On 2024-02-16 18:28:02 +0000, olcott said:

On 2/16/2024 12:10 PM, Mikko wrote:

On 2024-02-16 17:35:12 +0000, olcott said:

Ȟ.q0 ⟨M⟩ ⊢* embedded_H ⟨M⟩ ⟨M⟩ ⊢* Ĥ.qy     // M applied to ⟨M⟩ halts

Is Ȟ intended to mean the same as Ĥ?

embedded_H always means the first state of the Linz H

I.e., it means the same as H.q0.

--
Mikko

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On 2024-02-16 17:35:12 +0000, olcott said:

On 2/16/2024 11:25 AM, Mikko wrote:

On 2024-02-16 15:07:17 +0000, olcott said:

On 2/16/2024 4:09 AM, Mikko wrote:

On 2024-02-16 05:42:40 +0000, olcott said:

// Turing machine H ---  H applied to ⟨H⟩
// --- Do you halt on your own Turing Machine description ?
H.q0 ⟨H⟩ ⟨H⟩ ⊢* H.qy // H applied to ⟨H⟩ halts
H.q0 ⟨H⟩ ⟨H⟩ ⊢* H.qn // H applied to ⟨H⟩ does not halt

The conditions after the // are incorrect: H must be applied to a
pair, not just <H> or some other singlet.

I am merely using different notational conventions that
are easier to understand because they are more conventional.
Linz uses Wm as the finite string Turing machine description
of some arbitrary machine M.

Your conventions are not easier to understand. I do understand what
H applied to <H> <H> means but not what H applied to <H> means.

*I rewrote everything*

I am merely using different notational conventions that are easier to understand because they are more conventional. Linz uses Wm as the
finite string Turing machine description of some arbitrary machine M.

// *Verbatim Linz Turing machine H --- M applied to w*
// --- Does M halt on w?
H.q0 Wm w ⊢* H.qy // M applied to w halts
H.q0 Wm w ⊢* H.qn // M applied to w does not halt

// *Linz Turing machine H --- M applied to w* (different encoding)
// --- Does M halt on w?
H.q0 ⟨M⟩ w ⊢* H.qy // M applied to w halts
H.q0 ⟨M⟩ w ⊢* H.qn // M applied to w does not halt

// *Linz Turing machine H --- M applied to ⟨M⟩ is own description*
// --- Does M halt on its own Turing Machine Description?
H.q0 ⟨M⟩ ⟨M⟩ ⊢* H.qy // M applied to ⟨M⟩ halts
H.q0 ⟨M⟩ ⟨M⟩ ⊢* H.qn // M applied to ⟨M⟩ does not halt

I am applying the Linz H' and Linz Ĥ in reverse order first transforming
H into Olcott Ȟ as the one parameter version of Linz H where a machine
is applied to its own Turing machine description.

embedded_H ⟨M⟩ ⟨M⟩ means H.q0 ⟨M⟩ ⟨M⟩ shown above.

// *Olcott Turing machine Ȟ --- M applied to ⟨M⟩ its own description* // --- Does M halt on its own Turing Machine Description?
Ȟ.q0 ⟨M⟩ ⊢* embedded_H ⟨M⟩ ⟨M⟩ ⊢* Ĥ.qy // M applied to ⟨M⟩ halts
Ȟ.q0 ⟨M⟩ ⊢* embedded_H ⟨M⟩ ⟨M⟩ ⊢* Ĥ.qn // M applied to ⟨M⟩ does not halt

// *Olcott Turing machine Ȟ --- Ȟ applied to ⟨Ȟ⟩ its own description* // --- Do you halt on your own Turing Machine Description?
Ȟ.q0 ⟨Ȟ⟩ ⊢* embedded_H ⟨Ȟ⟩ ⟨Ȟ⟩ ⊢* Ĥ.qy // Ȟ applied to ⟨Ȟ⟩ halts
Ȟ.q0 ⟨Ȟ⟩ ⊢* embedded_H ⟨Ȟ⟩ ⟨Ȟ⟩ ⊢* Ĥ.qn // Ȟ applied to ⟨Ȟ⟩ does not halt
Ȟ applied to ⟨Ȟ⟩ simply correctly transitions to Ĥ.qy

Linz Turing machine Turing machine Ĥ applied to ⟨Ĥ⟩ is the self- contradictory form of Olcott Turing machine Ȟ applied to ⟨Ȟ⟩

// *Linz Turing machine Ĥ --- Ĥ applied to ⟨Ĥ⟩ its own description*
// --- Do you halt on your own Turing Machine Description?
Ĥ.q0 ⟨Ĥ⟩ ⊢* embedded_H ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qy ∞ // Ĥ applied to ⟨Ĥ⟩ halts
Ĥ.q0 ⟨Ĥ⟩ ⊢* embedded_H ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qn // Ĥ applied to ⟨Ĥ⟩ does not halt
Ĥ applied to ⟨Ĥ⟩ cannot correctly transition to Ĥ.qy or Ĥ.qn
because Ĥ applied to ⟨Ĥ⟩ is self contradictory.

