NOTATION xm a K P A ^ iff 2 1 K P a n d 21 K V 21 f=s 3 t ^ iff there exists a valuation 5* such that 21 |=s* y? and s(k) = s*(k) forallfc^i. Let (/? G Form, let a^0,..., airi_1 € A, and let 5 be a valuation. We write 21 K p[aiOJ-5a2n-i] if 21 |=5* tp for the valuation 5* that is obtained from s by replacing s(j) with s*(j) = a3 for j = i 0 ,... ,i n - i and leaving s*(j) = s(j) for j ^ i0,... ,i n _i. One can show that if all free variables of (p are among Vi0,... ,Vin_1 and 5o, «i are valuations, then 21 |=S0 ' P k v »Oin-i] iff 2l|=ao y?[aio,... ,ai n _J. In particular, if p is a sentence, then the satisfaction relation 21 f=s y? does not depend on s. If 21 (=s p for every valuation s, then we write 21 f= (/?. When working with specific formulas, we shall often use more suggestive, self- explanatory terminology. For example, if 3Jt = (M, E) is a model of Ls and a G M, then we shall write u Wl \= 3x(x G a)" instead of "SD T |= 3vi(t i G v0)[a]." A theory in a language L is any set of sentences of this language. For example, ZFC is a theory in L5. Let T be a theory in a language L. We say that a model 21 of L is a model ofT and write 21 |= T if 21 |= p for each (p e T. A theorem of T is a sentence tp of L that has a formal proof from the sentences of L. We write T h (p to indicate that (p is a theorem of T. By Godel's Completeness Theorem, (p is a theorem of T if and only if 21 |= (^ for all models 21 of T.
Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 1997 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.
