Sujet : Re: Verified fact that Ĥ.H ⟨Ĥ⟩ ⟨Ĥ⟩ and H ⟨Ĥ⟩ ⟨Ĥ⟩ have different behavior ZFC
De : richard (at) *nospam* damon-family.org (Richard Damon)
Groupes : comp.theory sci.logicDate : 10. Mar 2024, 21:48:33
Autres entêtes
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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.
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So you are saying that some Turing machines are not sound?
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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.
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But the "Pathological Relationship" is ALLOWED.
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ZFC simply tossed out the Russell's Paradox question as unsound
expressly disallowing the "Pathological Relationship".
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So you are saying that some Turing machines are not real Turing machines?
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I am only claiming that both H and Ĥ.H correctly say YES
when their input halts and correctly say NOT YES otherwise.
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well the halting problem requires them to correctly say NO, so you haven't solved it
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All decision problem instances of program/input such that both
yes and no are the wrong answer toss out the input as invalid.
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all decision problems are defined so that all instances are valid or else they are not defined properly
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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