[FOM] Question about hard-core independence

[John Steel]

  All of the statements given so far are decided by very weak
large cardinal axioms, presumably, though we don't know which
way they go yet. I think that is why they are unsatisfying as
examples of "hard-core independence". They are probably not
independent of the sum of what we have good reason to believe.
(Of course, it could turn out that your S is some arithmetic sentence
which is provably independent of all the large cardinal axioms we know of,
but if you find such an S, then ...Wow!)

  I personally don't think V=L is independent of the sum of what we have
good reason to believe, since it is decided by large cardinal axioms.

  Turning "the sum of what we have good reason to believe" into the basis
for a precisely defined concept of hard-core independence seems hopeless.

  I took the original question as a way of asking whether taking "set" to
mean constructible set removes all ambiguity from the language of set
theory. (That may not have been how Prof. Gaifman was thinking of it.
It does seem implicit in the latest formulation involving
independence methods which "manipulate directly the semantics".) I think
adding V=L may very well remove all ambiguity, but at the cost of speaking
a less expressive language. I do think there could be a way to show that
the full language of set theory is ambiguous without getting into abstract
theories of meaning.  We do that all the time, by exhibiting the distinct
possible meanings. (Einstein didn't need a theory of meaning to show that
"A is simultaneous with B" is ambiguous.)

   Whether there is actually any such ambiguity seems like an open
question, to me. The proving conditional generic absoluteness theorems at
the $\Sigma^2_n$ level for incompatible theories which are provably (from
large cardinal assumptions) true in (necessarily different)  set-generic
extensions is one approach to exhibiting an ambiguity.



John Steel




On Mon, 13 Jan 2003, Haim Gaifman wrote:

> 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
>
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