Ackermann, W.: Zur Widerspruchsfreiheit der Zahlentheorie, Math. Ann. Vol. 117, 1940 pp. 162–194 Google Scholar

Aczel, P.: Two notes on the Paris independence result. In: Model Theory and Arithmetic(Lect. Notes Math. 890), Springer-Verlag, 1981 pp. 21–31

Adamowicz, Z.: Open inductions and the true theory of rationals, Preprint 289, Polish Acad. Sci., 1984 Google Scholar

Adamowicz, Z.: Axiomatization of the forcing relation with an application to PA, Fund.Math. Vol. 120, 1984 pp. 167–186 Google Scholar

Adamowicz, Z.: Algebraic approach to ∃1-induction. In: Proceedings of the third Easterconference on model theory (Gross Koris, 1985), Humboldt Univ. Berlin, 1985 pp. 5–15

Adamowicz, Z.: Open induction (abstract), Journ. Symb. Log. Vol. 50, 1985 pp. 260 Google Scholar

Adamowicz, Z.: Some results on open and diophantine induction. In: Logic Colloquium'84, North Holland, 1986 pp. 1–20

Adamowicz, Z.: Open induction and true theory of rationals, Journ. Symb. Log. Vol. 52,1987 pp. 793–801 Google Scholar

Adamowicz, Z.: End-extending models of I ∆ 0+ exp + BΣ1 , preprint 431, Inst. Math.Polish Acad. Sci., 1988, p. 38 Google Scholar

Adamowicz, Z.: A recursion theoretic characterization of instances of BΣn provable inПn +1 (N) ,Fund. Math. Vol. 129, 1988 pp. 231–236 Google Scholar

Adamowicz, Z.: Parameter free induction, the Matiyasevič Theorem and BΣ 1. In: Logic Colloquium '86, North Holland, 1988 pp. 1–8

Adamowicz, Z., Bigorajska, T.: Functions provably total in IΣ 1 , Fund. Math., Vol. 132, 1989, pp. 189–194 Google Scholar

Adamowicz, Z., Kossak, R.: Note on an intermediate induction scheme. In: Seminarberichte 71, Proceedings of the Fourth Easter Conference on Model Theory, Gross-Köris, 1986 pp. 1–5

Adamowicz, Z., Kossak, R.: A note on BΣn and an intermediate induction schema, Zeitschr. Math. Log. Grundl. Math. Vol. 34, 1988 pp. 261–264 Google Scholar

Adamowicz, Z., Morales-Luna, G.: A recursive model for arithmetic with weak induction, Journ. Symb. Log. Vol. 50, 1985 pp. 49–54 Google Scholar

Addison, J. W.: Separation principles in hierarchies of classical and effective descriptiveset theory, Fund. Math. Vol. 46, 1958 pp. 123–135 Google Scholar

Adler, A.: Extensions of non-standard models of number theory, Zeitschr. Math. Log.Grundl. Math. Vol. 15, 1969 pp. 289–290 Google Scholar

Aho, A. V., Hopcropt, J. E., Ullman, J. I.: The Design and Analysis of Computer Algorithms, Addison-Wesley, 1974

Ajtai, M.: Σ 1 1-formulae on finite structures, Annals Pure Appl. Logic, Vol. 24, 1983 pp.1–48 Google Scholar

Ajtai, M.: The complexity of the pigeonhole principle, 29–th Symp. on Foundations of Comp. Sci., 1988 pp. 346–355 Google Scholar

Ajtai, M.: Parity and the pigeonhole principle. In: Feasible Mathematics, Buss, Scott eds.,Birkhäuser, 1990 pp. 1–24 Google Scholar

Artemov, S. N.: Arifmetičeski polnye modal'nye teorii. (Arithmetically complete modaltheories) In: Semiotics and information science, No. 14, Akad. Nauk SSSR, Vsesojuz.Ins. Nauč. i Tehn. Inf., 1980 pp. 115–133 (translation in Amer. Math. Soc. Transl. (2)Vol. 135, 1987 pp. 39–54)

Artemov, S. N.: Priloženia modal'noj logiki v teorii dokazatelstv. (Applications of modallogic in proof theory) In: Nekl. logiki i ich prilozenia, Moskva, Akad. Nauk SSSR, 1982

Artemov, S. N.: On modal representations of extensions of Peano arithmetic, C. R. Math.Acad. Sci, Soc. Roy. Canada Vol. 6, 1984 pp. 129–132

