Is there a sentence that proves itself is not provable?

[Scott Mayers]

@PeteOlcott

I'm trying to interpret:

Because formal systems of symbolic logic inherently express and represent the deductive inference
model formal proofs to theorem consequences can be understood to represent sound deductive
inference to true conclusions without any need for other representations such as model theory.

Are you suggesting that 'truth' about something is itself sufficiently defined through formal (deductive) systems without concern to the initial inputs being used?

No. Truth is deduction from true premises to true conclusions as expressed
as the valid use of rules-of-inference to theorem consequences.

So when the input is true then the output is true and that is all there is to the essence of truth.

But then you have nothing new to add value to logic that isn't already understood. In other words, your 'thesis' is empty of meaning because you are only confirming the standard meaning of validity. What makes any view of yours distinctly novel? What are you intending to contribute with your paper if it isn't unique?

What is "sound" is just a logical theorem with both 'true' inputs (from the real world) and a 'valid' system of reasoning that the system requires being both consistent and complete (OF the real world).

Are you not trying to say that the incompleteness theorems are invalid with respect to reality? That was the impression I got. But if you are not, then what is your whole point?

[PeteOlcott]

@PeteOlcott

I'm trying to interpret:

Are you suggesting that 'truth' about something is itself sufficiently defined through formal (deductive) systems without concern to the initial inputs being used?

No. Truth is deduction from true premises to true conclusions as expressed
as the valid use of rules-of-inference to theorem consequences.

So when the input is true then the output is true and that is all there is to the essence of truth.

But then you have nothing new to add value to logic that isn't already understood.

Except that the incompleteness and undefinability proofs have always been wrong. That is the ONLY thing that I add.
Gödel's G and Tarski's x are unprovable ONLY because they are untrue.
It is like saying that all of mathematics is incomplete because 3 > 5 cannot be proven in PA.

[Scott Mayers]

No. Truth is deduction from true premises to true conclusions as expressed
as the valid use of rules-of-inference to theorem consequences.

So when the input is true then the output is true and that is all there is to the essence of truth.

But then you have nothing new to add value to logic that isn't already understood.

Except that the incompleteness and undefinability proofs have always been wrong. That is the ONLY thing that I add.

But this is what I thought you meant. If they are wrong, either (A) there is no such possible theorem of these types, to be considered as being true universally, OR (B) that there may be one but that the ones that have been expressed thus far are incorrect.

If you hold (A), then for all possible systems of formal logic, not one formal system exists that completely accounts for all 'truths' in totality. This IS the reasoning of the arguments presented for 'incompletness/undecidability', though. That IF given some ideal (maybe only a mythical) logically formal system exists, it cannot have a theorem that can hold true of reality speaking about completeness or consistency. IF 'true' such theorems are by default to be 'trusted' as "true". But if false, then there is no 'formal system' that suffices to completely cover all truths in the domain of that universal reality.

The proofs were saying that there CAN be a universal truth BEYOND 'formal systems' if logic includes non-deductive formalism. What the formalists are then asserting about logic for all of reality is that no formal system exists that can cover all truths. What we have left are non-deductively 'consistent' systems. ...or systems that include para-consistent logic, such as those that allow for contradiction as a function third option at minimal: a 'trivial' system,...which is already agreed to by the incompleteness theorems.

So you seem to be just mistakenly interpreting the theorists proposing those "theorems" as suggesting their theorems (as 'theories') are true universally OUTSIDE of their formal systems being used to prove them. They are just showing that there is no 'formal' system that can exhaustively cover all "truths" universally. This too then means you agree because the 'incompleteness' theorems are themselves unable to BE 'complete' unless you INCLUDE non-formally inconsistent or contradictory types of logic.

So unless you can prove this is not what they meant, the theorems OF the formal systems that demonstrate incompleteness are 'sound' (true of reality) OR, if they are 'not-sound', then they are 'incomplete' systems themselves. You can't actually disprove their theorems 'sound' without proving their theorems as 'complete', the opposite of their theorems to imply 'incompleteness'. The circularity locks in their own position as fit with your own regardless.

