Refuting Incompleteness and Undefinability

[PeteOlcott]

You claim to have falsified Gödel's incompleteness theorem. That theorem applies to ALL formal systems that can express elementary number theory, whether type theory or set theory or category theory or any other foundational approach.

Axioms are True:
The elementary statements which belong to T are called the elementary
theorems of T and said to be true.
(Haskell Curry, Foundations of Mathematical Logic, 2010).

Formal proofs express deductive Inference:
The chain of symbolic manipulations in the calculus corresponds to
and represents the chain of deductions in the deductive system.
(R. B. BRAITHWAITE 1962).

∴ Formal proofs to theorem consequences represent sound deductions to true conclusions.
Formalized as: ∀F ∈ Formal_Systems ∀x ∈ WFF(F) (True(F, x) ↔(F ⊢ x))

The above universal truth predicate system decides that many undecidable decision
problems are undecidable simply because they are erroneous.

LP ↔ ~True(LP) becomes LP ↔ ~⊢LP
Thus the Liar Paradox is only true if it is provably not true.

Self-contradictory expressions of language are treated as deductive
inference with contradictory premises, unsound.

[wtf]

You claim to have falsified Gödel's incompleteness theorem. That theorem applies to ALL formal systems that can express elementary number theory, whether type theory or set theory or category theory or any other foundational approach.

Axioms are True:

Is that a yes or a no?

Like I say I'm not going to play this game with you. I'm only pointing out that you can't give a straight answer to very obvious objections to your work; and that you apparently lack the self-awareness and humility to say, "Now that you mention it, I better retract the claims I can't defend, and go do my homework."

I don't know the consequences of my claim.

Then I'll take it that you are grateful to me, or would be if you thought about it, for pointing out several areas of weakness in your theory. Areas where you need to pay some attention before making grandiose claims that you can't support. No shame in not knowing things. But you can't assert claims then refuse to defend them. If you're intellectually honest you have to admit you haven't made your case.

All the best.

[PeteOlcott]

In this thread I am only prepared to refute the Tarski Undefinability Theorem ...

AND this simplified refutation of the 1931 Incompleteness Theorem. ∃F ∈ Formal_Systems (∃G ∈ Language(F) (G ↔ ~(F ⊢ G))).

[PeteOlcott]

The consequence of my work would be that the Tarski Undefinability Theorem, Gödel's 1931 Incompleteness Theorem, Turing's 1931 Halting Problem proof and many other undecidable decision problems would be shown to be erroneous.

Do you retract your earlier claims about Gödel and Turing? And which other undecidable decisions problems to you claim are erroneous? What do you think of Hilbert's tenth problem?

Not at all. I can refute Tarski and the simplified essence of Gödel in this same thread.
As I said before my refutation of Turing through the proxy of my refutation of Peter Linz
will have to wait until my universal UTM interpreter is fully operational. The detailed
design of it and my refutation are 100% complete, only c++ coding remains.

I didn't bother looking at this before so I never considered Hilbert. All that I was focusing
on for 22 years is refuting the Liar Paradox, the Halting Problem, Proof and the Incompleteness
Theorem. In the last two years I also noticed Tarski Undefinability Theorem. From this
I noticed that a single universal Truth Predicate would complete all of my goals simultaneously.

The Halting Problem was the most difficult and the most unequivocally solved. It turns out
that all of the conventional halting problem proof counter-examples have an execution
trace that is quite different than everyone expected. It only took me 3000 hours studying
the same two pages of the Peter Linz text since 2004 to notice this.

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

[PeteOlcott]

Then I'll take it that you are grateful to me, or would be if you thought about it, for pointing out several areas of weakness in your theory.
All the best.

I am totally unaware of any weaknesses that you or anyone else ever pointed out.

All of the "weaknesses" that anyone ever pointed out have so far only been simply a lack of understanding on their part because I have not yet sufficiently explained what I mean in a way that they could understand.

It took me 22 years to convert my intuitions into correct formal specifications.

[Logik]

You claim to have falsified Gödel's incompleteness theorem. That theorem applies to ALL formal systems that can express elementary number theory, whether type theory or set theory or category theory or any other foundational approach. When I asked you to comment on the obvious consequences of your claim, you changed the subject and started a new thread.

If Pete isn't going to - I am actually willing to claim this and defend it, but I am too lazy to start another thread. So I will try to do it in a paragraph or 2 below.