Artemov, S. N.: Nearifmetičnost’ istinostnyh predikatnyh logik dokazatelnosti. (Nonarithmeticity of truth predicate logics of provability.) Doklady Akad. Nauk SSSR Vol.284, 1985 pp. 270–271 (translation in Soviet. Math. Dokl. Vol. 32, 1985 pp. 403–405)

Artemov, S. N.: O modal'nyh logikah axiomatizujuščih dokazatelnost’. (Modal logicsaxiomatizing provability.), Izv. Akad. Nauk SSSR, Ser. Math. Vol. 49, 1985 pp. 1123–1154

Baaz, M., Pudlák, P.: Kreisel's conjecture for L∃ 1. In: Arithmetic, Proof Theory and Computational Complexity, eds. Clote, P. and Krajíček, J., Oxford Univ. Press 1992,pp. 30–60

Baker, T. P., Gill, J., Solovay, R.: Relativizations of the P=NP question, SIAM Journ.Comput., Vol. 4, 1975 pp. 431–442 Google Scholar

Balcázar, J. Z., Díaz, J., Gabarró, J.: Structural Complexity I, Springer-Verlag 1988, Eatcs 11, 191 p.

Balcázar, J. Z., Díaz, J., Gabarró, J.: Structural Complexity II, Springer-Verlag 1990, Eatcs 22, 283 p.

Barnes, D. W., Monro, G. P.: A simple model for a weak system of arithmetic, Bull.Austral. Math. Soc. Vol. 11, 1974 pp. 321–323 Google Scholar

Barwise, J., Schlipf, J. S.: On recursively saturated models of arithmetic. In: Model Theory and Algebra; A Memorial Tribute to Abraham Robinson, Springer-Verlag, 1975 pp. 42–55

Bell, J. L., Machover, M.: A course in mathematical logic. North Holland Publ. Comp. 1977, p. 599

Bellisima, F.: On the modal logic corresponding to diagonalizable algebra theory, Boll.Un. Math. Ital. B(5) Vol. 15, 1978 pp. 915–930 Google Scholar

Beltyukov, A. P.: Decidability of the universal theory of natural numbers with additionand divisibility. In: Zap. naucn. sem. Leningr. otdel. Mat. in-ta V. A. Steklova 60, 1976 pp. 15–27 Google Scholar

Beltyukov, A. P.: A computer description and a hierarchy of initial Grzegorczyk classes,(Russian), Zap. Nauc. Sem. LOMI Vol. 88, 1979 pp. 30–46, (Translation in Journ. Soviet Math. Vol. 20, 1982)Google Scholar

Bencivenga, E.: Finitary consistency of a free arithmetic, Notre Dame Journ. Formal Log. Vol.25, 1984 pp. 224–226 Google Scholar

Benda, M.: On strong axiom of induction in set theory and arithmetic. In: Mathematical Logic in Latin America, Amsterdam, North-Holland, 1980, pp. 43–56

Benda, M.: On Harrington's partition relation, Journ. Comb. Theory, Ser. A Vol. 28, 1980 pp. 338–350 Google Scholar

Bendová, K., Hájek, P.: A logical analysis of the truth-reaction paradox, Comm. Math.Univ. Carol. Vol. 23, 1982 pp. 667–688 Google Scholar

Bennet, Ch.: On some orderings of extensions of arithmetic. Doctoral dissertation, University of Goteborg (Sweden)-Department of Phil, 1986 pp. 100 pages

Bennet, Ch.: Lindenbaum algebras and partial conservativity, Proc. Amer. Math. Soc.Vol. 97, 1986 pp. 323–327 Google Scholar

Bennett, J. H.: Ph. D. disertation, Princeton Univ. 1962

Berarducci, A.: The interpretability logic of PA, Journ. Symb. Log., Vol. 55, pp. 1059–1089 Google Scholar

Berarducci, A., Intrigila, B.: Combinatorial principles in elementary number theory,1990 (preprint)

Bergstra, J. A., Tucker, J. V.: Hoare's logic and Peano's arithmetic, Theoret. Comput.Sci. Vol. 22, 1983 pp. 265–284 Google Scholar

Berline, C: Idéaux des anneaux de Peano (d'après Cherlin). In: Model Theory and Arithmetic, (Lect. Notes Math. 890), Springer-Verlag, 1982 p. 32

