[FOM] strongly minimal and minimal structures

Dave Marker marker at math.uic.edu
Wed Apr 21 19:32:51 EDT 2010

My answer below, while correct is lacking in detail.

>1. Given a first order language L and its extension L', and an
>L'-structure M, if M as a L'-structure is strongly minimal, is it
>still strongly minimal as an L-structure?

Yes.   Any set definable in L is also definable in L' and hence
finite or cofinite.

To be more precise.  Let T and T' be the L and L' theories
of M respectively.  Let N be a saturated model of T.
Then N can be expanded to make it a model of T'
(see ex 4.5.35 in my book).
Since T' is strongly minimal, N is minimal and since N is
saturated, N is strongly minimal.

Why is there a saturated model of T?
Here are three options:
* T' is strongly minimal so any sufficiently large model of T'
(and hence the L-reduct) is saturate
* T is a redcut a strongly minimal theory T', hence there
omega-stable, so there are saturated models of every cardinality
* The conclusion we are trying to prove is absolute so it is harmless
to assume GCH--25 years ago I'd have listed this argument first :)

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