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.

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