Berline, C, McAloon, K., Ressayre, J. P. (Eds): Model theory and arithmetic.Comptes rendus d'une action thématique programmée du C. N. R. S. sur la theoriedes modèles et Parithmetique. In: (Lect. Notes Math. 890), Springer-Verlag, 1981, p.306

Bernard, I C: The fixed point theorem for diagonalizable algebras, Studia Logica Vol. 34, 1975 pp. 239–251 Google Scholar

Bernardi, C: On the equational class of diagonalizable algebras, Studia Logica Vol. 34, 1975 pp. 322–331 Google Scholar

Bernardi, C: The uniqueness of the fixed-point in every diagonalizable algebra (Thealgebraization of the theories which express Theor VIII.), Studia Logica Vol. 35, 1976 pp. 336–343 Google Scholar

Bernardi, C: On the relation provable equivalence and on partitions in effectively inseparable sets, Studia Logica Vol. 40, 1981 pp. 29–37 Google Scholar

Bernardi, C: A shorter proof of a recent result by R. Di Paola, Notre Dame Journ. Formal Log. Vol. 25, 1984 pp. 390–393 Google Scholar

Bernardi, C, D'Aquino, P.: Topological duality for diagonalizable algebras, Notre Dame Journ. Formal Log. Vol. 29, 1988 pp. 345–364 Google Scholar

Bernardi, C, Montagna, F.: Equivalence relations induced by extensional formulae:classification by means of a new fixed point property, Fund. Math. Vol. 124, 1984 pp. 221–233 Google Scholar

Bernays, P.: Über das Induktionsschema in der rekursiven Zahlentheorie. In: Kontrolliertes Denken. Untersuchungen zum Logikkalkül und zur Logik der Einzelwissenschaften. Festschrift für Wilhelm Britzelmayr, Verlag Karl, Alber, Herder, Freiburg, 1951 pp. 10–17

Bernays, P.: The original Gentzen consistency proof for number theory. In: Intuitionismand Proof Theory, North Holland: Amsterdam, 1970 pp. 409–417

Bezboruah, A., Shepherdson, J.: Gödels second incompleteness theorem for Q, Journ.Symb. Log. Vol. 41, 1976 pp. 503–512 Google Scholar

Blass, A.: The intersection of nonstandard models of arithmetic, Journ. Symb. Log. Vol.37, 1972 pp. 103–106 Google Scholar

Blass, A.: On certain types and models for arithmetic, Journ. Symb. Log. Vol. 39, 1974 pp. 151–162 Google Scholar

Blass, A.: End extensions, conservative extensions, and the Rudin-Frolik ordering, Trans.Amer. Math. Soc, 1977 No.225 pp. 325–340 Google Scholar

Blass, A.: Amalgamation of nonstandard models of arithmetic, Journ. Symb. Log. Vol.42, 1977 pp. 372–386 Google Scholar

Blass, A. R.: Conservative extensions of models of arithmetic, Arch. Math. Log. Vol. 20,1980 pp. 85–94 Google Scholar

Börger, E.: Computability, Complexity, Logic, North-Holland P. C. 1989, xx + 592 pp.

Boolos, G.: A note on Beth's theorem, Bull. Acad. Polon. Sci. Vol. 21, 1973 pp. 1–2 Google Scholar

Boolos, G.: On Kalmar's consistency proof and a generalization of the notion of omega-consistency, Arch. Math. Log. Vol. 17, 1975 pp. 3–7 Google Scholar

Boolos, G.: On deciding the truth of certain statements involving the notion of consistency, Journ. Symb. Log. Vol. 41, 1976 pp. 779–781 Google Scholar

Boolos, G.: On deciding the provability of certain fixed point stataments, Journ. Symb.Log. Vol. 42, 1977 pp. 191–193 Google Scholar

Boolos, G.: Reflection principles and iterated consistency assertions, Journ. Symb. Log.Vol. 44, 1979 pp. 33–35 Google Scholar

Boolos, G.: The unprovability of consistency. An essay in modal logic, Cambridge Univ Press: Cambridge GB, 1979, p. 184

Boolos, G.: Provability in arithmetic and a schema of Grzegorczyk, Fund. Math. Vol. 106, 1980 pp. 41–45 Google Scholar

