XPost: comp.theory, sci.logic, sci.math   
   From: mikko.levanto@iki.fi   
      
   On 17/01/2026 23:08, olcott wrote:   
   > For nearly a century, discussions of arithmetic have quietly   
   > relied on a fundamental conflation: the idea that   
   > “true in arithmetic” meant “true in the standard model of ℕ.”   
   > But PA itself has no truth predicate, no internal semantics,   
   > and no mechanism for assigning truth values. So what was   
   > called “true in arithmetic” was always meta-theoretic truth   
   > about arithmetic, imported from an external model and never   
   > grounded inside PA.   
      
   The expression "true in arithmetic" must at least include everything   
   that can be proven about every set that has the properties tha   
   1. it contains the empty set   
   2. for every set it contains it also contains the successor of that set   
   3. it does not contain any subset that has the proerties 1 and 2.   
   The arithmetic symbol 0 is interpreted to mean the empty set and   
   the arithmetic function successor is interpreted to mean the successor   
   function of sets defined so that the successor of X = X ∪ {X}.   
      
   --   
   Mikko   
      
   --- SoupGate-Win32 v1.05   
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