Likewise you claim to have disproved Turing's proof of the unsolvability of the Halting problem; and when I asked you to point out the error in the proof, you did not do so. You did say you've been studying the proof for 15 years. Turing's beautifully simple proof is about an hour's worth of work for a CS undergrad: Five minutes to read it and fifty-five minutes to grok it. If you've studied the proof, point out the error. Become famous. You'll be overthrowing 83 years of orthodoxy in computer science.

I don't think he has done that. Because I am going to use the Halting problem to disprove Godel.

My premise is trivial. There is a conceptual misalignment between Mathematicians and Computer Scientists in that the the notions of "provability" and "decidability" are NOT isomorphic.

Pete said this and I disagree.

Axioms are True:
The elementary statements which belong to T are called the elementary
theorems of T and said to be true.

Axioms are not True. Axioms are Truisms. In a Curry-Howard system provability necessarily means decidability.
If you accept my premise above then I shall go on to insist that axioms cannot be ASSUMED true, they need to be DECIDED true.

The reason for this is simple: Cost-asymmetry.

Observe how it's trivial for your brain to assert the identity 1 = 1, but it gets a little harder for 8888888888888888888888888888881 = 888888888888888888888888888881. The bigger the number (the more information) -> the more work required to assert the identity of two things.

x = x => True is not an axiom. It's a proposition. You need to implement a Turing machine/algorithm for the equality() function such that
x = x becomes equality(x,x) and DECIDE the answer not ASSUME the answer.

In this framework then the proof for the identity axiom becomes:

Code: Select all

for all x in X:
   if equality(x, x) == False:
      proof_invalid()
proof_valid()

The above algorithm is trivial. It also doesn't halt if X is an infinite set.

The provability/decidability debate really boils down to finitism vs infinitism.

The clash is simple. Mathematicians ignore the proof-of-work required (computational complexity). Computer scientists don't. We pay the piper.

P.S 8888888888888888888888888888881 = 888888888888888888888888888881 is False

[PeteOlcott]

If you accept my premise above then I shall go on to insist that axioms cannot be ASSUMED true, they need to be DECIDED true.

I speak of Truth generically as applied to both formal and natural language.
Truth is ultimately anchored in a set of expressions of language that are defined to be true.
If not then how do you prove that a dog is not a type of cat?

Dogs and cats are defined axiomatically to have a set of properties. Some of these
properties are in common and some are mutually exclusive. The mutually exclusive
properties of cats and dogs are the axioms from which the theorem {cats are not dogs}
can be inferred.

[Logik]

I speak of Truth generically as applied to both formal and natural language.

That would be a mistake then. "Truth" has a number of different and incompatible conceptions.

Formal languages are closed systems.
Natural languages are open systems.

To pretend they are the same beast requires ignoring their differences. One is just an abstract system for manipulating symbols, the other is 'the way people use language' e.g utility.

Truth is ultimately anchored in a set of expressions of language that are defined to be true.

Defined by whom and for what purpose?

If not then how do you prove that a dog is not a type of cat?

I don't. By definition a dog is one type of thing. A cat is another type of thing.
If they were the same type of thing I would not have defined two types.

Dogs and cats are defined axiomatically to have a set of properties.

First they are recognized/conceptualized as different things.
Then they are defined as different things.

BUT, I could recognize/conceptualize cats and dogs differently. I could conceptualize them as types of types and call them animals.
And I can go ahead and define what properties an "animal" has.

In the object-oriented programming paradigm this is known as multiple inheritance: https://en.wikipedia.org/wiki/Multiple_inheritance

Some of these
properties are in common and some are mutually exclusive. The mutually exclusive
properties of cats and dogs are the axioms from which the theorem {cats are not dogs}
can be inferred.

You are still alternating between two paradigms: axiom/theorem (Mathematics) and decidable/undecidable (Computation).

Conversely from the above, the common properties of cats and dogs are the axioms from which the theorem {cats}, {dogs} ∈ {Animals}

I don't need to infer the theorem "cats are not dogs" because that's backwards thinking.

I look at the world. I see two things. I recognize that in some ways they are different and in some ways they are the same.

If I DECIDE that they are different things then I will form two conceptions/categories and define two types: Cats and Dogs.
And so I could say "Here is a cat and here is a dog".

If I DECIDE they are the same thing then I will form one conception/category and define one type: Animals
And so I could say "Here are two animals".

Depending on the context, purpose of conversation and intentions of the interlocutors I may want to USE either taxonomy.
So I can DECIDE either way.

