XPost: sci.math, comp.theory   
   From: polcott333@gmail.com   
      
   On 1/25/2026 5:19 AM, Mikko wrote:   
   > On 24/01/2026 16:01, olcott wrote:   
   >> On 1/24/2026 2:20 AM, Mikko wrote:   
   >>> 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.   
   >>   
   >> When the basis of "true" is proof theoretic semantics   
   >> internal to the formal system relative to its own axioms   
   >> and not truth conditional in a separate model outside   
   >> of the system undecidability ceases to exist.   
   >   
   > No, it does not. It does not matter what you call it, a sentence   
   > that cannot be neither proven nor disproven is undecidable because   
   > that is what the word means. An example is Gödel's sentence in   
   > Peano arithmetics.   
   >   
      
   When a truth predicate gets the input "What time is?"   
   this input is rejected as not truth-apt.   
      
   When PA gets an expression that cannot be proven or   
   refuted using its own axioms then this expression is   
   not within its domain.   
      
      
   --   
   Copyright 2026 Olcott

   
      
   My 28 year goal has been to make 
   
   "true on the basis of meaning expressed in language"
   
   reliably computable for the entire body of knowledge.

   
      
   This required establishing a new foundation
   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)