XPost: comp.theory, sci.math   
   From: news.dead.person.stones@darjeeling.plus.com   
      
   On 05/01/2026 11:18, Tristan Wibberley wrote:   
   > On 04/01/2026 18:55, Mike Terry wrote:   
   >> Gödel's concerns there being a statement G such that neither G nor ¬G   
   >> has a derivation in the system.  There is no reference to "truth" in   
   >> that and I'd say his proof is essentially syntactical in nature.   
   >   
   >  From Curry and Feys very brief mention of the distinction I think   
   > Gödel's system P is a semantical system (it has numbers as objects   
   > distinct from their presentation - which allows him to just make it all   
   > the more complicated). Does that mean his proof must be semantical?   
      
   I'd say not necessarily...   
      
   The system P is motivated by wanting to discuss claims about arithmetic, and   
   we can think of   
   statements in P as being about numbers.  For example Godel describes variables   
   as representing   
   natural numbers.  But also we can regard P as a purely formal system with no   
   given meaning for its   
   symbols.  The question would be whether Godel's proof only works if we   
   interpret the statements as   
   having their arithmetic interpretation, or does the proof work even if we give   
   no interpretation for   
   the terms?  [Of course, manipulation of terms and constructions of derivations   
   must follow the rules   
   of the system, but those do not rely on interpreting terms as natural numbers.]   
      
   If Godel's paper had said "P contains a statement that is TRUE but unprovable"   
   that would be a   
   semantic claim, because to say whether a statement is true we need to   
   understand its meaning.  But   
   for Godel, incompleteness meant there's an "undecidable" sentence that can be   
   neither proved nor   
   disproved within the system.  That requires only understanding how to   
   manipulate strings of symbols   
   according to the syntactic rules for constructing formal proofs within the   
   system.   
      
   >   
   > Also he relies on a meta-system which means embedding, does that force   
   > the proof to be semantical even if derivations in P are syntactical?   
      
   This is trickier.  It's been a Long Time since I looked at any of this, and   
   I'm not going to have   
   the time it would take to refresh my understanding.  I suppose the point is   
   whether or not the proof   
   relies on the /meaning/ of terms and symbols as being numbers for the proof to   
   work.  It's ok that   
   Godel /uses/ the natural numbers in his meta-system to discuss properties of P   
   - for example in   
   defining a mapping from N to terms of P taking n to the /numeral/ for n.  This   
   isn't assuming any   
   meaning for the terms of P, so this much doesn't make it semantical.  But...   
   the proof is long and   
   I'm not as familiar with the details as I once was [which even then was well   
   less than 100% :( ], so   
   really we need a professor or someone who works routinely with all this to   
   help us!  Sci.logic used   
   to have some very knowledgable contributors, but I fear they've left over time.   
      
   >   
   > I haven't got a handle on semantical vs syntactical.   
   >   
   Yeah, I have difficulties sometimes.  E.g. with Propositional Logic we can   
   define the length of a   
   sentence as the number of symbols it contains, and that is clearly   
   "syntactic".  Similarly for the   
   "rank" of a sentence, being the depth of nesting of logical operators.  In   
   both cases we can write a   
   program to inspect the string involved and simply tell us the answer, fair   
   enough.  But we can also   
   write a program that simply inspects a sentence and tells us whether that   
   sentence is a tautology or   
   not, based just on its structure, so is "tautology" a syntactic or semantic   
   concept?  Authors   
   reasonably sidestep the question, because in the end the answer doesn't matter   
   for their subject   
   development...  [Lots of people insist it is a semantic notion, but it can be   
   defined without   
   assigning any meaning to the sentence in question - just look at the symbols   
   and calculate!  I   
   believe Smullyan defines "tautology" in exactly that way...  So I'm unclear on   
   this - perhaps it   
   depends on the wording that introduces the term.]   
      
      
   Mike.   
      
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