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From: Richard Damon <richard@damon-family.org>
Newsgroups: comp.theory,sci.logic
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Date: Sun, 10 Mar 2024 12:48:33 -0700
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On 3/10/24 12:16 PM, olcott wrote:
> On 3/10/2024 2:09 PM, immibis wrote:
>> On 10/03/24 19:32, olcott wrote:
>>> On 3/10/2024 1:08 PM, immibis wrote:
>>>> On 10/03/24 18:17, olcott wrote:
>>>>> ZFC simply tossed out the Russell's Paradox question as unsound.
>>>>
>>>> So you are saying that some Turing machines are not sound?
>>>>
>>>>>>> Both H ⟨Ĥ⟩ ⟨Ĥ⟩ and Ĥ.H ⟨Ĥ⟩ ⟨Ĥ⟩ correctly decide that:
>>>>>>> (a) Their input halts H.qy
>>>>>>> (b) Their input fails to halt or has a pathological
>>>>>>> relationship to itself H.qn.
>>>>>>
>>>>>> But the "Pathological Relationship" is ALLOWED.
>>>>>>
>>>>> ZFC simply tossed out the Russell's Paradox question as unsound
>>>>> expressly disallowing the "Pathological Relationship".
>>>>
>>>> So you are saying that some Turing machines are not real Turing 
>>>> machines?
>>>>
>>>>> I am only claiming that both H and Ĥ.H correctly say YES
>>>>> when their input halts and correctly say NOT YES otherwise.
>>>>
>>>> well the halting problem requires them to correctly say NO, so you 
>>>> haven't solved it
>>>
>>> All decision problem instances of program/input such that both
>>> yes and no are the wrong answer toss out the input as invalid.
>>>
>>
>> all decision problems are defined so that all instances are valid or 
>> else they are not defined properly
>>
> 
> Not in the case of Russell's Paradox.

Which is just Red Herring, as it has nothing to do with Halting.

It Turns out that for Set Theory, the fact that the rules for Naive Set 
Theory allowed for results based Set Descriptions that are not actually 
contructable, as their construction leads to a paradox, for Computation 
Theory, the ability for Algortihms (aka Turing Machines) to contain 
OTHER Algorithms (but never themselves, as that leads to in infinite 
machine) but could take descriptions of themselves as input, leads to a 
very useful system.

The "problem" you see, of having some questions not being answered by an 
algorithm given data representing the specific instance of the question, 
isn't considered actually a problem, since it turns out to have a fairly 
reasonable explainion for its existance (which required transfinite math 
to be discovered) which is that there are MANY more possible problems 
then possible machine to decider them, being Aleph-1 possible problems, 
but only Aleph-0 possible deciders.

Thus, currently the fact of uncomputable problems isn't considered an 
issue by most, except crackpots like you.

> 
>> all turing machine/input pairs are valid instances of the halting problem
>