From: mikko.levanto@iki.fi   
      
   dart200 kirjoitti 26.11.2025 klo 4.38:   
   > On 11/25/25 5:19 PM, Python wrote:   
   >> Le 26/11/2025 à 02:09, dart200 a écrit :   
   >>> On 11/25/25 4:31 PM, Python wrote:   
   >>>> Le 26/11/2025 à 01:29, dart200 a écrit :   
   >> ..   
   >>>> Encoding Gödel’s proof inside Gödel numbers doesn’t defeat the   
   >>>> theorem —   
   >>>> that is the theorem.   
   >>>   
   >>> sorry, i mean if u encode literally godel's paper into godel's   
   >>> numbers...   
   >>>   
   >>> doesn't that produce a proof within the system, for the statement   
   >>> that has no proof within the system? ? ?   
   >>   
   >> No — encoding Gödel’s paper as Gödel numbers does not produce a proof   
   >> inside the system, and it cannot possibly do so.   
   >>   
   >> you get just a big natural number, representing these syntactic strings.   
   >>   
   >> But the theory (PA, ZFC, whatever formal system) does not   
   >> automatically recognize that number as a proof.   
   >>   
   >> Because for a number to count as a proof inside the system, it must be:   
   >>   
   >> ✔ A well-formed derivation   
   >> ✔ From axioms, using inference rules   
   >> ✔ Such that each step is checkable arithmetically   
   >>   
   >> Gödel proves (in meta-math) that the system cannot prove G if it is   
   >> consistent.   
   >> Nothing in the system can know why Gödel outside the system can see that.   
   >>   
   >> Encoding Gödel’s meta-reasoning into a massive integer doesn’t   
   >> activate the meta-reasoning inside the system.   
   >>   
   >> The formal system looks at the number and sees:   
   >>   
   >> just a number, with no significance unless it matches the definition   
   >> of a proof.   
   >>   
   >> It does not read the argument.   
   >> It does not interpret his English/German sentences.   
   >> It does not adopt his epistemic insight.   
   >>   
   >> Meta-reasoning ≠ internal derivation.   
   >>   
   >> Gödel numbering is like ZIP-compressing a PDF.   
   >> Feeding it into PA doesn't make PA understand the PDF.   
   >>   
   >> Gödel’s construction is self-referential, but the system remains blind   
   >> to the meta-narrative explaining that construction. The blind spot is   
   >> exactly why incompleteness holds.   
   >   
   > what if we could put a hard limit to the degree of incompeletness found   
   > within a formal system?   
      
   There are infinitely many undecidable sentences in Peano arithmetic.   
   The set of all sentences is countable.   
      
   --   
   Mikko   
      
   --- SoupGate-Win32 v1.05   
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