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By Angus Macintyre, et al

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W,5 We begin by proving iMv,,n,(~,y) where C stands for any conjunct of E2 A H except M V tn (u,y). If x < p + 1 and C = M ( z , ~ ) , then we prove ( 3 ) with7(ii),(iv). If x p + 1, then ( 3 ) x ,A with C = M ( z , ~ ) follows from 7(vi). The same argument also works for x,X i Mn,,X,(z,~)and i Mv,,n,(u,g) because we have v’ > p + 1. Use 7(vii) when C is one of the conjunct8 of H. E. DONER, A. MOSTOWSKI and A. TARSKI 30 Let G be the conjunction of all the conjuncts in the matrix of the right hand side of (41, except MvSn(u,y).

So t h a t by t h e axiom Q w - 0 . @EX<->+, 6. Given f u l l s e p a r a t i o n and a f o r n u l a 4 l e t x = { y s { a l l + } where y i s not f r e e i n 9. Then @EX+>+. Conversely l e t A be a set and $(y) a formula. assumption t h e r e i s a set x such t h a t +(~)<->QEx,f o r each YEA. t h a t x s { f l 1 i n which case x i s uniquely determined by YEA. By We may assume By s t r o n g c o l l e c t i o n t h e r e i s a f u n c t i o n f defined on A such t h a t ( ~ ~ A ) ( 9 ( y ) < - > g ~ f ( y ) )By .

Because of Vaught's theorem, it is only necessary to show that every element of a structure of canonical type is definable in that structure. We begin with the following remark, which applies to any simply ordered structure I : (1) For if If a , b c UnI, [ a , b ) < b is definable in 1. [a,b) = a formula G i x ) , < ow, then a Fv(CiX))= {x). , It follows that 8: defined by a k G P ) I b 1 . and also THE ELEMENTARY THEORY OF WELL-ORDERING --A M E T A M A m T I W STUDY-in Now suppose a i s defined by F For any formula E, Fu(E) = { x ) , I, 47 and l e t and ' b we have UnI, c Now l e t = X 'F A iff B C $b'l (F') (G,) b' * = b, so t h a t b is defined i n I by B .

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