Eliminating Undecidability and Incompleteness in Formal Systems

[PeteOlcott]

When the formal notion of truth is defined in this way, all
previously undecidable logic sentences become decidable:

When Closed WFF x of formal system F is considered:
True----------its a theorem of F: ---------------(F ⊢ x)
False----------its negation is a theorem of F: (F ⊢ ¬x)
¬True---------its not a theorem of F: ----------(F ⊬ x)
¬Boolean-----its neither True nor False in F: (F⊬x ∧ F⊬¬x)

When-so-ever any closed WFF contains a ¬Boolean term the whole
WFF evaluates to ¬True thus maintaining consistency within the
{axioms of truth} adaptation to the notion of a formal system.

Eliminating Undecidability and Incompleteness in Formal Systems
https://www.researchgate.net/publicatio ... al_Systems

The above adaptations show how this logic sentence is ¬Boolean thus ¬True:
∃F ∈ Formal_Systems ∃G ∈ WFF(F) (G ↔ ((F ⊬ G) ∧ (F ⊬ ¬G)))

Making the following paragraph ¬Boolean, thus ¬True
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.
(Raatikainen 2018)

[Logik]

If your system does what it claims it does, you effectively solved computational complexity and you should claim your million dollars: https://en.wikipedia.org/wiki/Millenniu ... _versus_NP

Otherwise, the concept of Oracle Machine (a.k.a Human) remains: https://en.wikipedia.org/wiki/Oracle_machine

[PeteOlcott]

If your system does what it claims it does, you effectively solved computational complexity and you should claim your million dollars: https://en.wikipedia.org/wiki/Millenniu ... _versus_NP

Otherwise, the concept of Oracle Machine (a.k.a Human) remains: https://en.wikipedia.org/wiki/Oracle_machine

If my system does what I say it does has nothing to do with computation complexity,
it does directly pertain to refuting the halting problem proof though.

My system does what it says that it does by recognizing and rejecting semantically
incorrect ¬Boolean terms thus eliminating undecidability and incompleteness
in all formal systems that can express its axioms of truth.

On December 13, 2018 between 6:00 PM and 8:00 PM I created the algorithm that shows
exactly how the Peter Linz H would decide halting for the Peter Linz input pair(Ĥ, Ĥ).

http://liarparadox.org/Peter_Linz_HP(Pages_318-319).pdf

Every single detail of all of the relevant programming code for these two machines
is fully specified. I merely have to complete the C++ code for the UTM that will
execute them. My refutation of the Linz Proof can be generalized to the other HP
proofs on the basis of a newly discovered detail that pertains to all of these proofs.

[Logik]

If my system does what I say it does has nothing to do with computation complexity,
it does directly pertain to refuting the halting problem proof though.

But the halting problem is all about computational complexity! And it has a bunch of different proofs.

The most intuitive one goes like this:

A is an algorithm whose purpose is to find a solution to problem P by brute-force.
There is no algorithm B which can correctly answer the question: Will A find a solution to P?

The halting problem is not a problem for finite problem-space. We KNOW that finite state machines halt. The key is in the word "finite".

So I have no idea what conception of the halting problem you are "solving" if it's outside of the domain of complexity.

[PeteOlcott]

If my system does what I say it does has nothing to do with computation complexity,
it does directly pertain to refuting the halting problem proof though.

But the halting problem is all about computational complexity! And it has a bunch of different proofs.

The most intuitive one goes like this:

A is an algorithm whose purpose is to find a solution to problem P by brute-force.
There is no algorithm B which can correctly answer the question: Will A find a solution to P?

The halting problem is not a problem for finite problem-space. We KNOW that finite state machines halt. The key is in the word "finite".

So I have no idea what conception of the halting problem you are "solving" if it's outside of the domain of complexity.

I an only refuting the standard self-referential proof. I have spent 22 years
and many thousands of hours focusing on one single issue: pathological self-reference.
Take a look at the two pages of the Peter Linz text and see what I mean.

