my forty years on his shoulders - Department of Mathematics

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32systems, most of them of the form T,T’, where T,T’ have the same nonlogicalaxioms, and where T is based on classical predicate calculus, whereas T’ is basedon intuitionistic predicate calculus. For example, see (Kreisel 68a 344), (Kreisel68b Section 5), (Myhill 74), (Friedman 73), (Leivant 85).A much stronger result holds for PA over HA. Every Π 0 2 sentenceprovable in PA is provable in HA. The first proofs of this result were from theproof theory of PA via Gentzen (see (Gentzen 1969), (Schütte 1977)), and fromGödel’s so called Dialectica or functional interpretation, in (Gödel 1958), (Gödel1972).However, for other pairs for which the negative interpretation shows thatthey have the same provable Π 0 1 sentences - say classical and intuitionisticsecond order arithmetic - one does not have the required proof theory. In thiscase, the Dialectica interpretation has been extended by Spector in (Spector 1962),and the fact that these two systems have the same provable Π 0 2 sentences thenfollows.Nevertheless, there are many appropriate pairs for which the negativeinterpretation works, yet there is no proof theory and there is no functionalinterpretation.In (Friedman 1978), we broke this impasse by modifying Gödel’s negativeinterpretation via what is now called the A translation. Also see (Dragalin 1980).We illustrate the technique for PA over HA, formulated with primitive recursivefunction symbols.Let A be any formula in L(HA) = L(PA). We define the A-translation ϕ A ofthe formula ϕ in L(HA), in case no free variable of A is bound in ϕ. Take ϕ A to be
33the result of simultaneously replacing every atomic subformula ψ of ϕ by (ψ ∨A). In particular, ⊥ gets replaced by what amounts to A.The A translation is an interpretation of HA in HA. I.e., if ϕ A is defined,and HA proves A, then HA proves ϕ A . Also, obviously HA proves A → ϕ A .Now suppose (∃n)(F(n,m) = 0) is provable in PA, where F is a primitiverecursive function symbol. By Gödel’s negative interpretation, ¬¬(∃n)(F(n,m) =0) is provable in HA. Write this as ((∃n)(F(n,m) = 0) → ⊥) → ⊥.By taking the A translation, with A = (∃n)(F(n,m) = 0), we obtain that HAproves((∃n)(F(n,m) = 0 ∨ (∃n)(F(n,m) = 0)) → (∃n)(F(n,m) = 0)) → (∃n)(F(n,m) = 0.((∃n)(F(n,m) = 0) → (∃n)(F(n,m) = 0)) →(∃n)(F(n,m) = 0.(∃n)(F(n,m) = 0).This method applies to a large number of pairs T/T’ as indicated in(Friedman 1973) and (Leivant 1985).(Godel 1958) and (Godel 1972) present Gödel’s so called Dialecticainterpretation, or functional interpretation, of HA. Here HA = Heytingarithmetic, is the corresponding version of PA = Peano arithmetic withintuitionistic logic. It can be axiomatized by taking the usual axioms and rules ofintuitionistic predicate logic, together with the axioms of PA as usual given. Ofcourse, one must be careful to present ordinary induction in the usual way, andnot use the least number principle.In Gödel’s Dialectica interpretation, theorems of HA are interpreted asderivations in a quantifier free system T of primitive recursive functionals of
Page 1: 1MY FORTY YEARS ON HIS SHOULDERSbyH
Page 4 and 5: 4theorem is demonstrably implied by
Page 6 and 7: 6At the outer limits, normal mathem
Page 8 and 9: 8“Thus, according to Gödel, the
Page 11 and 12: 11(Rosser 1936) is credited for sig
Page 13 and 14: 13(Davis 1973), (Matiyasevich 1993)
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Page 17 and 18: 17v) connectives ¬,∧,∨,→,↔
Page 19 and 20: 19PROV[x 1 /#(A)] → PROV[x 1 /PR(
Page 21 and 22: 21FORMAL SECOND INCOMPLETENESS (PA(
Page 23 and 24: 23function symbols, together with t
Page 25 and 26: 25remarks by R. Parikh, it is likel
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Page 29 and 30: 29Harrington 1977), and are proved
Page 31: 31¬ as ¬.∧ as ∧.→ as →.
Page 35 and 36: 35We believe that the Spector devel
Page 37 and 38: 37P.J. Cohen proved that if ZF is c
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Page 41 and 42: 41vertices of T i .It is natural to
Page 43 and 44: 43An extremely interesting conseque
Page 45 and 46: 45THEOREM 9.5. (Friedman 2005). The
Page 47 and 48: 47Showing that all such statements
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Page 55 and 56: 55statements represents an inevitab
Page 57 and 58: 57Monographs, 24, Oxford: Clarendon
Page 59 and 60: 59Vol. 41, No. 3, September pp. 209
Page 61 and 62: 61Friedman, H., Robertson, N., and
Page 63 and 64: 63Society, 8:437-479, 1902.Hilbert,
Page 65 and 66: 65Mostowski, A. 1952. Sentences und
Page 67 and 68: 67Scott, D.S. 1961. “Measurable c
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