XPost: sci.math, comp.theory   
   From: mikko.levanto@iki.fi   
      
   On 23/01/2026 12:22, olcott wrote:   
   > On 1/23/2026 3:13 AM, Mikko wrote:   
   >> 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.   
      
   > There is a   
   > "true on the basis of meaning expressed in language"   
   > and I figured out how to make it computable over the   
   > body of knowledge.   
      
   Except that "true on the basis of meaning expressed in language" is   
   nmt computable and does not cover all of the body of knowldge.   
      
   --   
   Mikko   
      
   --- SoupGate-Win32 v1.05   
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