XPost: comp.theory, sci.logic   
   From: flibble@red-dwarf.jmc.corp   
      
   On Mon, 04 Aug 2025 13:29:04 -0500, olcott wrote:   
      
   > 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.   
      
   Your Ĥ is not a halt decider as defined by the Halting Problem so has   
   nothing to do with the Halting Problem.   
      
   /Flibble   
      
   --- SoupGate-Win32 v1.05   
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