Boolos, G.: ω-consistency and the diamond, Studia Logica Vol. 39, 1980 pp. 237–243 Google Scholar

Boolos, G.: Provability, truth and modal logic, Journ. Phil. Logic Vol. 9, 1980 pp. 1–7 Google Scholar

Boolos, G.: On systems of modal logic with provability interpretations, Theoria (Lund) Vol. 46, 1980 pp. 7–18 Google Scholar

Boolos, G.: Extremally undecidable sentences, Journ. Symb. Log. Vol. 47, 1982 p. 191

Boolos, G.: On the nonexistence of certain normal forms in the logic of provability, Journ.Symb. Log. Vol. 47, 1982 pp. 638–640 Google Scholar

Boolos, G.: The logic of provability, Amer. Math. Mon. Vol. 91, 1984 pp. 470–480 Google Scholar

Boolos, G.: 1-consistency and the diamond, Notre Dame Journ. Formal Log. Vol. 26, 1985 pp. 341–347 Google Scholar

Boolos, G., Jeffrey, R.: Computability and Logic, 2-nd edition, Cambridge Univ. Press, 1980, p. 262

Boolos, G., Mcgee, V.: The degree of the set of sentences of predicate provability logicthat are true under every interpretation, Journ. Symb. Log. Vol. 52, 1987 pp. 165–171 Google Scholar

Boughattas, S.: L'arithmétique ouverte et ses modèles non-standards, These de 3e cycle,Université Paris VII, Paris, 1987

Boughattas, S.: L'induction pour les formules ouvertes n'est pas finiment axiomatisable, Comptes Rendusde l'Académie des Sciences de Paris, Vol. 308 Série I, 1989 pp. 297–300 Google Scholar

Brandt, U.: Index sets in the arithmetical hierarchy, Annals Pure Appl. Logic Vol. 37, 1988 pp. 101–110 Google Scholar

Britton, J. L.: Integer solutions of systems of quadratic equations. In: Math. Proc.Cambridge Phil. Soc. Vol. 86, 1979 pp. 385–389 Google Scholar

Brown, D. K., Simpson, S. G.: Which set existence axioms are needed to prove theseparable Hahn-Banach theorem?, Annals Pure Appl. Logic Vol. 31, 1986 pp. 123–144 Google Scholar

Buchholz, W.: An independence results for П1 1 -CA + BI, Annals Pure Appl. Logic Vol.33, 1987 pp. 131–155 Google Scholar

Buchholz, W., Feferman, S., Pohlers, W., Sieg, W.: Iterated Inductive Definitions and Subsystems of Analysis: Recent Proof-Theoretical Studies. (Lect. Notes Math. 897), Springer-Verlag, 1981, p. 383

Buchholz, W., Pohlers, W.: Provable well orderings of formal theories for transfinitelyiterated inductive definitions, Journ. Symb. Log. Vol. 43, 1978 pp. 118–125 Google Scholar

Buchholz, W., Wainer, S.: Provably computable functions and the fast growing hierarchy. In: Logic and Combinatorics, Amer. Math. Soc: Arcata, 1987 pp. 179–198 Google Scholar

Buss, S. R.: The polynomial hierarchy and fragments of bounded arithmetic. In: 17–th Annual ACM Symposium on Theory of Computing, 1985 pp. 285–290 Google Scholar

Buss, S. R.: Bounded arithmetic (revised version of Ph. D. thesis), Bibliopolis, 1986

Buss, S. R.: The polynomial hierarchy and intuitionistic bounded arithmetic. In: Structurein Complexity Theory (Lect. Notes Comp. Sci. 223), Springer-Verlag, 1986 pp. 77–103

Buss, S. R.: Polynomial size proofs of the propositional pigeonhole principle, Journ. Symb.Log. Vol. 52, 1987 pp. 916–927 Google Scholar

Buss, S. R.: A conservation result concerning bounded theories and the collection axiom, Proc. Amer. Math. Soc. Vol. 100, 1987 pp. 109–116 Google Scholar

Buss, S. R.: The Boolean formula value problem is in ALOGTIME. In: 19–th Annual ACM Symp. on Theory of Computing, 1987 pp. 123–131 Google Scholar

Buss, S. R.: Axiomatizations and conservation results for fragments of bounded arithmetic, In: Logic and Computation, Contemporary Mathematics 106, Amer. Math. Soc, 1990 pp. 57–84 Google Scholar

