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From: Richard Damon <richard@damon-family.org>
Newsgroups: comp.theory,sci.logic
Subject: =?UTF-8?Q?Re=3A_A_simulating_halt_decider_applied_to_the_The_Peter_?=
 =?UTF-8?Q?Linz_Turing_Machine_description_=E2=9F=A8=C4=A4=E2=9F=A9?=
Date: Tue, 28 May 2024 22:04:16 -0400
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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:
>>>>
>>>> He STARTS with a proof that one specific (but arbitrary) machine 
>>>> gets the wrong answer.
>>>>
>>>
>>> *Not exactly, you are misreading this*
>>>
>>> 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
>>
>>   *** a single Turing Machine ***
>>
>> not singular
>>
>>>
>>> ...
>>>
>>> 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
>>
>>   *** some Turing Machine ***
>>
>> Note singular
>>
>>>
>>> Ordinary existential quantification looks for at least one
>>> element not exactly one element:
>>
>> But you can look for at least one by looking for one without an 
>> assumption that it is unique.
>>
> 
> 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)
> 
>>>
>>> Does at least one Turing machine exist of the infinite set
>>> of all Turing machines ...
>>
>> Right, so if we can prove that none of them are correct, you have 
>> shown that some is contradicted.
>>
>>>
>>> So like I have always said, the second ⊢* specifies
>>> an infinite set of Turing machines.
>>>
>>
>> As a SPECIFICATION of the domain of selection, yes.
>>
>> As a SPECIFICATION of what THIS ONE machine does, no.
>>
> 
> 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.