Re: Working out the details of the steps of Ĥ.H ⟨Ĥ⟩ ⟨Ĥ⟩ <Ĥ> ⊢* Ĥ.Hqn

On 3/8/24 10:40 PM, olcott wrote:

On 3/9/2024 12:25 AM, Richard Damon wrote:

On 3/8/24 9:55 PM, olcott wrote:

Ĥ.q0 ⟨Ĥ⟩ ⊢* ⊢* Ĥ.Hqy ∞ // Ĥ applied to ⟨Ĥ⟩ halts
Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.Hq0 ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.Hqn   // Ĥ applied to ⟨Ĥ⟩ does not halt
>
Expecting Ĥ.H ⟨Ĥ⟩ ⟨Ĥ⟩ to correctly report on the behavior of
Ĥ ⟨Ĥ⟩ is a little nuts because Ĥ contradicts Ĥ.H ⟨Ĥ⟩ ⟨Ĥ⟩.

>
But that is the job it signed up for when it tried to call itself a Halt Decider.
>

 There is no correct answer that Ĥ.H ⟨Ĥ⟩ ⟨Ĥ⟩ can possibly
provide that corresponds to the behavior of this Ĥ ⟨Ĥ⟩.
Therefore it must have a basis for its wrong answer.
 

WHY?
What is the actual grounds for that statement?
If it doesn't give the right answer, it just isn't a Halt Decider, and thus isn't actually constrained by the definition of one.
H either IS or it IS NOT a Halt Decider. Being "close" doesn't make it one.
Once you admit it isn't going to be a Halt Decider, you need to admit that and then decide what you are going to try to make it to meet your need. (If you have one).
The original task for a Halt Decider was Theorem proving and Knowledge gathering, where you needed to either be 100% accurate, or it didn't help. (you don't prove a theorem with a 99% accuracy, only 100%)
For the goal you have stated, you also need 100% or you have nothing. If you are willing to accept approximate and slightly flawed decisions, you don't need to refute the Halting Theorem, as it alread allows for those to exist.
Just like if you want to be able to refute most falsehood, you don't need the Truth Predicate, as most major false statement are provably false, (if the person will look at logic, and if not, having the predicate wouldn't help anyway).