Buss, S. R.: A note on bootstrapping Intuitionistic Bounded Arithmetic, proc Proof Theory Conference, Leeds, England, 1990, to appear BUSS S. R.: The undecidability of κ-provability, Annals of Pure and Appl. Logic 53, 1991 pp. 75–102 Google Scholar

Buss, S. R.: A conservation result concerning bounded theories and the collection axiom, Proc. Amer. Math. Soc. Vol. 100, 1987, pp. 916–927 Google Scholar

Buss, S. R.: Propositional consistency proofs, Annals of Pure and Appl. Logic 52, 1991 pp. 3–29 Google Scholar

Buss, S. R.: On model theory for intuitionistic bounded arithmetic with applications toindepende results. In: Feasible Mathematics, Buss, Scott eds., Birkhäuser, 1990, pp.27–48 Google Scholar

Buss, S. R.: The witness function method and provably recursive fragments of Peanoarithmetic In: Logic Methodology and Philosophy of Science '91, to appear

Buss, S. R. et al.: Weak formal systems and connections to computational complexity,Student written lecture notes for a course at UC Berkeley, 1988

Buss, S. R.J., Krajíček, J., Takeuti, G.: Provably total functions in bounded arithmetic theoriesR i 3, U i 2. and V i 2. In: Arithmetic, Proof Theory and Computational Complexity, eds. Clote, P. and Krajíček, J., Oxford Univ. Press 1992, pp. 116–161

Cantini, A.: Majorizing provably recursive functions in fragments of PA, Arch. Math. Log. Vol. 25, 1985 pp. 21–31 Google Scholar

Cantor, A.: Minimal proper elementary extensions of N with respect to definable relations Dissertation, Univ. S. Carolina, 1972

Carlson, T. J., Simpson, S. G.: A dual form of Ramsey's theorem, preprint, Penns. Univ, 1983

Carnap, R.: Logische Syntax der Sprache, Springer-Verlag, 1934

Carstens, H. G.: The theorem of Matijasevič is provable in Peano's arithmetic by finitely many axioms, Logique & Anal., N. S. Vol. 20, 1977 pp. 116–121 Google Scholar

Carstens, H. G.: Über die Kompliziertheit numerischer Modelle, Dissertation, Münister, 1972

Carstens, H. G.: Reducing hyperarithmetic sequences, Fund. Math. Vol. 89, 1975 pp. 5–11 Google Scholar

Carstens, H. G.: ∆0 2-Mengen, Arch. Math. Log. Vol. 18, 1976 pp. 55–65 Google Scholar

Caviness, B. F.: On canonical forms and simplification, Journ. ACM Vol. 17, 1970 pp. 385–396 Google Scholar

Cegielski, P.: La théorie élémentaire de la multiplication, C. R. Acad. Sci. Paris Ser. A-B Vol. 290, 1980 pp. A935–A938 Google Scholar

Cegielski, P.: Théorie élémentaire de la multiplication des entiers naturels. In: Model Theory and Arithmetic, (Lect. Notes Math. 890), Springer-Verlag, 1981 pp. 44–89

Cegielski, P.: La théorie élémentaire de la divisibilité est finiment axiomatisable, C. R. Acad. Sci. Paris Ser. I Math. Vol. 299, 1984 pp. 367–369 Google Scholar

Cegielski, P., McAloon, K., Wilmers, G.: Modeles recursivement satures de Paddition et de la multiplication des entiers naturels. In: Logic Colloquium '80 (Studies in Logic and the Foundations of Mathematics 108), North-Holland, 1982 pp. 57–68

Celluci, C: Proprietadi uniformitae 1–coerenza dell'aritmetica del primoordine, Matematiche Vol. 32, 1977 pp. 35–49 Google Scholar

Cenzer, D., Clote, P., Smith, R. L., Soare, R. I., Wainer, S. S.: Members of countable П0 1 classes, Annals Pure Appl. Logic Vol. 31, 1986 pp. 145–163 Google Scholar

Chaitin, G. J.: Göde's theorem and information, Int. Journ. Theor. Phys. Vol. 21, 1982 pp. 941–954 Google Scholar

Chatzidakis, Z.: La représentation en termes de faisceaux des modèles de la théorie élémentaire de la multiplication des entiers naturels. In: Model theory and arithmetic (Paris, 1979–1980) (Lect. Notes Math. 890), Springer-Verlag, 1981 pp. 90–110

