent proof .
Antinomy
...term often used in logic and epistemology, when describing a
paradox or unresolvable contradiction.
https://www.newworldencyclopedia.org/entry/Antinomy
Gödel, Kurt 1931.
On Formally Undecidable Propositions of Principia Mathematica And
Related Systems
https://mavdisk.mnsu.edu/pj2943kt/Fall%202015/Promotion%20Application/Previous%20Years%20Article%2022%20Materials/godel-1931.pdf
On this basis we define a much more powerful F in a formal system
having its own unprovability operator: ⊬
Is such an operator actually computable? or possible to know the
answer of in general?
Prolog does it.
YOu are just showing you lack of understanding of how things work.
YOU JUST DON'T UNDERSTAND HOW LOGIC WORKS.
Mere empty rhetoric utterly bereft of any supporting reasoning.
mindless idiots consider rhetoric much more convincing that correct reasoning. 40% of the electorate believed the lies about election fraud
even though there was almost no evidence of any fraud that could have possibly change the results.
What I am talking about is the philosophical foundations of correct reasoning. This is not at all the same things as studying a textbook and logic and fully understand every detail of this book.
This latter view is a narrower perspective.
You can't just postulate that something exists and then use its
existance to prove something.
All correct reasoning begins with premises.
The eliminates the need for the complexity of arithmetization and
diagonalization.
So?
It simplifies the problem enough that the interaction between the
elements of the problem is not masked by too many extraneous details.
G := (F ⊬ G) means G is defined to be another name for (F ⊬ G)
https://en.wikipedia.org/wiki/List_of_logic_symbols
∃G ∈ F (G := (F ⊬ G))
There exists a G in F that proves its own unprovability in F
Within this much more powerful F a proof of G in F requires a sequence
of inference steps in F that prove that they themselves do not exist.
But since this is a DIFFERENT G, it doesn't disprove that Godel's G is
actually True but Unprovable.
It meets Gödel's equivalence requirements stated above.
Again, you fall into the trap of your own strawman.
You can't argue that a statement can't be correct if you have replaced
the statement with something it isn't.
You are just proving your stupiditiy.
An IQ more then two standard deviations above the mean is by no means
any sort of stupid and you know it. You are flatly dishonest in your denigration.
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