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From: olcott <polcott333@gmail.com>
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 10:37:18 -0500
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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)


-- 
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer