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Finally, we sometimes write PrT and ConT, when no canonical representation is given, to denote the predicates used when some specific representation is in mind. iii. If xuu 0 has u free, possibly v0 free, and no other variable free, and x satisfies PA C p u 0 “U is a sentence of L“, we x defines truth for some structure in the language I,. if, for a given language L , let Mod(dVo Mod(& 0 assert that +. is the conjunction of the following: a. v Sentences ~ i ~ , X(~VJ,YJ,~ ~ i ~ ( ,uo) t-t xf$; , u , ) ~ x ( ‘ $ ; ,vo) ) c.

Chqose N = u &ED M,. n The Arithmetised Completeness Theorem has other applications. For some of these cf. Smoryn'ski 1 9 7 3 A&B, Srnoryn'ski 1 9 7 7 , M c A t o o n 1 9 7 8 4 , and Srnoryn'ski 1982C. 7. POWERFUL ARITHMETISATIONS (THE 1970s 111) The Arithmetised Completeness Theorem is a fairly powerful tool. As we saw, it has many applications and it gives a variety of end extensions-- particularly, C -elementary ones. But it does not seem to yield fully elementary end extensions. , from it. the fully formalised Ramsey Theorem However, Ramsey's Theorem can be proven in PA and elementary end extensions exist.

I n proof theory (in the Orey Compactness Theorem-- cf. Fefemun 1960-- and in Smory6skC 1982C) this power is occasionally necessary; in model theory it-particularly the uniformity-- is generally not necessary. 9. Definition. Let M be a model of PA and N a structure for some primitive recursively encodable language L . We say that N is s t r o n g l y definable in M iff i. IN( = IM[ ii. e. there is a formula Tr(uo;ul,t)Z) and some b E /MI such that for all $u0... V n- 1 and a . =Go a iff M l=Tr(F;r$',(zo n-1 n- 1 ...

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