XPost: comp.theory, sci.logic   
   From: polcott333@gmail.com   
      
   Diagonalization only arises when one assumes that a   
   Turing machine decider must report on its own behavior   
   instead of the behavior specified by its machine description.   
      
   Everyone assumes that these must always be the same.   
   That assumption is proven to be incorrect.   
      
   When one assumes a halt decider based on a UTM then   
   the simulated input remains stuck in recursive simulation   
   never reaching simulated states ⟨Ĥ.∞⟩ or ⟨Ĥ.qn⟩.   
      
   Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.embedded_H ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.∞   
   Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.embedded_H ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qn   
      
   (a) Ĥ copies its input ⟨Ĥ⟩   
   (b) Ĥ invokes embedded_H ⟨Ĥ⟩ ⟨Ĥ⟩   
   (c) embedded_H simulates ⟨Ĥ⟩ ⟨Ĥ⟩   
   (d) simulated ⟨Ĥ⟩ copies its input ⟨Ĥ⟩   
   (e) simulated ⟨Ĥ⟩ invokes simulated embedded_H ⟨Ĥ⟩ ⟨Ĥ⟩   
   (f) simulated embedded_H simulates ⟨Ĥ⟩ ⟨Ĥ⟩   
   on and on never reaching any simulated final state of ⟨Ĥ.qn⟩   
      
   When embedded_H aborts its simulation and transitions to Ĥ.qn   
   on the basis that its simulated input cannot possibly reach its own   
   simulated final halt state of ⟨Ĥ.qn⟩ embedded_H is correct.   
      
   This causes embedded_H itself to halt, thus contradicting its result   
   *only if a Turing machine decider can be applied to its actual self*   
   and not merely its own machine description.   
      
   --   
   Copyright 2025 Olcott "Talent hits a target no one else can hit; Genius   
   hits a target no one else can see." Arthur Schopenhauer   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)