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From: Mikko <mikko.levanto@iki.fi>
Newsgroups: comp.theory
Subject: Re: D correctly simulated by H cannot possibly halt --- templates and infinite sets
Date: Fri, 31 May 2024 11:51:22 +0300
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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)


-- 
Mikko