By Steven Buechler

Balance concept all started within the early Nineteen Sixties with the paintings of Michael Morley and matured within the 70s via Shelah's learn in model-theoretic class idea. this present day balance idea either impacts and is motivated by means of quantity conception, algebraic crew idea, Riemann surfaces and illustration concept of modules. there's little version idea this present day that doesn't contain the tools of balance concept. the purpose of this ebook is to supply the scholar with a short path from easy version thought to investigate in balance conception, to arrange a pupil for study in any of ultra-modern branches of balance conception and to offer an advent to class thought with an exposition of Morley's Categoricity Theorem.

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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.