[continued from previous message]   
      
   >> No.  I'm not sure what you mean by "a 'satisfactory' problem" because   
   >   
   > /A number which is a description number of a circle-free machine will be   
   > called a satisfactory number. In §8 it is shown that there can be no   
   > general process for determining whether a given number is satisfactory   
   > or not/ [Tur36 p241]   
   >   
   > after turing spends §1-§7 defining turing machine, §8 is where he proves   
   > the "halting theorem" as you say. in that proof he's really proving   
   > there's no way to prove a number "satisfactory" or more technically   
   > "circle-free"   
   >   
   > big miss there buddy, the "satisfactory" paradox (inability to build a   
   > general decider for "satisfactory" numbers) he describes §8 (on p247) in   
   > is literally keystone contradiction he bases the rest of his undecidable   
   > proof   
   >   
   >> Turing uses the term "satisfactory" only in relation to numbers.   
   >> However, he is not supporting Godel's incompleteness theorem, he is   
   >   
   > he is 100% supporting godel's result   
   >   
   > /If the negation of what Godel has shown had been proved, i.e. if, for   
   > each U, either U or —U is provable, then we should have an immediate   
   > solution of the Entscheidungsproblem. For we can invent a machine H   
   > which will prove consecutively all provable formulae. Sooner or later H   
   > will reach either U or —U. If it reaches U, then we know that U is   
   > provable. If it reaches —U, then, since K is consistent (Hilbert and   
   > Ackermann, p. 65), we know that U is not provable/ [Tur36 p259]   
   >   
   > what he's doing is saying that if godel had proven otherwise (to   
   > incompleteness) then there'd be some machine which would prove all   
   > provable formulae   
   >   
   > because of this there must be some method to ensure such isn't   
   > construct-able, and ultimately §11 is tying such a machine to the   
   > previously disproveb notion (of §8) that a decider D that could   
   > determine whether a given number is satisfactory:   
   >   
   > /We are now in a position to show that the Entscheidungsproblem cannot   
   > be solved. Let us suppose the contrary. Then there is a general   
   > (mechanical) process for determining whether Un(𝓜 ) is provable. By   
   > Lemmas 1 and 2, this implies that there is a process for determining   
   > whether 𝓜 ever prints 0, and this is impossible, by §8. Hence the   
   > Entscheidungsproblem cannot be solved/ [Tur26 p262]   
   >   
   > yes he is ultimately also disproving the entscheidungsproblem, but he's   
   > motivated in doing so specifically because it aligns with godel's   
   > previous result, and his paper ultimately strengthens godel's result,   
   > even if the specific method *is* quite different   
   >   
   >> using what we now call the halting theorem to derive results about   
   >> computation numbers.  In section 11 he says "It should perhaps be   
   >> remarked what I shall prove is quite different from the well-known   
   >> results of Gödel".   
   >>   
   >>> the halting problem variant was   
   >>> first described by kleene and/or davis   
   >>   
   >> The term was indeed coined later, but the result is right there in the   
   >   
   > not just coined later, but also described later. turing specifically   
   > tied his support of incompleteness to the "satisfactory number" paradox,   
   > not halting paradox   
   >   
   >> paper along with two proofs.  He was inventing the whole subject on the   
   >> fly so it not surprising that we now use other terms, but a theorem by   
   >> another name shall smell as sweet.   
   >   
      
      
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   Copyright 2025 Olcott   
      
   My 28 year goal has been to make   
   "true on the basis of meaning" computable.   
      
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