By J. N. Crossley, Anil Nerode (auth.), Prof. Dr. Rohit Parikh (eds.)

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CONTENTS
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Bibliography
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Tw F We claim that M, M F s Tw Suppose not =t , for any R-terms Then there is a partial homomorphism from ~ = s = t TW 1= s Tw~ s = t if and only if for all countable R- t • Then by Lemma 4, all R-structures t. Assume TW TW ~ s ! t[f] containing M will be a countable R-structure, and To this end, let M have Then let M 1= s t • M be a Rng(f) , in the appropriate M F s ! t[fl. We now wish to complete the proof that the relation n~ 1/ s,t. is t S countable elementary substructure of sense. N~ s M,N A special case of (the proof of) Lemma 18.

Tions. fornrulae considered in his system All* (The ~­ are more than sufficient to define infinite paths of infinite recursive finitely branching trees). NB. c. letters for formulae while we use Roman capitals (for nearly everything). 4. Transformation oft -founded trees; sharpening (*) in para. 3 by:(**) imposing (additional) reQuirements on mappings from to R2 -derivations (of A) ) when (*) is true. Rl -derivations (of A) In the literature the distinc- tion between (*) and (*¥) is sometimes expressed by nonnal form and normalization theorems (when cut free rules are called 'normal').

A(x 1 ,x 2»,f1"3 (x 3 ),···,f1"n (x n » (xl),f (x 2 »,f (x 3 ), ... ,f (x +1» 1"1 1"2 1"3 1" n+1 n appropriate 1". Suppose this is straight from the defini- Using this, we have B(B(f N. ' 1 (xl), ... ,f = 1"n (x +1» n for The following Lemma is the analog to Lemma 16, for the R-calculus. LEMMA 2. If (f } C1 aSSignment, h then is a partial homomorphism fr om M onto = hex) is an N-assignment, fa(g(x» f[J(Val l (s,g» = Va1 2 (s,h) , for R-terms for variables of type x g is an M- of type a, cr • H,N, l f} are fixed.

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