XPost: comp.theory, sci.logic   
   From: F.Zwarts@HetNet.nl   
      
   Op 07.aug.2025 om 15:39 schreef olcott:   
   > On 8/7/2025 4:11 AM, Fred. Zwarts wrote:   
   >> Op 07.aug.2025 om 05:15 schreef olcott:   
   >>>   
   >>> Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.embedded_H ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.∞,   
   >>> if Ĥ applied to ⟨Ĥ⟩ halts, and   
   >>>   
   >>> Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.embedded_H ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qn   
   >>> if Ĥ applied to ⟨Ĥ⟩ does not halt.   
   >>>   
   >>> The Linz proof has Ĥ take its own machine description   
   >>> as input (this is fine) yet requires Ĥ.embedded_H to   
   >>> directly report on its own behavior. No TM can ever   
   >>> do that.   
   >> That is your mistake. It does not require to report on its own   
   >> behaviour, but on the behaviour of Ĥ.   
   >   
   > That *is* its own behavior.   
      
   That is your mistake. It is not its own behaviour, but that of Ĥ (which   
   resembles its own behaviour). That is a fundamental difference.   
   When it is unable to analyse the behaviour that resembles its own   
   behaviour, it fails due to a limitation in its analysis. A decider is   
   required to analyse any behaviour, even if it resembles its own behaviour.   
      
   > Ĥ.embedded_H *is* an aspect of Ĥ.   
   > Ĥ.embedded_H does correctly report on the behavior   
   > specified by its input ⟨Ĥ⟩ ⟨Ĥ⟩ as measured by its   
   > correct simulation of this input.   
   >   
   >> It is irrelevant whether the behaviour of Ĥ resembles that of embedded_H.   
   >   
   >   
   >   
      
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