[FOM] Independence without forcing

John Steel steel at math.berkeley.edu
Sun Jul 13 15:39:50 EDT 2003

One version of this problem is: is there a mathematically natural
sentence phi such that

Con(ZFC) --> Con(ZFC + V=L + phi)


Con(ZFC)--> Con(ZFC + V=L + not-phi)

are both provable in ZFC.

(Presumably some ``unnatural" self-referential sentence phi will have
these properties--is that true?)

   I think this is a pretty well-known question. It calls for a
new method--the independence cannot be proved by forcing, nor by
comparing consistency strengths. It is, however, a somewhat odd
question, because those two methods are so powerful, and we seem to
have no candidates for a phi to which they will not apply. Often
enough, one has a problem, but no method to solve it. Sometimes
one has a method in search of a problem. In the case of the
question above, we are looking for both the method and the problem
it solves!

John Steel

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