[Logik]

I an only refuting the standard self-referential proof. I have spent 22 years
and many thousands of hours focusing on one single issue: pathological self-reference.
Take a look at the two pages of the Peter Linz text and see what I mean.

I am struggling to compute consequences for your claims towards anything that is even remotely empirical (e.g outside of the universe of abstract formalism).

Pathological self-reference sounds like performative contradiction? e.g uttering the English sentence: "I don't exist".

But then, you are talking about decidability in formal systems.

There is no way to decide the sentence "I exist" either. So what gives on the "eliminating undecidability" ?

[PeteOlcott]

I an only refuting the standard self-referential proof. I have spent 22 years
and many thousands of hours focusing on one single issue: pathological self-reference.
Take a look at the two pages of the Peter Linz text and see what I mean.

I am struggling to compute consequences for your claims towards anything that is even remotely empirical (e.g outside of the universe of abstract formalism).

Pathological self-reference sounds like performative contradiction? e.g uttering the English sentence: "I don't exist".

But then, you are talking about decidability in formal systems.

There is no way to decide the sentence "I exist" either. So what gives on the "eliminating undecidability" ?

The formalization of the strong Liar Paradox: "This sentence is not true".
LP := ~True(LP)
LP ↔ (⊬LP)

Are two ways that I have formalized it:

The first one requires a non standard logical operator to implement actual self-reference
rather than the approximation of self-reference provided by material equivalence.
It is not true because it is infinitely recursive.

Axiom of Truth(3) proves that the second one is simply false.

[Logik]

The first one requires a non standard logical operator to implement actual self-reference
rather than the approximation of self-reference provided by material equivalence.

What do you consider to be a "standard" and "non-standard" logical operator?

Would you not say that this is a valid way of expressing recursion:

Code: Select all

def f(x):
  f(x-1)

[PeteOlcott]

The first one requires a non standard logical operator to implement actual self-reference
rather than the approximation of self-reference provided by material equivalence.

What do you consider to be a "standard" and "non-standard" logical operator?

Would you not say that this is a valid way of expressing recursion:

Code: Select all

def f(x):
  f(x-1)

https://en.wikipedia.org/wiki/List_of_logic_symbols
(1) It must be the symbolic logic of predicate logic. Thus is cannot be any arbitrary programming language.

I am speaking math, I am not speaking computer science even though
https://en.wikipedia.org/wiki/Curry%E2% ... espondence
Since most math people do not understand that I must stick to predicate logic.

(2) I have never seen this symbol used in any predicate logic expression ":=" It is shown on the above link as "definition"

[Logik]

https://en.wikipedia.org/wiki/List_of_logic_symbols
(1) It must be the symbolic logic of predicate logic. Thus is cannot be any arbitrary programming language.

Why are you imposing arbitrary constraints?

If you are talking about symbolic logic then you cannot be talking about the Chomsky hierarchy.
Because in ANY Chomsky language you need to define the semantics of your very operators as part of your language itself.
Nothing is a given. Nothing is "standard".

That's the whole point of recursion and compiler bootstrapping.

[PeteOlcott]

https://en.wikipedia.org/wiki/List_of_logic_symbols
(1) It must be the symbolic logic of predicate logic. Thus is cannot be any arbitrary programming language.

Why are you imposing arbitrary constraints?

If you are talking about symbolic logic then you cannot be talking about the Chomsky hierarchy.
Because in ANY Chomsky language you need to define the semantics of your very operators as part of your language itself.
Nothing is a given. Nothing is "standard".

That's the whole point of recursion and compiler bootstrapping.

You keep saying that knowing that is nothing was standard this communication
(using standard English) would be impossible. Why do you keep contradicting yourself?
The constraints are for the purpose of communicating with math people,
they are not arbitrary.

Higher order predicate logic with types would be Chomsky level-2.
The semantics of its operators can be defined in a tiny subset of "c".

I merely assume that this has already been done because it has already
be proven to have been done. No sense delving into extraneous details.
The finite string transformation rules are identical to those used in math.

A
A → B
-------
∴ B

I think that for most of the logic operators their full semantics
is specified by a tiny little truth table.

