[FOM] AI-completeness (and settling CH)

Robert Lindauer rlindauer at gmail.com
Fri May 11 04:40:51 EDT 2007

> 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.

I'm envisioning the following "proof structure"

T_0 decides P
T_0 extended by S = T_1
For any S-like extension of T_1 T_(n) decides P The same way T_0 does.

-> A T_(n), T_n-> P ->-> P

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).

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.


Robbie Lindauer

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