Tiny models of categorical theories (Q1204109)

For \(T\) a theory, let \(| T|\) denote the number of \(L\)-formulas modulo \(T\)-equivalence. A model \(M\) of \(T\) is called tiny, if \(| M|<| T|\). In Chapter 2 the authors give a new proof of the following result of Lovey: If \(T\) is categorical, non-totally transcendental and nontrivial, then \(T\) has no tiny model. Their proof essentially simplifies the original one. In Chapter 3 the authors investigate the following statement \(\Phi\): ``If \(T\) is strongly minimal and \(M\) is a tiny model of \(T\), then \(M\) is countable.'' They show that this statement is independent of ZFC. First they show that \(\forall\kappa (\text{cf}(\kappa)>\omega\to \kappa^ \omega= \kappa)\) implies \(\Phi\). Then they use a forcing developed by Baumgartner to show that \(\neg\Phi\) is relative consistent with ZF.