[FOM] consistency and completeness in natural language

[Torkel Franzen]

Hartley Slater says:

  >But my earlier 
  >posting also amplified on why facts are necessarily consistent, and 
  >showed that if it is provable that P, then it is true that P, i.e. P, 
  >so the above involves a crucial use-mention error.  Only formulae are 
  >'provable' in S, not facts: 'there are infinitely many primes' not 
  >that there are infinitely many primes, '0=1' not that 0=1.

  Formulas are indeed what formal systems prove. Nevertheless we say
such things as "it is provable in ZFC that every Polish group H
satisfies WVC(H)", "S proves the consistency of T", and so on, all the
time. What we mean by saying "S proves that A" is that S proves a
standard formalization of A. This is usually unproblematic, but as we
know, there are also cases where it is essential exactly how A is
formalized, and cases ("the consistency of T") where genuine problems
arise about choosing a formula to refer to. In these respects, "It is
provable in T that.." is just like "He said that..", "It says in
the paper that...".

  As for "provable that" and truth, there is nothing in e.g. "it is
provable in PA+~Con(PA) that PA is inconsistent" that justifies the
conclusion that PA is inconsistent. I doubt that people avoid
"provable that" in such cases.

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Torkel Franzen