Re: A simulating halt decider applied to the The Peter Linz Turing Machine description ⟨Ĥ⟩

On 5/28/24 11:37 AM, olcott wrote:

On 5/28/2024 6:34 AM, Richard Damon wrote:

On 5/27/24 11:24 PM, olcott wrote:

On 5/27/2024 7:17 PM, Richard Damon wrote:

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He STARTS with a proof that one specific (but arbitrary) machine gets the wrong answer.
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*Not exactly, you are misreading this*
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The domain of this problem is to be taken as the set of all Turing
machines and all w; that is, we are looking for a single Turing machine
that, given the description of an arbitrary M and w, will predict
whether or not the computation of M applied to w will halt

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  *** a single Turing Machine ***
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not singular
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...
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Proof: We assume the contrary, namely that there exists an algorithm,
and consequently some Turing machine H, that solves the halting problem
https://www.liarparadox.org/Peter_Linz_HP_317-320.pdf

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  *** some Turing Machine ***
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Note singular
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Ordinary existential quantification looks for at least one
element not exactly one element:

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But you can look for at least one by looking for one without an assumption that it is unique.
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 The Peter Linz words refer to ordinary existential quantification
anchored in the infinite set of Turing machines.
 ∃H  ∈ Turing_Machines
∀x  ∈ Turing_Machines_Descriptions
∀y  ∈ Finite_Strings
such that H(x,y) = Halts(x,x)
 

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Does at least one Turing machine exist of the infinite set
of all Turing machines ...

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Right, so if we can prove that none of them are correct, you have shown that some is contradicted.
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So like I have always said, the second ⊢* specifies
an infinite set of Turing machines.
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As a SPECIFICATION of the domain of selection, yes.
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As a SPECIFICATION of what THIS ONE machine does, no.
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 My formalization already handles this.
 *Here is the same thing applied to H/D pairs*
∃H  ∈ C_Functions
∀D  ∈ x86_Machine_Code_of_C_Functions
such that H(D,D) = Halts(D,D)
 

Do you REALLY mean to relax your specification to H just needing to be a "C Function"?
Note, that might allow my cheat to get H to answer THIS PARTICULAR problem, but it still won't let H handle ALL machines. I don't think the limitation of asking D only about the input D fixes all the other non-computable problems.