I see that you don't disagree with my observation that
your convetions are not easier to understand.

I also see that you can't find better notations.

--
Mikko

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On 2024-02-18 15:51:30 +0000, olcott said:

On 2/18/2024 12:58 AM, immibis wrote:

On 18/02/24 02:47, olcott wrote:

On 2/17/2024 4:24 PM, immibis wrote:

On 17/02/24 16:33, olcott wrote:

On 2/16/2024 11:58 PM, immibis wrote:

On 17/02/24 06:03, olcott wrote:

On 2/16/2024 10:58 PM, immibis wrote:

On 17/02/24 05:24, olcott wrote:

On 2/16/2024 10:17 PM, immibis wrote:

On 17/02/24 04:41, olcott wrote:

On 2/16/2024 9:33 PM, Richard Damon wrote:

On 2/16/24 10:24 PM, olcott wrote:

On 2/16/2024 9:02 PM, Richard Damon wrote:

On 2/16/24 9:38 PM, olcott wrote:

On 2/16/2024 7:31 PM, immibis wrote:

On 17/02/24 02:13, olcott wrote:

On 2/16/2024 7:07 PM, immibis wrote:

On 16/02/24 19:26, olcott wrote:

Likewise Tarski concluded that no truth predicate >>>>>>>>>>>>>>>>>>> can exist that correctly answers this question: >>>>>>>>>>>>>>>>>>>
Is this sentence: "this sentence is not true" true or false?

He is correct. It can't.

It never occurred to Tarski or Gödel that the domain of truth
predicates and formal proofs does not include self-contradictory
expressions.

So can a truth predicate exist that correctly answers the question, or
is Tarski correct to say it can't exist?

Using this same reasoning we can say math is incomplete >>>>>>>>>>>>>>>>>>> because there is no square-root of an actual banana. >>>>>>>>>>>>>>>>>>

no

ONLY when we restrict the domain of math functions to numbers
can we understand that there is not supposed to be any square
root of an actual banana.

The halting problem is solvable on some restricted domains. You are
invited to find some domains where the halting problem is solvable.

Until you understand how and why Tarski is incorrect >>>>>>>>>>>>>>>>

You dishonestly avoided the question. I repeat the question: Can a
truth predicate exist that correctly answers the question, or is Tarski
correct to say it can't exist?

A truth predicate exists in the domain of truth bearers. >>>>>>>>>>>>>>> Tarski was too stupid to understand this.

How do you know that a COMPUTABLE truth predicate exists? >>>>>>>>>>>>>

I can see the details of how this all works.
You have already agreed to these details.

WHERE?

You are just blowing smoke out of your ass.

You have shown that you just don't have the understanding of this sort
of material, after all, you have claimed that ENGLISH is a formal logic
system, which just shows how ignorant you are of what things actually
mean.

The only reason that any analytic expression of language >>>>>>>>>>> is true is that it is semantically linked through a finite >>>>>>>>>>> or infinite sequence of steps to the semantic meanings that >>>>>>>>>>> make it true.

Actually, the great innovation of mathematics is that the steps can be
formal - symbolic.
For example, if x+1=y is true, then x+2=y+1 is also true. It doesn't >>>>>>>>>> matter what x and y represent. x+2=y+1 is still true.

Semantic entailment can be and has been formalized for many decades. >>>>>>>>>

A semantically entails B if B is true in all models where A is true. >>>>>>>

That is not what I mean. There is one model of the
actual world.

Mathematics does not care about the actual world.

This is the only possible way to create the functional equivalent of a >>>>> human mind.

Cyc (pronounced /ˈsaɪk/ SYKE) is a long-term artificial intelligence >>>>> project that aims to assemble a comprehensive ontology and knowledge >>>>> base that spans the basic concepts and rules about how the world works. >>>>> https://en.wikipedia.org/wiki/Cyc

This is irrelevant to the halting problem.

