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From: Mikko <mikko.levanto@iki.fi>
Newsgroups: comp.theory
Subject: Re: My reviewers think that halt deciders must report on the behavior of their caller
Date: Fri, 6 Jun 2025 10:44:34 +0300
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On 2025-06-05 16:15:20 +0000, olcott said:

> On 6/5/2025 3:02 AM, Mikko wrote:
>> On 2025-06-05 02:13:32 +0000, dbush said:
>> 
>>> On 6/4/2025 10:09 PM, olcott wrote:
>>>> On 6/4/2025 8:43 PM, Richard Damon wrote:
>>>>> On 6/4/25 11:50 AM, olcott wrote:
>>>>>> On 6/4/2025 2:04 AM, Mikko wrote:
>>>>>>> On 2025-06-03 21:39:46 +0000, olcott said:
>>>>>>> 
>>>>>>>> They all say that HHH must report on the behavior of
>>>>>>>> direct execution of DDD()
>>>>>>> 
>>>>>>> No, they don't say that. A halting decider (and a partial halting
>>>>>>> decider when it reports) must report whether the direct execution
>>>>>>> of the computation asked about terminates. Unless that computation
>>>>>>> happens to be DDD() it must report about another behaviour instead
>>>>>>> of DDD().
>>>>>>> 
>>>>>>>> yet never bother to notice that the directly executed DDD() is
>>>>>>>> the caller of HHH(DDD).
>>>>>>> 
>>>>>>> To say that nobody has noticed that is a lie. Perhaps they have not
>>>>>>> mentioned what is irrelevant to whatever they said. In particular,
>>>>>>> whether DDD() calls HHH(DDD) is irrelevant to the requirement that
>>>>>>> a halting decider must report about a direct exection of the
>>>>>>> computation the input specifies.
>>>>>>> 
>>>>>> 
>>>>>> *People have ignored this for 90 years*
>>>>>> *People have ignored this for 90 years*
>>>>>> *People have ignored this for 90 years*
>>>>>> 
>>>>>> The only possible way that HHH can report on the
>>>>>> direct execution of DDD() is for HHH to report on
>>>>>> the behavior of its caller:
>>>>> 
>>>>> So?
>>>>> 
>>>>> It *IS* a fact that to be correct, it needs to answer about the direct 
>>>>> executiom of the program that input represents.
>>>>> 
>>>>> That is DEFINITION.
>>>>> 
>>>> 
>>>> Likewise with the definition of Russell's Paradox
>>>> until ZFC showed that this definition is complete
>>>> nonsense.
>>> 
>>> But unlike Russel's Paradox, which showed a contradiction in the axioms 
>>> of naive set theory, there is no contradiction in the axioms of 
>>> computation theory.
>> 
>> That is easy to fix. Just add the axiom that HHH correctly computes
>> whether DDD halts.
> 
> Incoherent axioms cannot be added.

There is nothing to prevent incoherent axioms. For example, the axiom of
comprehension of the Naive Set Theory is incoherent.

-- 
Mikko