FOM: Simpson's query on my response to Silver

[Torkel Franzen]

  Hartry Field says:

  >The hypothesis I was entertaining (and maintaining the coherence of, though
  >not endorsing) was that there is no more determinacy to the concepts of
  >'finite' and 'natural number' than the most powerful recursively axiomatized
  >first order mathematical theory we accept gives them; so that sentences
  >undecidable in this theory M (or more exactly, in all reasonable candidates
  >for it--this qualification is necessary because of the vagueness of
  >'accept') lack determinate truth value.

  The suggested hypothesis is not untypical of philosophizing about
the possible "indeterminacy" in basic mathematical concepts, in that
it itself rests on concepts that are not only possibly but demonstrably
indeterminate.

  In the present case you invoke "the most powerful recursively
axiomatized first order mathematical theory we accept", you invoke the
notion of a "reasonable candidate for [such a theory]", and you
quantify over all such "reasonable candidates". All of this doesn't on
the face of it make any obvious sense. Do you have any reasonable
candidates to suggest? What kind of acceptance are you talking about? How
definite is this range of "reasonable candidates" supposed to be?

  As the "hypothesis" now stands its coherence and indeed validity can
hardly be put in question, but only because it's unclear what the heck
you are talking about.

  >If GC is undecidable in PA, it may
  >well be undecidable in (all reasonable candidates for) M, so it might on the
  >hypothesis entertained lack determinate truth value.

  It is entirely obscure what you are claiming here as long as the
idea of "undecidability in all reasonable candidates for M"
hasn't been given any substance.

  >I took the subtext of
  >Silver's argument to be an objection to the coherence of this.  The response
  >of course is that though the undecidability of GC shows its truth in the
  >standard model, it may not be true in all models of (all candidates for) M.

  While "GC is true in the standard model" is an ordinary mathematical
statement, not on the face of it problematic, "GC is true in all models of
all candidates for M" doesn't even on the face of it have any definite
meaning, so again it's an open question what to make of this response.

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Torkel Franzen
Computer science, Lulea technical university