XPost: sci.logic, sci.math, comp.theory   
   From: user7160@newsgrouper.org.invalid   
      
   On 1/24/26 4:52 PM, Richard Damon wrote:   
   > On 1/24/26 5:31 PM, olcott wrote:   
   >> On 1/24/2026 4:25 PM, Richard Damon wrote:   
   >>> On 1/24/26 3:38 PM, olcott wrote:   
   >>>> On 1/24/2026 1:52 PM, Richard Damon wrote:   
   >>>>> On 1/24/26 2:25 PM, olcott wrote:   
   >>>>>> On 1/24/2026 1:23 PM, Richard Damon wrote:   
   >>>>>>> On 1/24/26 12:54 PM, olcott wrote:   
   >>>>>>>> On 1/24/2026 11:10 AM, Richard Damon wrote:   
   >>>>>>>>> On 1/24/26 10:44 AM, olcott wrote:   
   >>>>>>>>>>   
   >>>>>>>>>> The statement that G is true and unprovable in PA has   
   >>>>>>>>>> always been counter-factual. It has never actually been   
   >>>>>>>>>> true  PA and that is why it is unprovable in PA.   
   >>>>>>>>>   
   >>>>>>>>> Sure it is. At least it is a FACT that no natural number will   
   >>>>>>>>> statisfy that relationship, and there is no proof in PA of that   
   >>>>>>>>> fact.   
   >>>>>>>>>   
   >>>>>>>>   
   >>>>>>>> Have you ever heard of: "true in the standard model of arithmetic"?   
   >>>>>>>   
   >>>>>>>   
   >>>>>>> Sure, but they are not in Peano Arithmatic, but are (generally)   
   >>>>>>> 1st order variations of the Peano Axioms which lead to alternate   
   >>>>>>> number systems.   
   >>>>>>>   
   >>>>>>> Godel's proof is statd to be in a system with at least the   
   >>>>>>> properties of Peano Arithmatic, having the ability to show the   
   >>>>>>> properties of the "Natural Numbers"   
   >>>>>>>   
   >>>>>> Gödel’s incompleteness theorem only “works” if   
   >>>>>> one smuggles in an external notion of truth   
   >>>>>> (truth in ℕ) and then pretends it is an   
   >>>>>> internal notion of truth (truth in PA).   
   >>>>>> If we refuse to make that identification,   
   >>>>>> incompleteness evaporates.   
   >>>>>>   
   >>>>>   
   >>>>> But Truth in N is part of Peano Arithmatic, as Peano Arithmatic is   
   >>>>> a axiomiation to create the Natural Numbers.   
   >>>>>   
   >>>>   
   >>>> You have that backwards. Truth in ℕ requires PA   
   >>>> as part of it, and PA itself has no notion of   
   >>>> Truth in ℕ. PA only has proofs from its own axioms   
   >>>> that can be construed as truth in PA, not truth in ℕ.   
   >>>>   
   >>>   
   >>> Which means you don't understand what N actually is.   
   >>>   
   >>> Nothing can be "True in N" unless that truth comes from the Axioms of   
   >>> PA, as N is the result of PA.   
   >>>   
   >>   
   >> combined with the meta-math external model.   
   >   
   > Nope. N is just a set of object built in the Formal System defined by   
   > PA. 0 comes from Axiom 1 which states there is a 0.   
   >   
   > The rest come from the successor function where n+1 = S(n)   
   >   
   > And the induction property makes sure we get to the full set.   
   >   
   > No "meta-math" needed.   
   >   
   > You are just smoking your category errors and believing your own lies.   
   >   
   >>   
   >>> But then, your claim of not knowing what is true in the world you are   
   >>> creating somes on point for you.   
      
   dick u are a hypocrite eh???   
      
   --   
   arising us out of the computing dark ages,   
   please excuse my pseudo-pyscript,   
   ~ nick   
      
   --- SoupGate-Win32 v1.05   
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