If it is 'true' that X is wrong, then X is only not-wrong when it is 'false'.

[Scott Mayers]

The remaining alternative, continuing from the last post of mine above, is that (B) that there may be [a true "incompleteness" formal theorem that is true of reality (ie sound)] but that the ones that have been expressed thus far are incorrect.

Would you then accept this as 'possible'? That is, CAN there ever be a real "theorem" that speaks of all systems as being incomplete? If there can, then reality is 'incomplete' of containing a formal system of reasoning. If there CANNOT ever be such a theorem, then we default to accepting any theorem of a non-formal system that covers all possibilities as real as only possibly "complete". In other words, their formal theorems about the limitations require being proven wrong when they are proven right and vice versa.

So we still end up on the same conclusion: that a universal formal system can never suffice to cover all 'truths' about some greater inclusive world of possibilities.

[PeteOlcott]

But then you have nothing new to add value to logic that isn't already understood.

Except that the incompleteness and undefinability proofs have always been wrong. That is the ONLY thing that I add.

But this is what I thought you meant. If they are wrong, either (A) there is no such possible theorem of these types, to be considered as being true universally, OR (B) that there may be one but that the ones that have been expressed thus far are incorrect.

The Tarski proof is the simplest. Just read this one page paper that I have re-written at least 30 times
in the last 24 hours. The second page is merely Tarski quotes to prove that my understanding of what
he said is correct.

https://www.researchgate.net/publicatio ... Reexamined
I provide the truth predicates that he "proved" cannot possibly exist and show where he went wrong.

[Logik]

The Tarski proof is the simplest. Just read this one page paper that I have re-written at least 30 times
in the last 24 hours. The second page is merely Tarski quotes to prove that my understanding of what
he said is correct.

https://www.researchgate.net/publicatio ... Reexamined
I provide the truth predicates that he "proved" cannot possibly exist and show where he went wrong.

From the other thread. You need to define the semantics for ⊢.

Can this: ∃F∃G(G ↔ ~(F ⊢ G)) be shown to be satisfiable without Gödel numbers ?

If you are talking about Godel numbers then you are necessarily looking for a symbolic, not numeric satisfiability.

Godel is isomorphic to Turing. Let Godel sleep. Further - Curry-Howard isomorphism tells us that a "proof" is a working computer program. An algorithm so...

Strategy 1:
LHS ↔ RHS ↔ True
G ↔ True
F ⊢ G ↔ False

Strategy 2:
LHS ↔ RHS ↔ False
G ↔ False
F ⊢ G ↔ True

If you define "⊢" in a regular language for us ( https://en.wikipedia.org/wiki/Regular_language )
Then we can simply resort to any SAT algorithm:

https://en.wikipedia.org/wiki/Boolean_s ... ty_problem

[PeteOlcott]

The Tarski proof is the simplest. Just read this one page paper that I have re-written at least 30 times
in the last 24 hours. The second page is merely Tarski quotes to prove that my understanding of what
he said is correct.

https://www.researchgate.net/publicatio ... Reexamined
I provide the truth predicates that he "proved" cannot possibly exist and show where he went wrong.

From the other thread. You need to define the semantics for ⊢.

Can this: ∃F∃G(G ↔ ~(F ⊢ G)) be shown to be satisfiable without Gödel numbers ?

If you are talking about Godel numbers then you are necessarily looking for a symbolic, not numeric satisfiability.

Godel is isomorphic to Turing. Let Godel sleep. Further - Curry-Howard isomorphism tells us that a "proof" is a working computer program. An algorithm so...

Strategy 1:
LHS ↔ RHS ↔ True
G ↔ True
F ⊢ G ↔ False

Strategy 2:
LHS ↔ RHS ↔ False
G ↔ False
F ⊢ G ↔ True

If you define "⊢" in a regular language for us ( https://en.wikipedia.org/wiki/Regular_language )
Then we can simply resort to any SAT algorithm:

https://en.wikipedia.org/wiki/Boolean_s ... ty_problem

⊢ means the conventional application of rules-of-inference as specified by the logic symbols.
A
A → B
-------
B
∴ A ⊢ B

Your analysis of this: G ↔ ~(F ⊢ G)
and mine are the same yet they both depend on my generic truth predicate
axiom(3) to decide that the RHS is untrue (not necessarily false) in F.