It is indirectly relevant.
The Cyc project proves that English can be mathematically formalized.
This proves that semantics can be directly formalized in the formal
system with no need to separate syntax from semantics.

Whether we math can prove things about English is completely irrelevant
to whether there is a Turing machine that answers the halt status of
all Turing machines.

If math can proving things about English then math is expressive enough
that it can perform any analytical proof about anything.

If math can prove things about English like something about the lengths
of words or nesting levels of subordiante clauses that does not mean
that it can perform an analytical proof about anything.

--
Mikko

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On 2024-02-18 01:47:29 +0000, olcott said:

On 2/17/2024 4:24 PM, immibis wrote:

On 17/02/24 16:33, olcott wrote:

On 2/16/2024 11:58 PM, immibis wrote:

On 17/02/24 06:03, olcott wrote:

On 2/16/2024 10:58 PM, immibis wrote:

On 17/02/24 05:24, olcott wrote:

On 2/16/2024 10:17 PM, immibis wrote:

On 17/02/24 04:41, olcott wrote:

On 2/16/2024 9:33 PM, Richard Damon wrote:

On 2/16/24 10:24 PM, olcott wrote:

On 2/16/2024 9:02 PM, Richard Damon wrote:

On 2/16/24 9:38 PM, olcott wrote:

On 2/16/2024 7:31 PM, immibis wrote:

On 17/02/24 02:13, olcott wrote:

On 2/16/2024 7:07 PM, immibis wrote:

On 16/02/24 19:26, olcott wrote:

Likewise Tarski concluded that no truth predicate >>>>>>>>>>>>>>>>> can exist that correctly answers this question: >>>>>>>>>>>>>>>>>
Is this sentence: "this sentence is not true" true or false? >>>>>>>>>>>>>>>>

He is correct. It can't.

It never occurred to Tarski or Gödel that the domain of truth
predicates and formal proofs does not include self-contradictory
expressions.

So can a truth predicate exist that correctly answers the question, or
is Tarski correct to say it can't exist?

Using this same reasoning we can say math is incomplete >>>>>>>>>>>>>>>>> because there is no square-root of an actual banana. >>>>>>>>>>>>>>>>

no

ONLY when we restrict the domain of math functions to numbers >>>>>>>>>>>>>>>>> can we understand that there is not supposed to be any square >>>>>>>>>>>>>>>>> root of an actual banana.

The halting problem is solvable on some restricted domains. You are
invited to find some domains where the halting problem is solvable.

Until you understand how and why Tarski is incorrect >>>>>>>>>>>>>>

You dishonestly avoided the question. I repeat the question: Can a
truth predicate exist that correctly answers the question, or is Tarski
correct to say it can't exist?

A truth predicate exists in the domain of truth bearers. >>>>>>>>>>>>> Tarski was too stupid to understand this.

How do you know that a COMPUTABLE truth predicate exists? >>>>>>>>>>>

I can see the details of how this all works.
You have already agreed to these details.

WHERE?

You are just blowing smoke out of your ass.

You have shown that you just don't have the understanding of this sort
of material, after all, you have claimed that ENGLISH is a formal logic
system, which just shows how ignorant you are of what things actually
mean.

The only reason that any analytic expression of language
is true is that it is semantically linked through a finite
or infinite sequence of steps to the semantic meanings that
make it true.

Actually, the great innovation of mathematics is that the steps can be >>>>>>>> formal - symbolic.
For example, if x+1=y is true, then x+2=y+1 is also true. It doesn't >>>>>>>> matter what x and y represent. x+2=y+1 is still true.

Semantic entailment can be and has been formalized for many decades. >>>>>>>

A semantically entails B if B is true in all models where A is true. >>>>>

That is not what I mean. There is one model of the
actual world.

Mathematics does not care about the actual world.

This is the only possible way to create the functional equivalent of a
human mind.

Cyc (pronounced /ˈsaɪk/ SYKE) is a long-term artificial intelligence
project that aims to assemble a comprehensive ontology and knowledge
base that spans the basic concepts and rules about how the world works.
https://en.wikipedia.org/wiki/Cyc

This is irrelevant to the halting problem.

It is indirectly relevant.
The Cyc project proves that English can be mathematically formalized.

In a formal language every string either is in the language or is not.
But people people have different opinions of whether e.g. "Our mission
so to boldly go where no one has returned from." is syntactically
correct. People also disagree what "I never gave no money" means.
To formalize English means to replace it with something that looks
more or less similar, e.g., COBOL.

--
Mikko

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