Correctly answering this single question refutes the Linz proof:
Which state does Ĥ applied to ⟨Ĥ⟩ transition to correctly ?

The following simplifies the syntax for the definition of the Linz
Turing machine Ĥ, it is now a single machine with a single start state.
A copy of Linz H is embedded at Ĥ.qx.

Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.qx ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qy ∞
Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.qx ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qn

Because there are no finite number of steps of the pure simulation of
⟨Ĥ⟩ applied to ⟨Ĥ⟩ by embedded_H such that this simulated input meets
the Linz definition of halting:

computation that halts … the Turing machine will halt whenever it enters
a final state. (Linz:1990:234)

Therefore it is correct to say that the input to embedded_H specifies a
sequence of configurations that never halts.

https://www.researchgate.net/publication/358009319_Halting_problem_undecidability_and_infinitely_nested_simulation_V3

--
Copyright 2021 Pete Olcott

Talent hits a target no one else can hit;
Genius hits a target no one else can see.
Arthur Schopenhauer

On 1/29/2022 5:51 PM, olcott wrote:
> Correctly answering this single question refutes the Linz proof:
> Which state does Ĥ applied to ⟨Ĥ⟩ transition to correctly ?
>
>
> The following simplifies the syntax for the definition of the Linz
> Turing machine Ĥ, it is now a single machine with a single start state.
> A copy of Linz H is embedded at Ĥ.qx.
>
> Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.qx ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.⊢* ∞
> Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.qx ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qn
>
> Because there are no finite number of steps of the pure simulation of
> ⟨Ĥ⟩ applied to ⟨Ĥ⟩ by embedded_H such that this simulated input meets
> the Linz definition of halting:
>
> computation that halts … the Turing machine will halt whenever it enters
> a final state. (Linz:1990:234)

// Halting defined
Mathematically this is known as existential quantification:

∃n ∈ N number of simulated steps such that:
embedded_H simulates ⟨Ĥ⟩ applied to ⟨Ĥ⟩ ⊢* ⟨Ĥ⟩.qy or
embedded_H simulates ⟨Ĥ⟩ applied to ⟨Ĥ⟩ ⊢* ⟨Ĥ⟩.qn

>
> Therefore it is correct to say that the input to embedded_H specifies a
> sequence of configurations that never halts.
>

> https://www.researchgate.net/publication/358009319_Halting_problem_undecidability_and_infinitely_nested_simulation_V3
>
>

--
Copyright 2021 Pete Olcott

Talent hits a target no one else can hit;
Genius hits a target no one else can see.
Arthur Schopenhauer