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From: immibis <news@immibis.com>
Newsgroups: comp.theory
Subject: Re: D correctly simulated by H cannot possibly halt --- templates and
 infinite sets
Date: Fri, 31 May 2024 11:15:01 +0200
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On 31/05/24 10:51, Mikko wrote:
> On 2024-05-30 11:33:03 +0000, Richard Damon said:
> 
>> On 5/30/24 6:11 AM, immibis wrote:
>>> On 30/05/24 05:48, olcott wrote:
>>>> On 5/29/2024 9:55 PM, Richard Damon wrote:
>>>>> On 5/29/24 10:36 PM, olcott wrote:
>>>>>> On 5/29/2024 9:25 PM, Richard Damon wrote:
>>>>>>> On 5/29/24 9:55 PM, olcott wrote:
>>>>>>>> When the category is examined all at once then there is no need
>>>>>>>> to look at each individual element.
>>>>>>>
>>>>>>> So, which one or ones gave the correct answer for their input?
>>>>>>>
>>>>>>
>>>>>> *Formalizing the Linz Proof structure*
>>>>>> ∃H  ∈ Turing_Machines
>>>>>> ∀x  ∈ *Turing_Machines_Descriptions*
>>>>>> ∀y  ∈ Finite_Strings
>>>>>> such that H(x,y) = Halts(x,y)
>>>>>>
>>>>>> When we formalize it that way then some simulating halt deciders
>>>>>> get the correct answer.
>>>>>>
>>>>>> *Everyone else implicitly assumes this incorrect formalization*
>>>>>> ∃H  ∈ Turing_Machines
>>>>>> ∀x  ∈ *Turing_Machines*
>>>>>> ∀y  ∈ Finite_Strings
>>>>>> such that H(x,y) = Halts(x,y)
>>>>>>
>>>>>>
>>>>>
>>>>> Nope.
>>>>>
>>>>> You just don't understand the meaning of a "Description" in the 
>>>>> problem.
>>>>>
>>>>
>>>> A deciders compute the mapping FROM ITS INPUTS
>>>> to it own accept or reject state
>>>> Deciders cannot take ACTUAL TURING MACHINES AS INPUTS
>>>> Deciders can only take FINITE STRINGS AS INPUTS
>>>>
>>>
>>> If you want to be pedantic, you made the mistake.
>>> It's actually H(DescriptionOf(x),y) = Halts(x,y)
>>> DescriptionOf is an injective function that converts Turing machines 
>>> into finite strings.
>>
>> OR in his formal language:
>>
>>
>> ∃H  ∈ Turing_Machines, and for
>> ∀M  ∈ Turing_Machines, which have a description Wm, and
>> ∀w  ∈ Finite_Strings
>> such that H(Wm,w) = Halts(M,w)
>>
>> The problem is that there doesn't exist a FUNCTION "Description" as 
>> one machine can have an potentially infinite number of descriptions.
> 
> At this point we may assume the axiom of choice.
> 
>> It is more that there is an inverse function, Described where we could 
>> say that M = Described(Wm)
> 
> 
also the function "Description" doesn't have to 'know' all of the 
descriptions - one per machine is enough.