[FOM] AI-completeness (and settling CH)

[James Hirschorn]

On Friday 11 May 2007 04:40, Robert Lindauer wrote:
> > Such a scenario is hypothesized in some slides of John Steel (laguna.ps
> > on his web site). A sequence of theories T_0, T_1, ... is postulated
> > there which in the presence of large enough cardinals, assuming that they
> > are not destroyed by small forcing notions, decides the theory of
> > V_omega+2 or at least all the sentences whose truth value can be forced.
> > If the existence of such a sequence of theories can be proved, and if it
> > can be proved that any such sequence of theories agrees on CH, then (at
> > least up to my understanding) the truth value of CH will have been
> > decided, or at least reduced to the question of the consistency of the
> > needed large cardinals.
>
> Does this propose a kind of induction over theories?  If so, the range
> of the induction would still have to be outside of the range of any of
> the theories.
>

Yes, he does specify that the theories should be recursively axiomatizable. 

The other requirements on the T_n are that, under a suitable large cardinal 
hypothesis,
1) Each T_n can be forced over V.
2) T_n \subset T_n+1.
3) The T_n are generically complete for the \Sigma^2_n theory, i.e. any two 
forcing extensions of V satisfying T_n satisfy the same \Sigma_n sentences 
(referring of course to the Levy hierarchy) over V_\omega+2 (of the 
respective forcing extension). 

<snipped>
>
> For all S-extended T Theories P is true, therefore P is true.
>
> Two problems:
>
> First, there could be non-S-like extensions of T which decide
> otherwise (CH is indepedent of ZFC).

I think perhaps your point is that it must be demonstrated that theories not 
fulfilling the criterion can be ignored for purposes of resolving CH. Indeed, 
after giving it some more thought, it is not obvious to me a priori that if 
every sequence of theories as above decides CH the same way, then CH has been 
resolved once and for all. Of course, it would be nice if someone more 
knowledgable could shed some enlightenment on the matter.

In any case, we should probably keep in mind the current state of progress. So 
far (to my knowledge) there has only been one example of a theory T_1 
satisfying the criterion, and no T_2 has been proven satisfactory. 
The known example is T_1 = CH (or should it be ZFC + CH?). This is Woodin's 
celebrated absoluteness theorem that CH is generically complete for all 
\Sigma^2_1 sentences. Steel asks in his slides if T_2 = diamond works. There 
has been a huge amount of work towards a sequence of theories that would give 
a very different interpretation of the theory of V_\omega+2. In fact Woodin 
describes his book "The axiom of determinacy, Forcing axioms, and the 
nonstationary ideal" as a version of his above mentioned absoluteness 
theorem, for not CH. According to Dehornoy's article, it has been shown by 
Woodin that his axiom (*) is generically complete for all n, and thus only 
the criterion (1) above is missing for determining the theory of V_\omega+2.

My point about the current state of progress is that if hypothetically we ever 
arrive at a point where it has been demonstrated that all such theories agree 
on CH, then additional phenomena will likely have been exposed which together 
with the theory leave no question about CH. This is the sort of thing that 
happened with the theory of V_\omega+1 where PD has been proven true (well, it 
was proven to be a consequence of the existence of infinitely many Woodin 
cardinals, the consistency of which has been corroborated by inner models). 

>
> Second:  "any" or "all" appearing in the conclusion has a domain that
> is too large for discussion "all the possible CH-deciding theories".
> There would be no "mechanical way" of deciding whether or not T_(n)
> was an S-extension of T.
>

Perhaps I am missing the point, but I don't see any problem with quantifying 
over all such sequences of theories (though there may very well be one). Note 
that this scenario only concerns the theory of V_\omega+2 and does not even 
begin to address V_\omega+3.

James Hirschorn