XPost: comp.theory, comp.lang.c++, comp.lang.c   
   From: 643-408-1753@kylheku.com   
      
   On 2025-10-03, olcott  wrote:   
   > On 10/3/2025 4:50 PM, Kaz Kylheku wrote:   
   >> On 2025-10-03, olcott  wrote:   
   >>> This concept must be named or it cannot   
   >>> be referenced. I must reference it.   
   >>> It must be a simple and accurate name.   
   >>>   
   >>> WTF would you call it?   
   >>>   
   >>> The set of H/P pairs such that:   
   >>> For decider H and input P   
   >>> If H says halts then P loops   
   >>> If H says loops then P halts   
   >>> making H(P) always incorrect.   
   >>   
   >> This is not /your/ concept; it is something you should have   
   >> understood properly 20 years ago.   
   >>   
   >   
   > It isn't that I didn't understand this 22 years ago.   
   > I understood this the whole time. It is that 22 years   
      
   As little as five years go or so you still believed that there   
   is supposed to be a /single/ case that is undecidable.   
      
   > ago I spotted its bogus liar paradox structure.   
      
   Because you didn't understand it at the time, but given what you should   
   know now, you should see that it's not the same at all.   
      
   The diagonal case, though it contains a self-reference, is not a truth   
   statement and it is not asserting its own falsehood.   
      
   The details of the self-reference matter.   
      
   For instance, these sentence are just as self-referential as the   
   Liar Paradox, yet we can cheerfully assign them truth values:   
      
     This sentence has five words.  (True)   
     This sentence has four words.  (False)   
      
   Like the Liar sentence, these sentences are making a logical assertion.   
   Unlike the Liar sentence, they are not simply making a logical assertion   
   about their truth value; they are making an assertion about a property   
   different from their truth value.   
      
   By talking about something other than their own truth value, the   
   logical assertion is anchored to something external.   
      
   The following sentence is self-referential, yet talks about nothing but   
   its truth value:   
      
     This sentence is true.   
      
   this is still just internal: the logical assertion talking about its   
   own truth.  We can assign the value True or False to this sentence   
   without contradiction---but that is a problem almost if not just as bad   
   as not being able to assign a truth value!   
      
   So anyway, in the diagonal trick used for reasoning about halting,   
   the self-reference is much more like:   
      
     This sentence has four words.   
      
   than it is like:   
      
     This sentence is false.   
      
   The H decider isn't asserting something about the truth value of its   
   subject; it is anchored to a different property, halting,   
   which is like word count, or any other property.   
      
   The "opposite behavior" is something as if we could make   
   a sentence that changes its word count to something other than   
   what it has been counted by some procedure.   
      
   Suppose H is a word count procedure. Let the sentence D be:   
      
     In this sentence, 'Church-Turing' is two words, except if H says   
     that this sentence has an even number of words, in which case it   
     is one word.   
      
   Thus if we allow sentences to stipulate how many words comprise a   
   hyphenated compound which occurs in them, and if we allow such   
   stipulations to refer to word counting functions, then the word count   
   function becomes incomputable. :)   
      
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