FOM: Goedel: truth and misinterpretations

Torkel Franzen torkel at
Thu Oct 26 02:52:33 EDT 2000

V.Kanovei says:

  >There is no one concrete sentence 
  >there, all we know is that they do exist in plentitude.

  They're concrete enough, as pointed out by Martin Davis. To be sure,
known instances are pretty unwieldy.

  >As a mathematical fact I have no objection, 
  >assuming that "there exist infinitely many ... provable in ZFC" 
  >is understood as a single ZFC formula which describes the intended 
  >meaning of the sentence in a known proper way.

  This is quite beyond me. We don't normally express mathematical statements
"as a single ZFC formula", and it's not clear what you are requiring.

  >The mathematical statement correctly saying that 
  >(CH and not-Prov_{ZFC}CH) or (not-CH and not-Prov_{ZFC}not-CH) 
  >is true in the metalanguage of mathematicians about mathematics, 
  >i.e. there is a proof of it. 
  >Neither of the two parts is true in the meta-language UNTIL 
  >somebody has this established, so are laws of mathematics, 
  >and there are reasonable doubts that this can be reasonably 

  You're making a metaphysical counter-claim to the metaphysical claim
"CH is true or false". This doesn't really take us any further. We
need to understand what difference any of this makes to our thinking
in or about mathematics.

  >ZFC was taken because its axioms fit mathematical practice, 
  >this is almost an empiric observation despite it refers to 
  >nonmaterial objects.

  Yes, but does this mathematical practice have any kind of justification?
Could we use just any old axioms in mathematics with equal justification?


  Torkel Franzen, Luleå university

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