number-theory

4B Semantics and completeness 574A10 Theorem (Undecidability of TA2+p[n]) There is no algorithm that accepts arbitraryF, M, r as inputs and decides whether I, f[n] M:r.Proof Let F = 0 and T=- a--+a. For any M, if Hp[n] M: a-*a then M is closed andby 4A3, M has a of M* which is closed and has the propertyHoo eq M* : a-->aBut by the subject-construction theorem (2B2) it is easy to see that a closed /3[r]]-nfwith type a--+a in TA2 must have form for some x. Hence[n] M:a-*a > M=d[n] IThus a test for Hpp[n] would give a test for convertibility to I, contrary to standardundecidability results (e.g. Barendregt 1984 Thm. 6.6.2 or HS 86 Cor. 5.6.1).4B Semantics and completenessIn this section TAZ+p will be interpreted in an arbitrary A-model and its rules provedsound and complete, and the same will be done for TA,1+6n in extensional A-models.The concept of A-model has been motivated, defined and thoroughly discussed inBarendregt 1984 Ch. 5 (especially Def. 5.2.7), and in HS 86 Ch. 11. There areseveral definitions of A-model in these sources, all equivalent, and the one presentedbelow has been chosen mainly because it emphasizes the concepts needed in thecompleteness proof. It is from HS 86 Def. 11.3.4B1 Definition (Semantic environments) Given a non-empty set D, a semantic environmentin D is any function E that assigns to each term-variable v a memberE(v) of D. If E is a semantic environment, x a variable and d e D, the semanticenvironment[d/x]Eis defined to be the same as E except that it assigns d to x. (By the way, if E(x) = dthen [d/x]E = E.)4B2 Definition (A-models) A A-model is a triple.9 = (D, , I I) where D is a set withat least two members, is a 2-place function such thatd1,d2 E D dl-d2 E D,and I I is a function that assigns to each E and each term M a member of D calledIMIE, such that(i) WE = E(x) ,for all term-variables x,(n) IPQIE =(iii) [PI[d/x]E for all d E D,(iv) IMIE, = IMIE2 if E1(v) = E2(v) for all v E FV(M),(v) IMIE = I NI E if M =a N,(vi) IAX-PIE = IAr-QIE if IPI [d/x]E = 1Q°[d/x]E for all d e D.