FOM: a question re. completeness
Stephen G Simpson
Michael Detlefsen 3 Feb 1998 10:54:27 writes:
> T is consistency-complete iff for every sentence s of the language
> of T that is not provable in T, if s were provable in T, T would be
I would say that this isn't a very precise definition, one of the
difficulties being that it contains a counterfactual conditional. Let
me try to make it a little more precise, as follows.
Background: We know of many "mathematically natural" statements in the
language of PA that are independent of PA. The first such
independence result was due to Paris-Harrington. But all of these
independent sentences that we know of seem to satisfy (*)
(*) either PA |- S -> Con(PA), or PA |- (not S) -> Con(PA) .
(Here Con(T) means "T is consistent".) Of course we can diagonalize
to make up artificial sentences S in the language of PA which do not
obey (*), but they aren't "mathematically natural", at least I don't
know of any that are. So this seems to be an interesting property of
PA. Furthermore, it's a property that is not shared by ZFC, because
for example neither the continuum hypothesis nor its negation implies
Con(ZFC), and ditto for a host of other independence results proved by
The property in question, which T = PA has and T = ZFC doesn't have,
For every "mathematically natural" proposition S in the language of
T, either T |- S -> Con(T) or T |- (not S) -> Con(T).
This seems to be a foundationally interesting property of T = PA and
possibly other important f.o.m. systems.
Questions: Can we state property C in a more correct and/or more
comprehensive and/or more precise way? Am I overlooking something?
What's going on here?
I'm sure that many people here on the FOM list can do better than I
did above. I wouldn't be surprised if property C is already well
known to many people, in some precise version.
I guess this ties in somehow with the recent Friedman/Steel discussion
of "pictures" and CH. Can anyone spell out this connection?