XPost: comp.theory, sci.math   
   From: polcott333@gmail.com   
      
   On 1/1/2026 10:45 PM, Richard Damon wrote:   
   > On 1/1/26 11:22 PM, olcott wrote:   
   >> On 1/1/2026 9:45 PM, Richard Damon wrote:   
   >>> On 1/1/26 10:33 PM, olcott wrote:   
   >>>> On 1/1/2026 8:45 PM, André G. Isaak wrote:   
   >>>>> On 2026-01-01 16:48, Richard Damon wrote:   
   >>>>>> On 1/1/26 6:13 PM, Tristan Wibberley wrote:   
   >>>>>>> On 01/01/2026 22:40, Richard Damon wrote:   
   >>>>>>>   
   >>>>>>>> But it IS a theorem of the base system, as it uses ONLY the   
   >>>>>>>> mathematical   
   >>>>>>>> operations definable in the base system. What makes you think it   
   >>>>>>>> isn't a   
   >>>>>>>> Theorem in the base system.   
   >>>>>>>   
   >>>>>>> It has no derivation in the base system, if it had you wouldn't   
   >>>>>>> think   
   >>>>>>> the base system were incomplete.   
   >>>>>>>   
   >>>>>>   
   >>>>>> It has no PROOF in the base system.   
   >>>>>   
   >>>>> Which means it is not a theorem of the base system. A theorem is a   
   >>>>> statement which can be proven in a particular system.   
   >>>>>   
   >>>>   
   >>>> This is the kind of clarity that we need.   
   >>>> True in the base system essentially means   
   >>>> a theorem of the base system.   
   >>>   
   >>> Which s I explained, it is by at least the very normal definition.   
   >>>   
   >>> It is a statement of fact in the base system.   
   >>>   
   >>> And, that fact in the base system has been proven by a proof in some   
   >>> system that knows of the base system.   
   >>>   
   >>   
   >> Has always been irrelevant.   
   >   
   > Nope. Got a reference?   
   >   
   >> Truth in the base system has always   
   >> actually been theorems of the base system.   
   >   
   > But only if "Theorem" includes things proven to be true in the system   
   > even if the proof is in another.   
   >   
   > Truth DOES need to be based on the axioms of the base system, but allows   
   > the truth to be established by an infinite chain of reasoning, unlike   
   > proofs that need to be finite.   
   >   
   >>   
   >> That is the way that   
   >> "true on the basis of meaning expressed in language"   
   >> has always worked. When math diverged math erred.   
   >   
   > Nope. Not unless you mean by "meaning" to include the infinite chain for   
   > reasoning.   
   >   
   > Note, "Formal Systems" don't work the way you want, as their "semanitcs"   
   > are defined from the axioms and the operations of the system, possible   
   > continued for an infinite chain of operations.   
   >   
   > Your problem is you just don't comprehend how infinity works, because   
   > you mind is just to small.   
   >   
   >>   
   >>> If you want to limit a "Theorem" to only be a something provable in   
   >>> the base system then it is merely a True Statement in the base   
   >>> system, which the system can not be proven.   
   >>>   
   >> So when we directly encode all semantics   
   >> in the formal language such that   
   >> ∀x ∈ F (Provable(F,x) ≡ True(F,x))   
   >> Then incompleteness ceases to exist   
   >>   
   >   
   > Nope, because you CAN'T do that unless you system can't support the   
   > Natural Numbers.   
   >   
      
   What do you think is missing from   
   "true on the basis of meaning expressed in language"   
   about natural numbers?   
   add/subtract/multiply/divide is all there   
      
   > Sorry, you just aren't allowed to ASSUME something like that.   
   >   
   > Your world is just exploded into a totally inconsistent mess.   
      
      
   --   
   Copyright 2025 Olcott

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

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