[FOM] Question about hard-core independence

[Haim Gaifman]

Obviously, I was trying to find a definition
that would rule out uninteresting arithmetical
examples obtained by Goedelian techniques, and
I did not succeed.

Actually, your construction leaves something open.
 Your  assumption:

>   (1) We know T+Con(T) to be consistent,

should be interpreted as having a proof of this in some
 theory. The theory T to which I applied
the notion of hard-core independence is ZFL.
Your construction yields a statement whose
independence is derived not from con(ZFL)---as I
specified---but from con(ZFL+con(ZFL)).
But this is bad enough.

Furthermore, Harvey Friedman provided an arithmetical example
(using Goedelian results) in which the relative
independence is derivable in PA from Con(ZFL):
 Let S be any extension
of ZFL obtained by adding a statement for which we
have no reason to believe either that it is provable or
disprovable in ZFL. Then the sentence:
R(ZFL) ==> Con(S) has the desired properties, where
R(ZFL) is the Rosser sentence for ZFL.

This creates the following situation:
Take any set theory T  you are prepared
to adopt for the purposes of doing math.
Then there exists an arithmetical statement of
the form: D&~D', where D and D' are diophantine
sentences (that is, claiming the existence
of  solutions to certain diophantine equations), such that
we have no idea about its truth and such that,
if T is consistent, then T does not decide it.
This however does not add doubts about
the reality, or uniqueness, of the standard model of natural
numbers. If I have doubts, they do not stem from
such examples, especially after I have seen
the proof of such independence results.

On the other hand the set-theoretic independence
results do raise doubts about the reality of
"Cantor's Universe", or about its uniqueness.
The doubts depend on the methods these
claims are proved, rather than on the claims
themselves. The methods manipulate
directly the semantics rather than the syntax.
Can we have a more precise
characterization of such "hard-core independence"?
One can of course rule out, by definition,
examples that can be reduced to
arithmetical statements.. But I do not find this a
satisfactory solution.

My first question still remains: is there
a "hard-core" independence result
for ZFL? But now the notion is not
precisely defined. It indicates the
kind of non-arithmetical result
obtained for ZFC.

Haim Gaifman