My last two paragraphs taken within the context of the Tarski quotes prove
that Tarski's conclusion that his x is undecidable in his theory because there
are no possible truth predicates that could exist in his theory that would
correctly decide x is refuted by my truth predicates that would decide x
correctly in his theory.

[Logik]

⊢ means the conventional application of rules-of-inference as specified by the logic symbols.
A
A → B
-------
B
∴ A ⊢ B

Yes. I am asking you to define the meaning of the symbol →

How would a language compiler parse "→"?

Give me a specification of some sort. Truth table?
Example of how you translate "→" into any Turing-complete language?

Anything other than a trivial English definition.

A → B
God → Universe

[PeteOlcott]

⊢ means the conventional application of rules-of-inference as specified by the logic symbols.
A
A → B
-------
B
∴ A ⊢ B

Yes. I am asking you to define the meaning of the symbol →

How would a language compiler parse "→"?

Give me a specification of some sort. Truth table?
Example of how you translate "→" into any Turing-complete language?

Anything other than a trivial English definition.

A → B
God → Universe

https://en.wikipedia.org/wiki/Truth_tab ... mplication

This is the formal YACC BNF specification for the language compiler THAT DOES
parse every WFF of MTT into an abstract syntax tree.
https://www.researchgate.net/publicatio ... y_YACC_BNF

MTT then translates this into a directed acyclic graph evaluation tree to
be evaluated in the same sort of way as Prolog queries.

[Scott Mayers]

⊢ means the conventional application of rules-of-inference as specified by the logic symbols.
A
A → B
-------
B
∴ A ⊢ B

Yes. I am asking you to define the meaning of the symbol →

How would a language compiler parse "→"?

Give me a specification of some sort. Truth table?
Example of how you translate "→" into any Turing-complete language?

Anything other than a trivial English definition.

A → B
God → Universe

(A → B) is identical to (not-A or B). In essence, the premises are either false and B either true or false, or B necessarily follows when A is true.

[Scott Mayers]

Except that the incompleteness and undefinability proofs have always been wrong. That is the ONLY thing that I add.

But this is what I thought you meant. If they are wrong, either (A) there is no such possible theorem of these types, to be considered as being true universally, OR (B) that there may be one but that the ones that have been expressed thus far are incorrect.

The Tarski proof is the simplest. Just read this one page paper that I have re-written at least 30 times
in the last 24 hours. The second page is merely Tarski quotes to prove that my understanding of what
he said is correct.

https://www.researchgate.net/publicatio ... Reexamined
I provide the truth predicates that he "proved" cannot possibly exist and show where he went wrong.

You still had not answered the question I asked in one or your many opened thread on this one topic: Why are you opening multiple threads on the same identical thesis?

You refused to prove your case THAT the incompleteness theorems are incorrect and so seem to be just pushing your paper as though true by ambushing your paper in multiple threads. Something is clearly amiss when you won't answer me directly on this. What's your intentions here?

[PeteOlcott]

But this is what I thought you meant. If they are wrong, either (A) there is no such possible theorem of these types, to be considered as being true universally, OR (B) that there may be one but that the ones that have been expressed thus far are incorrect.

The Tarski proof is the simplest. Just read this one page paper that I have re-written at least 30 times
in the last 24 hours. The second page is merely Tarski quotes to prove that my understanding of what
he said is correct.

https://www.researchgate.net/publicatio ... Reexamined
I provide the truth predicates that he "proved" cannot possibly exist and show where he went wrong.

You still had not answered the question I asked in one or your many opened thread on this one topic: Why are you opening multiple threads on the same identical thesis?

You refused to prove your case THAT the incompleteness theorems are incorrect and so seem to be just pushing your paper as though true by ambushing your paper in multiple threads. Something is clearly amiss when you won't answer me directly on this. What's your intentions here?

