[FOM] Question about hard-core independence

[Torkel Franzen]

Haim Gaifman says:

  >Question: Is there a hard-core independence result for
  >ZF + V=L ?
  >The definition of hard-core independence and the motivation
  >for the question are as follows:
  >An independence claim for a first-order theory T is a claim
  >of the form: neither A, nor not-A is provable in T. Let a
  >*hard-core* independence for T be an independence claim such
  >that:
  >(I) The claim can be derived from the consistency of T, and
  >(II) we have no good reasons for believing either in the
  >truth of A, or in the truth of not-A.
  >Note that this notion depends on our epistemic state.
  >The notion can be weakened by requiring in (I) that we have
  >strong reasons for believing the independence claim (instead of a proof
  >of it from the consistency of T).

  We can produce such hard-core independence results of an uninteresting
kind by means of Gödel flummery. Suppose

  (1) We know T+Con(T) to be consistent,
  (2) C is any arithmetical statement for which we have no good reasons for
      believing either in the truth of C or in the truth of not-C.
      (This could for instance be a statement of the form "k has exactly
       two prime factors".)

  Now let B be the sentence

           Con(T+Con(T)) => C

  For this B we have

  (3) B is compatible with T+Con(T), by (1) and Gödel's theorem.
  (4) We have no good reasons for believing either in the truth of B
      or in the truth of not-B, by (1) and (2).

  Now choose A so that T proves the true statement

  (5)      A <=> B & ~Provable-in-T(A)

  For this A we have

  (6) T does not prove A, for then T would also prove Provable-in-T(A),
      so by (5) T would prove not-A and be inconsistent
  (7) Thus we know that A is true if and only if B is true, so we have
      no good reasons for believing either in the truth of A or in the
      truth of not-A.
  (8) T does not refute A, for then T would prove "A is refutable in T"
      and "if B then A is provable in T", so T would prove
     "if B then T is inconsistent", so T+Con(T) would refute B,
      contrary to (3).