This is Binary classification 101 stuff: https://en.wikipedia.org/wiki/Binary_classification
It is at the core of the concept of "precision".

1 bit of information = 1 distinction.

Animals => 1 bit of information
Cat-animals and Dog-Animals -> 2 bits of information.

Methinks you are going too far in what logic/truth is and what it's for. I am on the same page as Scott Aronson ( https://www.scottaaronson.com/blog/?p=710 ) the Turing machine is at the foundation of human thought. What we have is a formal model of the human mind.

Logic is simply a tool for executing algorithms in one's mind.
Natural language is for communicating between minds.

[PeteOlcott]

Gödel's first incompleteness theorem says that no axiomatic theory that's powerful enough to include the arithmetic of the natural numbers can be both consistent and complete.

Do you understand that in order for anyone to take your work seriously, you would inevitably have to respond to exactly the questions I'm asking? Or perhaps you went one claim too far about ‎Gödel and would like to walk it back. Once you claim ‎Gödel's incompleteness theorem is erroneous, you must immediately confront the logical consequences.

Here are the two pages of Linz that I studied off and on since 2004.
http://liarparadox.org/Peter_Linz_HP(Pages_318-319).pdf

And what is the flaw in the proof? And what is the flaw in Turing's extremely simple original proof?

If the following sentence is unsatisfiable in any formal system what-so-ever then
the counter-example basis for Gödel's 1931 proof cannot possibly exist.
∃F ∈ Formal_Systems ∃G (WFF(F,G) ∧ (G ↔ ~(F ⊢ G))) // parses correctly

No I do not have to even be aware of the consequences much less confront them.
The only consequence that I will directly address is that refuting the 1931 Theorem
eliminates proof that some formal system are necessarily inconsistent or incomplete.

I address my Peter Linz refutation that same way. The counter-example basis
of the halting problem is shown to be erroneous. In the case of the Peter Linz
proof I show exactly how H can correctly decide halting for its (Ĥ, Ĥ) input pair.

[wtf]

No I do not have to even be aware of the consequences much less confront them.

Well ok then! I'm out.

[PeteOlcott]

No I do not have to even be aware of the consequences much less confront them.

Well ok then! I'm out.

I am not trying to know anything at all about any detail of mathematics or logic
that does not pertain to formalizing the notion of truth. That refuting Incompleteness
would have myriads of deeply recursive consequences of consequences is extraneous
to whether or not and how incompleteness can be refuted. I am surprised that you
can't see this.

Since the rest of humanity has failed in understanding the crucial essence of the
underlying inter-relationships I must maintain an enormously sharper single-minded
focus to have any chance at all of success.

[wtf]

I am surprised that you
can't see this.

Since the rest of humanity has failed in understanding the crucial essence of the
underlying inter-relationships I must maintain an enormously sharper single-minded
focus to have any chance at all of success.

I take it as great praise indeed that you are surprised I can't see your point. After all, the entire rest of humanity can't see it. You're not surprised at that. But you're surprised I can't see it either. You must consider me far wiser than the rest of humanity to even have a chance of getting it. I'm honored.

[PeteOlcott]

I am surprised that you
can't see this.

Since the rest of humanity has failed in understanding the crucial essence of the
underlying inter-relationships I must maintain an enormously sharper single-minded
focus to have any chance at all of success.

I take it as great praise indeed that you are surprised I can't see your point. After all, the entire rest of humanity can't see it. You're not surprised at that. But you're surprised I can't see it either. You must consider me far wiser than the rest of humanity to even have a chance of getting it. I'm honored.

Try and show how knowing the side-effects of refuting the incompleteness theorem
could have any impact on whether or not it is solved assuming that a single correct
criterion measure has already been established to determine whether or not it is solved.
Within stochastic reasoning the more evidence the better. This is not stochastic reasoning.

[Speakpigeon]

As I see it, we can make any number of formal methods and, obviously, true formulas will depend on the kind of proof method used.
So the question is whether any one of these methods is the one and only one which is correct. Up until such a day as we can decide which method is correct, if even there is one, grandiose claims about formal logic can be dismissed out of hand.
EB

[Logik]

So the question is whether any one of these methods is the one and only one which is correct. Up until such a day as we can decide which method is correct, if even there is one, grandiose claims about formal logic can be dismissed out of hand.

Observe that you have simply moved the problem elsewhere.

In what formal system would you arbitrate the "correctness" or "incorrectness" of your methodology? What is the objective standard for "correctness" ?

Science uses predictive utility as arbiter.