There are different aspects to this question:
Once we define what Truth is we can prove that expressions of language such as the Liar Paradox
that assert their own unprovability are simply untrue (yet perhaps not false) eliminating the basis
for the Tarski Undefinability Theorem and thus refuting it.

This is summed up right here: Also I have begun to translate this into simple English:
https://www.researchgate.net/publicatio ... Reexamined
I have rewritten this paper 50 times since yesterday morning.

Then when we are totally done with all that we can see how the same
refutation of Tarski applies equally to any incompleteness proof.

I initially tried to explain this all at once then I had decompose it into
pieces so that people would be be able to follow what I am saying one
smaller piece at a time.

[Scott Mayers]

The Tarski proof is the simplest. Just read this one page paper that I have re-written at least 30 times
in the last 24 hours. The second page is merely Tarski quotes to prove that my understanding of what
he said is correct.

https://www.researchgate.net/publicatio ... Reexamined
I provide the truth predicates that he "proved" cannot possibly exist and show where he went wrong.

You still had not answered the question I asked in one or your many opened thread on this one topic: Why are you opening multiple threads on the same identical thesis?

You refused to prove your case THAT the incompleteness theorems are incorrect and so seem to be just pushing your paper as though true by ambushing your paper in multiple threads. Something is clearly amiss when you won't answer me directly on this. What's your intentions here?

There are different aspects to this question:
Once we define what Truth is we can prove that expressions of language such as the Liar Paradox
that assert their own unprovability are simply untrue (yet perhaps not false) eliminating the basis
for the Tarski Undefinability Theorem and thus refuting it.

This is summed up right here: Also I have begun to translate this into simple English:
https://www.researchgate.net/publicatio ... Reexamined
I have rewritten this paper 50 times since yesterday morning.

Then when we are totally done with all that we can see how the same
refutation of Tarski applies equally to any incompleteness proof.

I initially tried to explain this all at once then I had decompose it into
pieces so that people would be be able to follow what I am saying one
smaller piece at a time.

I appreciate you updating your approach in order to communicate this better. So I'll be patient but still believe that you likely have mistook the interpretations of the theorems. Note I opened a distinct thread to discuss historical aspects of the topic. It may help you there to experiment communicating some of the background from your understanding for others so that ''we' can hopefully come to common ground on this particular argument you have. I initially proposed an earlier related problem about the limitations of rational numbers to the real numbers which may help to express the significance of these theorems. While I may disagree with your position up front, maybe I might at least understand better what you mean in time. Good luck and keep up your effort to express it better.

[PeteOlcott]

I appreciate you updating your approach in order to communicate this better. So I'll be patient but still believe that you likely have mistook the interpretations of the theorems. Note I opened a distinct thread to discuss historical aspects of the topic. It may help you there to experiment communicating some of the background from your understanding for others so that ''we' can hopefully come to common ground on this particular argument you have. I initially proposed an earlier related problem about the limitations of rational numbers to the real numbers which may help to express the significance of these theorems. While I may disagree with your position up front, maybe I might at least understand better what you mean in time. Good luck and keep up your effort to express it better.

I haven't spent 10,000 hours over 22 years on a mistake. You have to actually read my two page paper and see what I said.

[Scott Mayers]

We derive these three universal Truth predicate axioms:
(1) ∀F ∈ Formal_Systems ∀x ∈ WFF(F) (True(F, x) ↔ (F ⊢ x))
(2) ∀F ∈ Formal_Systems ∀x ∈ WFF(F) (False(F, x) ↔ (F ⊢ ~x))
(3) ∀F ∈ Formal_Systems ∀x ∈ WFF(F) (~True(F, x) ↔ ~(F ⊢ x)

A "well-formed-formula" is just a syntactical convention for a system. For the non-initiated, this means nothing but looks confusing without an explanation.

The concept of a 'sequent' using the symbol, "⊢" with something to the left of is, means 'given' the list of inputs, what follows the "⊢" is the conclusion. For a 'theorem', this requires representing ONLY the conclusion to demonstrate its tautological nature. Thus, for example,

⊢ P or not-P


means, "it is a theorem that P or not-P is true in the understood system." The way you expressed these are not universal truths but conditional truths.