XPost: comp.theory, sci.math, sci.lang   
   From: tristan.wibberley+netnews2@alumni.manchester.ac.uk   
      
   On 29/11/2025 23:19, olcott wrote:   
      
   > Gödel, Kurt 1931.   
   > On Formally Undecidable Propositions of Principia Mathematica And   
   > Related Systems   
      
   Do you have a reference to the original and also English translation of   
   his 1938 paper "On Formally Undecidable Propositions of Principia   
   Mathematica And Related Systems II"?   
                                   ^^   
      
   His 1931 paper says he'll followup with a completed proof and   
   generalisation to more systems - so I think that's what we have to look   
   at to understand what people refer to as his first incompleteness proof   
   and theorem. I've been told (albeit by a chatbot) that the title and   
   year above is what I should look for.   
      
      
   > If you think that I am wrong then don't fucking guess   
   > show exactly what his sentence actually says without   
   > the ruse of Gödel numbers in a language has its own   
   > self-reference operator and provability operator.   
      
   You've gone off the deep end there.   
      
      
   > I say it says this:   
   > G := (F ⊬ G) // G says of itself that it is unprovable in F   
      
      
   It says that G is not a theorem of F, and perhaps it does so   
   epitheoretically because of the use of ":=" which often nominates a   
   substitution to apply to get an object of F, and that would /almost/   
   trivially make it true, albeit not for all possible F.   
      
   "[fact] in [a system]" conventionally can mean [fact] for all definition   
   extensions of [a system] when mathematicians are talking because they   
   add definitions when using the system and examine the consequences "/in/   
   the system". The prepositions are ambiguous across specialisms, clearly.   
      
   There are some more ambiguities so reflecting and responding usefully on   
   Olcott's expression is difficult and nondeterministic.   
      
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   Tristan Wibberley   
      
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