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By Wilfrid Hodges

ISBN-10: 0521066360

ISBN-13: 9780521066365

ISBN-10: 0521304423

ISBN-13: 9780521304429

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P is a A-m-type over A in M i f (1) p is a set of formulas of the form p(Z;a) where a E A E IMI, Z = (xo,.. ,x , , , - ~ ) ,and p(Z;p) or ~ p ( 28) ; belongs to A. , p is finitely satisfiablein M : for every finite q E p , M C (3Z)A\(PEP p. xxxiii NOTATION If we are not interested in A, we just write p is a A-m-type in B. Types are denoted by p, Q and T . We write m-type instead of L-m-type, A-type instead of A-1-type, p-m-type instead of {p}-m-type, and A-( < K,)-type instead of A-m-type for some m < w .

The proof is by induction on B. For = 0 (*) says that if {V, #,} u r is consistent, then there is i such that {$,} u r is consistent, and this is correct. When B is a limit ordinal the claim follows directly from the induction hypothesis. We must now show the passage from p to B + 1. Assume by way of contradiction that Rm[{V,#,} u r, A , A] 2 p + 1 but max, P[{#,} u r, A , A] < B + 1. Since P[{#,} u r , A , A] < B + 1 for every i = 1,. . 1(3). Let q = q,, p = (n + 1) max,p,. A is infinite by aaumption, 80 p < A.

Several kinds of ranks were used, and most of them are particular caaes of P ( p , A , A), on which we concentrate. We investigate them 20 RANKS A N D INCOMPLETE TYPES [CH. 11,8 0 alao when there is no apparent application; more information is obtained in Chapter 111, Section 4, and Chapter 5, Section 7. Rm(p,A, A) is interesting mainly for A = 2, No,00 and A = L or A finite. What is the meaning of the rank Rm(p,A , A)? For finite p , we can say that it measures the complexity of the family of sets {a: a realizes p u { c p ( ~ ; 6))) for v E A , 6 E 6.

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Model Theory by Wilfrid Hodges

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