Logic Colloquium 1982: Proceedings by G. Lolli, etc.

By G. Lolli, etc.

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B. ii. the language L, ( PAlProvgC(;,‘jl’) A ProvpC(ul, L ‘Ji * If $uvo is any formula withu free and other free variables) such that PAC$vuo L formula Prov($)(v ‘U 0 ’ 1’ $(r$7,uo) }, by $: + ProvpC(uO~V1*(rX”), L ‘-x-). vo either free or vacuous (and no ”U is a sentence of L”, we define a v 2 ) , supposedly representing proofs overPC from axioms L(uO;ul Prov($) iii. + ) xT) ,u2) : 3 finite Q5 I U : $UU 0 , r m ~ U; I(ProvpC(ul L + )). We also define : Pr($)(uo;uz) ulProv($)(VL 0 ;v ,u ) , 1 2 suppressing mention of the language.

The schematic derivability of Ramsey's Theorem in PAL was established in Jockusch 1 9 7 2 . Incidentally, Jockusch gives exact bounds on the arithmetic complexities of the homogeneous sets in terms of those of the given partitions. I note that the proof via conservation yields explicit bounds, but confess I McAloon, in McAZoon 1978B, haven't checked how close they come to Jockusch's. derives upper bounds model theoretically. 1: SSy(No) is c-closed. 7 that SSy(M) is c-closed for nonstandard M.

G. Paris and Hurrington 1577 or Ketonen and S o ~ o u a y1581 ) . e. theory has, not only an arithmetically definable completion, but also an arithmetically definable (in fact, A,) model. construction can routinely be formalised within PA. The fact is that the whole The problem is that, although 39 Lectures on Nonstandard Models of Arithmetic the construction is routine, it is a notational morass. I turn now to the unin- spiring discussion of syntactic preliminaries. 6. Notational Conventions. i.

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