Gödel's 1931 incompleteness theorem is anchored in this false assumption
The conventional definition of incompleteness:
Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ))
This false assumption is corrected by this foundational axiom:
∀φ ∈ T ((T ⊢ φ) ∨ (T ⊢ ¬φ))
This tiny little change
that I only discovered after trying everything else correctly
denigrates the 1931 incompleteness theorem into a triviality.
On 1/11/23 10:37 PM, olcott wrote:
Gödel's 1931 incompleteness theorem is anchored in this false assumption
The conventional definition of incompleteness:
Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ))
That isn't an assumption, it is a DEFINITION, definitions CAN'T be false.
This false assumption is corrected by this foundational axiom:
∀φ ∈ T ((T ⊢ φ) ∨ (T ⊢ ¬φ))
Which isn't true in any but the simplest of fields. I guess that shows
how much you can understand, only the simplest of fields.
This tiny little change
that I only discovered after trying everything else correctly
denigrates the 1931 incompleteness theorem into a triviality.
No, you didn't "Discover that", that was a long held hope in
mathemtatics that Godel proved wasn't true.
On 1/12/2023 6:49 AM, Richard Damon wrote:
On 1/11/23 10:37 PM, olcott wrote:
Gödel's 1931 incompleteness theorem is anchored in this false assumption >> The conventional definition of incompleteness:
Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ))
That isn't an assumption, it is a DEFINITION, definitions CAN'T be false.
OK so when I define that cats are dogs I am necessarily correct?
When-so-ever any expression of language directly contradicts knowledge
in the body of knowledge than that expression of language is incorrect.
If an analytic expression of language is true or false there must be a complete set of semantic connections making it true or false otherwise
it is not a truth bearer.
Because formal systems are only allowed to have finite proofs formal
systems are not allowed to have infinite connections to their semantic
truth maker. Thus an expression is only true in a formal system iff it
is provable within this system. Otherwise this expression is untrue
which may or may not included false.
This false assumption is corrected by this foundational axiom:
∀φ ∈ T ((T ⊢ φ) ∨ (T ⊢ ¬φ))
Which isn't true in any but the simplest of fields. I guess that shows
how much you can understand, only the simplest of fields.
This tiny little change
that I only discovered after trying everything else correctly
denigrates the 1931 incompleteness theorem into a triviality.
No, you didn't "Discover that", that was a long held hope in
mathemtatics that Godel proved wasn't true.
Since G is unprovable in F G is simply untrue in F, that is just simply
that way that truth actually works. By what-ever means an expression is
only true iff this expression has a complete semantic connection to its truth maker axioms. There is no such connection in G.
If I am correct and everyone in the universe disagrees this has no
effect what-so-ever on the correctness of what I say.
--
Copyright 2023 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer
On 1/12/23 11:35 AM, olcott wrote:
On 1/12/2023 6:49 AM, Richard Damon wrote:
On 1/11/23 10:37 PM, olcott wrote:
Gödel's 1931 incompleteness theorem is anchored in this false
assumption
The conventional definition of incompleteness:
Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ))
That isn't an assumption, it is a DEFINITION, definitions CAN'T be
false.
OK so when I define that cats are dogs I am necessarily correct?
No, because both "cats" and "dogs" are previously defined terms.
"Incompleteness" (as a TECHNICAL) term isn't otherwise defined.
And this techincal definition even aligns with the "common" definition,
of missing something, as an Incomplete system IS missing something, the proofs of some truths in it.
When-so-ever any expression of language directly contradicts knowledge
in the body of knowledge than that expression of language is incorrect.
And how does it directly contradict knowledge?
If an analytic expression of language is true or false there must be a
complete set of semantic connections making it true or false otherwise
it is not a truth bearer.
Righg, and that definition does NOT say "finite"
Because formal systems are only allowed to have finite proofs formal
systems are not allowed to have infinite connections to their semantic
Right, **PROOFS** must be finite, doesn't say anything about Truth.
truth maker. Thus an expression is only true in a formal system iff it
is provable within this system. Otherwise this expression is untrue
which may or may not included false.
Where does this come from.
True means, as you said above, as A complete set of semantic connections (could be infinite) makeing it true.
PROVEN means having a KNOWN FINITE set of semantic connections making it proven.
Thus true within a formal system must have a finite proof because formal systems are not allowed to have infinite proofs.
On 1/12/2023 6:49 AM, Richard Damon wrote:
On 1/11/23 10:37 PM, olcott wrote:
Gödel's 1931 incompleteness theorem is anchored in this false assumption >>> The conventional definition of incompleteness:
Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ))
That isn't an assumption, it is a DEFINITION, definitions CAN'T be false.
OK so when I define that cats are dogs I am necessarily correct?
When-so-ever any expression of language directly contradicts knowledge
in the body of knowledge than that expression of language is incorrect.
If an analytic expression of language is true or false there must be a complete set of semantic connections making it true or false otherwise
it is not a truth bearer.
Because formal systems are only allowed to have finite proofs formal
systems are not allowed to have infinite connections to their semantic
truth maker. Thus an expression is only true in a formal system iff it
is provable within this system. Otherwise this expression is untrue
which may or may not included false.
This false assumption is corrected by this foundational axiom:
∀φ ∈ T ((T ⊢ φ) ∨ (T ⊢ ¬φ))
Which isn't true in any but the simplest of fields. I guess that shows
how much you can understand, only the simplest of fields.
This tiny little change
that I only discovered after trying everything else correctly
denigrates the 1931 incompleteness theorem into a triviality.
No, you didn't "Discover that", that was a long held hope in
mathemtatics that Godel proved wasn't true.
Since G is unprovable in F G is simply untrue in F, that is just simply
that way that truth actually works. By what-ever means an expression is
only true iff this expression has a complete semantic connection to its
truth maker axioms. There is no such connection in G.
If I am correct and everyone in the universe disagrees this has no
effect what-so-ever on the correctness of what I say.
On 1/12/23 9:36 PM, olcott wrote:
Thus true within a formal system must have a finite proof because formal
systems are not allowed to have infinite proofs.
The false statement that you have DIED on.
On 1/12/2023 8:39 PM, Richard Damon wrote:
On 1/12/23 9:36 PM, olcott wrote:
Thus true within a formal system must have a finite proof because formal >>> systems are not allowed to have infinite proofs.
The false statement that you have DIED on.
Expressions of language that are proven to have a connection to their
truth maker axioms are a subset of all truth and comprise the entire
body of analytical knowledge.
One cannot have an expression of language within a formal system that is
true within this formal system that has no connection to its truth maker axioms within this formal system.
Peano arithmetic has no idea that strawberry rhubarb pie is sweet, thus
the expression "strawberry rhubarb pie is sweet" is not true in Peano arithmetic.
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