[FOM] Question about hard-core independence

[Haim Gaifman]

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).

Suppose that T is axiomatizable by a set of axioms that
represent all that we accept as evidently true, under some
presupposed interpretation of the language.  A hard-core
independence for T may raise some doubt whether our intended
interpretation of the language really picks up a unique
model (or a unique "world"). This doubt may not be decisive,
but it may linger nonetheless.

Goedel's incompleteness results are not cases of hard-core
independence, since we assume the consistency of T and
therefore have strong reasons for believing in the truth of
the Goedel sentence.  The same goes (for the weaker version
of hard-core independence) for arithmetical statements that
are equivalent to the consistency of large cardinal assumptions.
The reasons for believing the consistency are the reasons
for believing the statement.

We have no hard-core independence for arithmetical
statements.  The set theoretic independence results obtained
via forcing constitute, on the whole, hard-core
independence.  It appears that they are due to a looseness
in the concept of the powerset. There is a lot of slack
between it and the rigid skeleton of the ordinals.  The
version of set theory in which there is no such slack, since
all sets are rigidly related to the ordinals, is ZF + V=L. It
appears that after the natural numbers this is the next
candidate for a tight model. Hence the question.

The existence of an inaccessible (or a Mahlo cardinal) is
not a case of hard-core independence for ZF+V=L, because its
relative consistency is not provable. But even for the
weaker notion it does not seem to be a good case, because the
reasons for believing in its consistency are too close to the
reasons for  believing in its truth. Of course, you can
believe in its consistency because you believe that it is
true in some inner model. But still, this does not seem
to me to be a good candidate.

Joel Hamkin tells me that he has a hard-core independence
for ZF+V=L in which the independent statement is of second
order (my question is about first-order statements).

Haim Gaifman