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Let λAo = No- We define, by recursion, an elementary chain of models Mi, i < ω, and elementary maps gι : Mi —> Mi+\ such that / C go C . . C gi C To begin, let K, = |Mo| and let λΛ'γ be a «+— saturated elementary extension. 7 there is an elementary map go D f taking M o into M[. Let λΛ\ be an elementary submodel of λ4[ of cardinality K containing both Mo and go(Mo). In general, let Λ^^+1 >- ΛΛi be a ft+—saturated model, gi+ι : Mi —> Λf/+1 an elementary map extending g^ and Mi+\ an elementary submodel of λΛ'iΛ_λ containing both Mi and g%{Mi).

Proof. Suppose that a and b are finite sequences realizing the same complete types in M, and c G M. Let φ{yx) G tpM(a>c) = Q be a formula isolating q. Λ/f (= φ(bd). Since the formula y? isolates a complete type tpM^c) — tpM(pd), proving the homogeneity of M. Homogeneous countable models of the same complete theory are not necessarily isomorphic, indeed, many of the above examples have a countable saturated model and a prime model which are not isomorphic. 5. If M and Λί are countable homogeneous models in the same language which realize the same elements of 5(0), then M =λf.

Properties (a), (b) and (c) are easy to verify. Let Y denote the set of sequences of O's and l's of length ω and for / G Y let pf = { φs : 5 is an initial segment of / }. Each pf is consistent (by (b)) and for distinct / and g in Y, pf U pg is inconsistent (by (c)). Consistent completions of the p/'s form 2H° many elements of 5 n (0), completing the proof. 1 it was proved that a small theory has a countable atomic model. The next proposition generalizes this result to potentially uncountable theories.