By William Craig

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One c'trn find some h,. ritid cp,. ,fiw e t i c h U # @, h,. reduces r 1 171 rind [cp,. to each ~ othcr. ( b ) Lcit tiny m < o. t < o. cp,,. cpl. for etrch r s t , cp,. is of some type ( r ) - ( s i )Isis,,,,. dom h,. , m ' } ,run h,. , m } , m d . (j ) . fi)r each r G t , h,. [ to e ( I ch 0 t h 0 r . Next. we turn to model theory. Using the pairing function p r of ch. 1. we define a I - I -correspondence between the relational structures and the lo-structures (which have been defined in chs.

6(e), and the definitions of d;, and d( respectively. 0 We now turn to derivations from E, which involve one o r more of the equalities 3(a), 3(b), 3(c). 7: (a) k ckdij= dij,fur k # i und k # j . (b) k ck(dij* x) = dij c,x,for k # i und k # j . (c) cidii= q t i i n . r . ( i . j ) t l 1. (d) k qk+'l d i j= d,;,for /< < mux (i,j ) . ; y)for i # jund-2 < i-j (f) k ck(dik* dk,J= qk+'l d,;,for k # i and k # j . dik dk,l= dik dij. (8) PrmJ (a) Assume that k # i and k # j . Case 1 : i =j . Then - - - - - C 2.

We let I L,, I be the language Lpq together with the interpretation IuI of its terms u. 5, the set cl uve" / L a ! At times we shall associate with the T just considered that v-ary is 171( X6,) l s j s l l , . Iw-operation whose value for each argument ( X8)G

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