FOM: scott sets

[Dave Marker]

Harvey asks
>"..given any consistent extension X of PA, what are the countable
standard parts of models of X? "

I know a partial answer to this. If T is a complete extension of PA
and S is a countable Scott set then S is the standard part of a model
of T iff the set of \Sigma_n consequences of T is in S for all n.
(This is due to Julia Knight.)

A consequence of this is that for any countable Scott set S and any real X
there is a complete extension of PA with a complete extension T of PA
with X recursive in T such that S is the standard system of a model of T.

Dave Marker