my forty years on his shoulders - Department of Mathematics

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20incompleteness given above using derivability conditions are rather subtle andinvolved.We recently addressed this problem in (Friedman 2007a), where wepresent new versions of Formal Second Incompleteness that are simple, andinformally imply Informal Second Incompleteness.These results rest on the isolation of simple formal properties shared byconsistency statements. Here we do not address any issues concerning proofs ofSecond Incompleteness.We start with the most commonly quoted form of Gödel's SecondIncompleteness Theorem - for the system PA = Peano Arithmetic.PA can be formulated in a number of languages. Of these, L(prim) is themost suitable for supporting formalizations of the consistency of PeanoArithmetic.We write L(prim) for the language based on 0,S and all primitive recursivefunction symbols. We let PA(prim) be the formulation of Peano Arithmetic forthe language L(prim). I.e., the nonlogical axioms of PA(prim) consist of theaxioms for successor, primitive recursive defining equations, and the inductionscheme applied to all formulas in L(prim).INFORMAL SECOND INCOMPLETENESS (PA(prim)). Let A be asentence in L(prim) that adequately formalizes the consistency of PA(prim), inthe informal sense. Then PA(prim) does not prove A.We have discovered the following result. We let PRA be the importantsubsystem of PA(prim), based on the same language L(prim), where we requirethat the induction scheme be applied only to quantifier free formulas of L(prim).
21FORMAL SECOND INCOMPLETENESS (PA(prim)). Let A be a sentence inL(prim) such that every equation in L(prim) that is provable in PA(prim), is alsoprovable in PRA + A. Then PA(prim) does not prove A.Informal second incompleteness for PA(prim) can be derived in the usualsemiformal way from the above formal second incompleteness for PA(prim).FORMAL CRITERION THEOREM 1. Let A be a sentence in L(prim) such thatevery equation in L(prim) that is provable in PA(prim), is also provable in PRA +A. Then for all n, PRA + A proves the consistency of PA(prim) n .Here PA(prim) n consists of the axioms of PA(prim) in prenex form with atmost n quantifiers.The above development can be appropriately carried out for systems withfull induction. However, there is a more general treatment which covers finitelyaxiomatized theories as well.We use the system EFA = exponential arithmetic for this more generaltreatment. EFA is the system of arithmetic based on addition, multiplication andexponentiation, with induction applied only to formulas all of whose quantifiersare bounded to terms. This is the same as the system IΣ 0 (exp) in (Hajek, Pudlak1993 p. 37).INFORMAL SECOND INCOMPLETENESS (general many sorted, EFA). Let L bea fragment of L(many) containing L(EFA). Let T be a consistent extension of EFAin L. Let A be a sentence in L that adequately formalizes the consistency of T, inthe informal sense. Then T does not prove A.
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 45 and 46: 45THEOREM 9.5. (Friedman 2005). The
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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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my forty years on his shoulders - Department of Mathematics

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