[PeteOlcott]

Because in ANY Chomsky language you need to define the semantics of your very operators as part of your language itself.
Nothing is a given. Nothing is "standard".

To be a pure mathematical formalist this would be true. I am a pure
mathematical formalist yet don't have time to delve into material that
is extraneous to the primary point that I am making.

My point can be entirely made on the less precise conventional
usage of these symbols by mathematicians.

http://liarparadox.org/Provable_Mendelson.pdf defines what this: "⊢" means.

[wtf]

I am a pure
mathematical formalist yet don't have time to delve into material that
is extraneous to the primary point that I am making.

Google translate renders that as: "I claim to be a mathematician but can't be bothered to learn any mathematics."

[PeteOlcott]

I am a pure
mathematical formalist yet don't have time to delve into material that
is extraneous to the primary point that I am making.

Google translate renders that as: "I claim to be a mathematician but can't be bothered to learn any mathematics."

No it is more like I say "I am go to eat some lunch" and people quibble endlessly
over what "to" really means, and have to go home when the restaurant closes
without even looking at the menu.

In computer science ignoring unimportant details this is called abstraction, you
make sure that you ONLY focus on the most relevant details carefully ignoring
any details that are not directly relevant.

It it not like no one has any idea what these logical operators mean:
What do I mean by logical operators: https://en.wikipedia.org/wiki/Truth_table

One thing that I am saying does need much better elaboration so I am posting this:
What do I mean by "Provable" This is what I mean: http://liarparadox.org/Provable_Mendelson.pdf


The following relevant details
(when added to any formal system capable of expressing them)
eliminate incompleteness and inconsistency from that formal system:

Here are my four axioms of Truth:
∀F ∈ Formal_Systems ∀x ∈ WFF(F) (True(F, x) ↔ (F ⊢ x))
∀F ∈ Formal_Systems ∀x ∈ WFF(F) (False(F, x) ↔ (F ⊢ ¬x))
∀F ∈ Formal_Systems ∀x ∈ WFF(F) (¬True(F, x) ↔ (F ⊬ x))
∀F ∈ Formal_Systems ∀x ∈ WFF(F) (¬Boolean(F, x) ↔ ((F ⊬ x) ∧ (F ⊬ ¬x)))

When-so-ever any closed WFF contains a ¬Boolean term the whole WFF
evaluates to ¬True thus maintaining consistency within the {axioms of truth}
adaptation to the notion of a formal system.

[wtf]

No it is more like I say "I am go to eat some lunch" and people quibble endlessly
over what "to" really means, and have to go home when the restaurant closes
without even looking at the menu.

No. Seriously, dude. When I pointed out to you earlier that examples like set theory with and without the axiom of choice, and geometry with and without the parallel postulate, refute your idea that axioms are true, you not only didn't engage with the point. You were unable to. You couldn't even look them up or formulate a coherent response. That's a bad look for someone claiming to have refuted the most important theorems in logic and computability theory.

Axioms aren't true or false. They're simply statements accepted without proof in order to get some axiomatic system off the ground. They can be anything you want. You can do geometry with the parallel postulate, in which case you get Euclidean geometry; or with the negation of the parallel postulate, in which case you get non-Euclidean geometry. You can do group theory with the assumption that your group operation is commutative, in which case you get the theory of Abelian groups; or with the assumption that it's not, in which case you get the theory of non-Abelian groups. Being Abelian isn't true or false in general; it's true or false of some particular group.

That you utterly fail to appreciate this shows a massive gap in your understanding.

Say I take the "axiom" that any nonzero number has a multiplicative inverse. That's true in the rational or real numbers; but false in the integers and natural numbers. Axioms are not true or false in themselves. They are only true or false when you pick a model for your axiomatic system.

It's baffling that you not only don't understand this point, but can't even coherently engage with it.

So when I point out that you claim to be a mathematician yet don't know the very first thing about math, this is exactly what I mean. In 22 years you haven't bothered to learn ANYTHING. Why not spend a few days or weeks trying to understand these examples I've given you?