XPost: sci.logic, sci.math, comp.theory   
   From: mikko.levanto@iki.fi   
      
   On 22/01/2026 18:40, olcott wrote:   
   > On 1/22/2026 2:21 AM, Mikko wrote:   
   >> On 21/01/2026 17:22, olcott wrote:   
   >>> On 1/21/2026 3:03 AM, Mikko wrote:   
   >>>>   
   >>>> No, it hasn't. In the way theories are usually discussed nothing is   
   >>>> "ture in arithmetic". Every sentence of a first order theory that   
   >>>> can be proven in the theory is true in every model theory. Every   
   >>>> sentence of a theory that cannot be proven in the theory is false   
   >>>> in some model of the theory.   
   >>>>   
   >>>>> only because back then proof theoretic semantics did   
   >>>>> not exist.   
   >>>>   
   >>>> Every interpretation of the theory is a definition of semantics.   
   >>>>   
   >>>   
   >>> Meta‑math relations about numbers don’t exist in PA   
   >>> because PA only contains arithmetical relations—addition,   
   >>> multiplication, ordering, primitive‑recursive predicates   
   >>> about numbers themselves—while relations that talk about   
   >>> PA’s own proofs, syntax, or truth conditions live entirely   
   >>> in the meta‑theory;   
   >>   
   >> Methamathematics does not need any other relations between numbers   
   >> than what PA has. But relations that map other things to numbers   
   >> can be useful for methamathematical purposes.   
   >>   
   >>> so when someone appeals to a Gödel‑style relation like   
   >>> “n encodes a proof of this very sentence,” they’re   
   >>> invoking a meta‑mathematical predicate that PA cannot   
   >>> internalize, which is exactly why your framework draws   
   >>> a clean boundary between internal proof‑theoretic truth   
   >>> and external model‑theoretic truth.   
   >>   
   >> Anyway, what can be provven that way is true aboout PA. You can deny   
   >> the proof but you cannot perform what is meta-provably impossible.   
   >   
   > Gödel’s sentence is not “true in arithmetic.”   
   > It is true only in the meta‑theory, under an   
   > external interpretation of PA (typically the   
   > standard model ℕ). Inside PA itself, the sentence   
   > is not a truth‑bearer at all.   
      
   There is no concept of "truth-bearer" in an uninterpreted theory because   
   there is not concept of "truth". The relevant concept is "sell-formed-   
   formula" and Gödels sentence is one. It may be true or false in an   
   interpretation.   
      
   Gädel's metatheory contains PA. In Gödel's interpretation PA is   
   interpreted in the same way as the PA part of the metathoéory.   
   Gödel proves that G of PA as interpreted in the metatheory is   
   true but cannot be proven in PA.   
      
   --   
   Mikko   
      
   --- SoupGate-Win32 v1.05   
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