XPost: comp.theory, sci.logic, sci.math   
   From: tristan.wibberley+netnews2@alumni.manchester.ac.uk   
      
   On 29/11/2025 20:51, olcott wrote:   
      
   > Try and show how Gödel incompleteness can be   
   > specified in a language that can directly encode   
   > self-reference and has its own provability operator   
   > without hiding the actual semantics using Gödel numbers.   
      
   An outline how I'd go about this kind of thing (not complete by far, and   
   also not checked properly, though I have thought about it a bit). I've   
   done this about Olcott's paraphrasing of Goedel's outline in PM rather   
   than the challenge stated above because Goedel's system P is weird and I   
   don't trust it at all. IMPORTANT! Goedels outline and Olcott's   
   paraphrase use a self-referential definition rather than universal   
   quantification! I'll have to cogitate more to waffle about that.   
      
   Note, it involves a formal system /extension/ rather than an /episystem/   
   and that's how we may interpret the challenge to do our task "in [the]   
   language".   
      
   Thinking about it a touch more carefully than before, assuming Olcott   
   very carefully formulated his G definition (I think F is nothing to do   
   with Goedel's system P, but is more akin to a modernisation of PM  which   
   goedel used for his similar outline). Note I have this time interpreted   
   F to refer to a basic system rather than the definition extension that I   
   previously supposed. This way we have a few points in the space of   
   systems to reason more clearly with.   
      
   You have to take this all with a pinch of salt, we're using ":=" and it   
   will take some formalisation which I'm not doing (and so will |/-).   
      
      
   Then consider Olcott's paraphrasing   
      
   G := (F |/- G)   
      
   The above must be an axiom of a definition extension I'll call "I" that   
   also adjoins G and I to the objects of F and we'll assume we've already   
   formalised extension within F. I'll call the resulting extended system   
   "I(F)" and suppose that's the form by which statements in F may name a   
   system "F" extended by I.   
      
   In that case I(F) is consistent: G is not derived of F.   
      
   We'd have a problem if I use a definition extension "J" that adjoins,   
   instead, objects G and J and an axiom   
      
   G := (J(F) |/- G).   
      
   "J(F)" names what I assumed Olcott had meant "F" to name previously. I   
   kind-of-conjecture that J(F) is contradictory because it seems obvious   
   that we can derive both |- G and |/- G which is how we'll recognise   
   contradiction. I hesitate to use the term "inconsistent" because I don't   
   trust the concept having found "indiscriminate" to be satisfactory and   
   more general for simpler systems.   
      
      
   Having drawn a very faint line on a block of stone before spending eons   
   carving and polishing and looking up to see what it took to do so   
   little, it is obvious why Olcott has not been swayed by the   
   conversations in these newsgroups.   
      
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   Tristan Wibberley   
      
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    * Origin: you cannot sedate... all the things you hate (1:229/2)