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60Friedman, H. 2006c. “273: Sigma01/optimal/size”. FOM Archives, March 28, 2006,http://www.cs.nyu.edu/pipermail/fom/2006-March/010279.htmlFriedman, H. 2006d. “n(3) < Graham's number < n(4) < TREE[3]”. FOM Archives,March 29, 2006, http://www.cs.nyu.edu/pipermail/fom/2006-March/010290.htmlFriedman, H. 2006e. “274: Subcubic Graph Numbers”. FOM Archives”. April 1, 2006,http://www.cs.nyu.edu/pipermail/fom/2006-April/010305.htmlFriedman, H. 2006f. “279: Subcubic Graph Numbers/restated”. FOM Archives, April 8,2006, http://www.cs.nyu.edu/pipermail/fom/2006-April/010362.htmlFriedman, H. 2006g. “Integer Thresholds in FFF”. FOM Archives, June 22, 2006,http://www.cs.nyu.edu/pipermail/fom/2006-June/010627.htmlFriedman, H. 2007a. “Formal statements of Gödel's second incompleteness theorem”.Abstract, #56, 8 pages,http://www.math.ohio-state.edu/%7Efriedman/manuscripts.htmlFriedman H. 2007b. “New Borel Independence Results”. May 7, 2007, 20 pages, proofsketch, #57,http://www.math.ohio-state.edu/%7Efriedman/manuscripts.htmlFriedman, H. 2009. "Upper Shift Fixed Points and Large Cardinals", "Upper Shift FixedPoints and Large Cardinals/correctionhttp://www.cs.nyu.edu/pipermail/fom/2009-October/014119.htmlhttp://www.cs.nyu.edu/pipermail/fom/2009-October/014123.htmlFriedman, H. 2010. Boolean Relation Theory and the Incompleteness Phenomena, to appear,Lecture Notes in Logic, Association for Symbolic Logic. Draft athttp://www.math.ohio-state.edu/%7Efriedman/manuscripts.htmlFriedman, H., Montalban, A., and Weiermann, W. In preparation.
61Friedman, H., Robertson, N., and P. Seymour P. 1987. “The Metamathematics of theGraph Minor Theorem”. Logic and Combinatorics, ed. S. Simpson, AMSContemporary Mathematics Series, vol. 65, 1987, 229-261.Gentzen, G. 1969. The collected papers of Gerhard Gentzen. Edited and translated intoEnglish by M.E. Szabo (Amsterdam: North-Holland).Gödel, K. 1929. “On the completeness of the calculus of logic”. In (Gödel 1986-2003Vol. I, 61-101).Gödel, K. 1931. “On formally undecidable propositions of Principia mathematica andrelated systems I”. In: (Gödel 1986-2003 Vol. I 145-195).Gödel, K. 1932a. “On the intuitionistic propositional calculus”. In: (Gödel 1986-2003Vol. I 223-225).Gödel, K. 1932b. “On completeness and consistency”. In: (Gödel 1986-2003 Vol. I 235-237).Gödel, K. 1933a. “On intuitionistic arithmetic and number theory”. In: (Gödel 1986-2003 Vol. I 287-295).Gödel, K. 1933b. “An interpretation of the intuitionistic propositional calculus”. In:(Gödel 1986-2003 Vol. I, 301-303).Gödel, K. 1934. “On undecidable propositions of formal mathematical systems”. In:(Gödel 1986-2003] Vol. I 346-371).Gödel, K. 1936. “On the lengths of proofs”. In: (Gödel 1986-2003 Vol. I 397-399).Gödel, K. 1938. “The consistency of the axiom of choice and of the generalizedcontinuum hypothesis”. In: (Gödel 1986-2003 Vol. II 26-27).Gödel K. 1939a. The consistency of the generalized continuum hypothesis, 1939, in:(Gödel 1986-2003 Vol. II 27).Gödel, K. 1939b. “Consistency proof for the generalized continuum hypothesis”. In:
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1MY FORTY YEARS ON HIS SHOULDERSbyH
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4theorem is demonstrably implied by
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6At the outer limits, normal mathem
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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)
Page 15 and 16: 15For degree 3, the existence of an
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
Page 27 and 28: 27The same remarks can be made with
Page 29 and 30: 29Harrington 1977), and are proved
Page 31 and 32: 31¬ as ¬.∧ as ∧.→ as →.
Page 33 and 34: 33the result of simultaneously repl
Page 35 and 36: 35We believe that the Spector devel
Page 37 and 38: 37P.J. Cohen proved that if ZF is c
Page 39 and 40: 39My ideas are not very well develo
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 51 and 52: 51Also consider the recursive unsol
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Page 57 and 58: 57Monographs, 24, Oxford: Clarendon
Page 59: 59Vol. 41, No. 3, September pp. 209
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
Page 69: 69389.*This research was partially
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