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From: joes <noreply@example.com>
Newsgroups: comp.theory
Subject: Re: D correctly simulated by H cannot possibly halt --- templates and
 infinite sets
Date: Sat, 1 Jun 2024 18:56:41 -0000 (UTC)
Organization: i2pn2 (i2pn.org)
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Am Sat, 01 Jun 2024 09:52:54 -0500 schrieb olcott:
> On 6/1/2024 3:20 AM, Mikko wrote:
>> On 2024-05-31 15:44:22 +0000, olcott said:
>>> On 5/31/2024 8:10 AM, Mikko wrote:
>>>> On 2024-05-28 16:16:48 +0000, olcott said:
>>>>

>>>>> When Ĥ is applied to ⟨Ĥ⟩
>>>>> Ĥ.q0 ⟨Ĥ⟩ ⊢* embedded_H ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qy ∞
>>>>> Ĥ.q0 ⟨Ĥ⟩ ⊢* embedded_H ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qn
>>>>>
>>>>> *Formalizing the Linz Proof structure*
>>>>> ∃H  ∈ Turing_Machines
>>>>> ∀x  ∈ Turing_Machines_Descriptions
>>>>> ∀y  ∈ Finite_Strings
>>>>> such that H(x,y) = Halts(x,x)
>>>>>
>>>>> *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)
>>>>>
>>>>> In both cases infinite sets are examined to see
>>>>> if any H exists with the required properties.
>>>>
>>>> That says nothing about correct simulation. It says
>>>> something abuout some D but not whether it is correctly
>>>> simulated. Also nothing is said about templates or
>>>> infinite sets. At the end is claimed that some
>>>> infinite sets are examined but not who examined, nor
>>>> how, nor what was found in the alleged examination.
>>>>
>>>
>>> *Formalizing the Linz Proof structure*
>>> ∃H  ∈ Turing_Machines
>>> ∀x  ∈ Turing_Machines_Descriptions
>>> ∀y  ∈ Finite_Strings
>>> such that H(x,y) = Halts(x,x)
>> 
>> The above is the counter hypothesis for the proof. Proof structore
>> is that a contradiction is derived from the counter hypthesis.
>> 
>>> The above disavows Richard's claim based on a misinterpretation of
>>> Linz that the Linz proof is about a single specific Turing machine.
>> 
>> Your ∃H declares H as a new symbol for a specific Turing machine.
>> Therefore everything that follows refers to that specific Turing 
machine.
>> There may be others that could be discussed the same way but they 
aren't.
>> 
> 
> ∃H  ∈ Turing_Machines
> There exists at least one H
> from the infinite set of all Turing_Machines
> 
> ∃!H  ∈ Turing_Machines
> There exists a single unique H
> from the infinite set of all Turing_Machines


>>>     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.
>> Note the words "a single Turing machine".
> 
> I know that he said that yet he meant this
> ∃H ∈ Turing_Machines *and didn't mean this* ∃!H ∈ Turing_Machines
> or he would be contradicting every other HP proof.
> 
>>> Linz <IS NOT> looking for a single machine that gets the wrong answer.
>>> Linz is looking for at least one Turing Machine that gets the right
>>> answer: ∃H ∈ Turing_Machines
>> 
>> Not at least one but exactly one. The Halting Problem asks for one
>> or a proof that there is none.
> 
> In other words when there are two machines that solve the halting
> problem then the halting problem IS NOT SOLVED?
I misunderstood this, too, but we want a single machine that solves the 
problem on its own, not multiple that each solve parts. There could be
many such machines.

-- 
joes