XPost: comp.theory, sci.logic, sci.math

A copy of Linz H is embedded at Ĥ.qx as a simulating halt decider (SHD).

Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.qx ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qy ∞
If the pure simulation of ⟨Ĥ⟩ ⟨Ĥ⟩ by embedded_H would reach its final state.

Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.qx ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qn
If the pure simulation of ⟨Ĥ⟩ ⟨Ĥ⟩ by embedded_H would never reach its final state.

When Ĥ is applied to ⟨Ĥ⟩
Ĥ copies its input ⟨Ĥ0⟩ to ⟨Ĥ1⟩ then embedded_H simulates ⟨Ĥ0⟩ ⟨Ĥ1⟩

Then these steps would keep repeating:
Ĥ0 copies its input ⟨Ĥ1⟩ to ⟨Ĥ2⟩ then embedded_H0 simulates ⟨Ĥ1⟩ ⟨Ĥ2⟩
Ĥ1 copies its input ⟨Ĥ2⟩ to ⟨Ĥ3⟩ then embedded_H1 simulates ⟨Ĥ2⟩ ⟨Ĥ3⟩
Ĥ2 copies its input ⟨Ĥ3⟩ to ⟨Ĥ4⟩ then embedded_H2 simulates ⟨Ĥ3⟩ ⟨Ĥ4⟩...

Because we can see that a correct simulation of the input to embedded_H
cannot possibly reach its final state we can see that this input never
halts.

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

Whether or not it is even possible for embedded_H to recognize this
infinitely nested simulation does not matter (for refuting Linz).

As long as embedded_H transitions to Ĥ.qn it refutes the Linz conclusion
that rejecting its input forms a necessary contradiction.
(bottom half of last page) https://www.liarparadox.org/Linz_Proof.pdf



After we have mutual agreement on the above point we can move on to how embedded_H would use finite string comparison to detect that its input specifies infinitely nested simulation.

Halting problem undecidability and infinitely nested simulation (V4)

https://www.researchgate.net/publication/359349179_Halting_problem_undecidability_and_infinitely_nested_simulation_V4

--
Copyright 2021 Pete Olcott

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

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