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From: Richard Damon <richard@damon-family.org>
Newsgroups: comp.theory,sci.logic
Subject: Re: D correctly simulated by H cannot possibly halt --- templates and
 infinite sets
Date: Wed, 29 May 2024 22:55:23 -0400
Organization: i2pn2 (i2pn.org)
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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.

H is asked about a specific Turing Machine, by passing it a complete 
representation of that machine in symbolic form.

Just as for most problems, other than those which happen to be defined 
in terms of the same symbol set as the Turing Machie uses.