XPost: comp.theory, sci.math, sci.lang   
   From: 046-301-5902@kylheku.com   
      
   On 2025-11-29, olcott  wrote:   
   > On 11/29/2025 3:39 PM, Kaz Kylheku wrote:   
   >> On 2025-11-29, olcott  wrote:   
   >>> On 11/29/2025 2:23 PM, Kaz Kylheku wrote:   
   >>>> On 2025-11-29, olcott  wrote:   
   >>>>> On 11/29/2025 11:53 AM, Kaz Kylheku wrote:   
   >>>>>> On 2025-11-29, olcott  wrote:   
   >>>>>>> Any expression of language that is proven true entirely   
   >>>>>>> on the basis of its meaning expressed in language is   
   >>>>>>> a semantic tautology.   
   >>>>>>   
   >>>>>> A tautology is an expression of logic which is true for all   
   >>>>>> combinations of the truth values of its variables and propositions,   
   >>>>>> which is, of course, regardless of what they mean/represent.   
   >>>>>   
   >>>>> I did not say tautology. I said semantic tautology.   
   >>>>> I am defining a new thing under the Sun.   
   >>>>   
   >>>> The existing tautology is already semantic. You have to know the   
   >>>> semantics (the truth tables of the logical operators used in the   
   >>>> formula, and the workings of quantifiers and whatnot) to be able to   
   >>>> conclude whether a formula is a tautology.   
   >>>>   
   >>>   
   >>> Try and show how Gödel incompleteness can be   
   >>> specified in a language that can directly encode   
   >>> self-reference and has its own provability operator   
   >>> without hiding the actual semantics using Gödel numbers.   
   >>   
   >> The numbers are essential, because Gödel Incompleteness is   
   >> about number theory.   
   >>   
   >   
   > The generalization Gödel incompleteness applies to   
   > every formal system that has arithmetic or better.   
      
   And there you are, trying to take the numbers out of it.   
      
   >> The Gödel Theorem involves a proof in which a certain number,   
   >> the "Gödel number" that may be called G, is asserted to have   
   >> a number-theoretical property.   
   >>   
   >   
   > G := (F ⊬ G) // G says of itself that it is unprovable in F   
      
   No, it doesn't; that is an outside interpretation of what it is saying.   
   Gödel's sentence says that a certain number isn't a theorem-number.   
      
   The interpretation that the number is the Gödel number of   
   that very sentence is made externally to the sentence.   
      
   Is there any part of your understanding that is accurate?   
      
   >> An example of a number-theoretical property is "25 is a perfect   
   >> square". Except we need it in more formal language.   
   >>   
   >> Gödel discovered that you can encode statements of number theory as   
   >> integers, and manipulate them (e.g. do derivation) by arithmetic.   
   >>   
   >   
   > That simply abstracts away the underlying semantics.   
   > G is unprovable in F because G is semantically unsound,   
      
   G is semantically sound, and can be adopted as an axiom.   
      
   > We can't see that with Gödel numbers.   
      
   A Gödel number can be decoded to recover the syntas of the formula.   
      
   In the case of the Gödel sentence, we don't need to do that; we   
   already know that the Gödel number decodes to that sentence.   
      
   >> Then it became obvious that whether or not a formula is a theorem   
   >> is a property of its Gödel number: a number-theoretical property.   
   >>   
   >> There are theorem-numbers and non-theorem-numbrers.   
   >>   
   >> The Gödel sentence says somethng like "The Gödel number   
   >> calculated by the expression G is not a theorem-number."   
   >>   
   >> But G turns out to be the Gödel number of that very sentence   
   >> itself.   
   >>>   
   >>>> Pick another word. Since only dimwitted crackpots like yourself will   
   >>>> want to discuss anything using that word, keep the syllable count low   
   >>>> and make sure there aren't too many off-centre vowels.   
   >>>   
   >>> Ad hominem the first choice of losers.   
   >>   
   >> I'm not making an argument; I'm suggesting a way of choosing   
   >> an alternative word, since "tautology" is taken.   
   >>   
   >>>>> *Semantic tautology is stipulated to mean*   
   >>>>   
   >>>> Reject; call it something else.   
   >>>>   
   >>>>> Any expression of language that is proven true entirely   
   >>>>> on the basis of its meaning expressed in language.   
   >>>>   
   >>>> You are gonna need to supply an example.   
   >>>   
   >>> The key is that a counter-example is categorically   
   >>> impossible.   
   >>   
   >> So you are saying every expression in a certain language   
   >> is proven true, so that its syntax admits no false sentences?   
   >   
   > It syntax admits anything that any human can   
   > say in any language comprised of symbols.   
      
   But that could be false. It is baffling by what you mean   
   bhy "counter-example is categorically impossible"; at ths point   
   it seems like a dodge from giving an example of sentence   
   that is proven true entierly on the basis of its meaning   
   expressed in language.   
      
   >> What language is that, and what are examples? What happens   
   >> when you try to make a false sentence?   
   >   
   > English, Second Order Predicate logic, C++...   
      
   How does C++ express a sentence that is proven entirely true   
   on the basis of its meaning expressed in a language;   
   do you need templates or Boost?   
      
   >> Is it possible to utter conjectures which later turn out false;   
   >> and if so, then what happens?   
   >   
   > Conjectures are not elements of the body of knowledge.   
      
   Some eventually are; but their syntax and meaning doesn't change.   
      
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