Church, A.: An independence question in recursive arithmetic. In: Colloque sur les Fondements des Mathématiques, les Machines Mathématiques, et leurs Applications (Collection de Logique Mathématique), Akad. Kiado: Budapest, 1965 pp. 21–26 Google Scholar

Cichon, E. A.: A short proof of two recently discovered independence results using recursion theoretic methods, Proc. Amer. Math. Soc. Vol. 87, 1983 pp. 704–706 Google Scholar

Cichon, E. A., Wainer, S. S.: Slow growing and Grzegorczyk hierarchies, Journ. Symb. Log. Vol. 48, 1983 p. 399 Google Scholar

Cinman, L. L.: On the complete induction axiom, Doklady Akad. Nauk SSSR, 1967 No.173 pp. 273–274

Cinman, L. L.: The role of the principle of induction in a formal arithmetical system, Mat. Sb. (NS), 1968 No.77 pp. 71–104 Google Scholar

Cleave, J. P.: A hierarchy of primitive recursive functions. (Russian translation in Problemy matematičeskoj logiki, Mir 1970, 94–113), Zeitschr. Math. Log. Grundl. Math. Vol. 9, 1963 pp. 331–345 Google Scholar

Cleave, J. P., Roseh, E.: ℰn-Arithmetic. In: Sets, Models a Recursion Theory, North-Holland, 1967 pp. 297–308

Clote, P.: A recursion theoretic analysis of certain generalizations of Ramsey's theoremand of the Gale-Stewart theorem, Dissertation, Duke Univ, 1979 Clote P.: Weak partition relations, finite games, and independence results in Peanoarithmetic. In: Model Theory of Algebra and Arithmetic (Lect. Notes Math. 834),Springer-Verlag, 1980 pp. 92–107

Clote, P.: Anti-basis theorems and their relation to independence results in Peano arithmetic. In: Model theory and arithmetic, (Lect. Notes Math. 890), Springer-Verlag, 1981 pp. 115–133

Clote, P.: A note on decidable model theory. In: Model Theory and Arithmetic, (Lect.Notes Math. 890), Springer-Verlag, 1981 pp. 134–142

Clote, P.: Applications of the low basis theorem in arithmetic. In: Recursion-theory week(Oberwolfach, 1984), (Lect. Notes Math. 1141), Springer-Verlag, 1985 pp. 65–88

Clote, P.: Partition relations in arithmetic. In: Proc. 6th Latin American Symp. on Math.Logic, Caracas, (Lect. Notes Math. 1130), Springer-Verlag, 1985 pp. 32–68

Clote, P.: Ultrafilters on definable sets of arithmetic. In: Logic Colloquium '84 (Studiesin Logic and the Foundations of Mathematics 120), North-Holland, 1986 pp. 37–58

Clote, P.: A note on the Mac Dowell-Specker theorem, Fund. Math. Vol. 127, 1987 pp.163–170 Google Scholar

Clote, P.: ALOGTIME and a conjecture of S. A. Cook. In: Logic in Computer Science,IEEE 1990, to appear in Annals of Mathematics and Artificial IntelligenceGoogle Scholar

Clote, P.: On polynomial size Frege proofs of certain combinatorial principles. In: Arithmetic, Proof Theory and Computational Complexity, eds. Clote, P. and Krajíček, J., Oxford Univ. Press 1992, pp. 162–184

Clote, P.: Bounded arithmetic and computational complexity. In:Structure in Complexity Theory 1990, IEEE, to appearGoogle Scholar

Clote, P., Hajek, P., Paris, J.: On some formalized conservation results in arithmetic, Arch. Math. Log. 30, 1990, pp. 201–218 Google Scholar

Clote, P., McAloon, K.: TWO further combinatorial theorems equivalent to the 1-consistency of Peano arithmetic, Journ. Symb. Log. Vol. 48, 1983 pp. 1090–1104 Google Scholar

Clote, P., Takeuti, G.: Exponential time and bounded arithmetic (extended abstract).In: Structure in Complexity Theory (Lect. Notes Comp. Sci. 223), Springer-Verlag, 1986 pp.125–143

Clote, P., Takeuti, G.: Bounded arithmetic for NC, ALOGTIME, L and NL, Annals of Pure and Appl. Logic 56, 1992 pp. 73–117 Google Scholar

Cook, S. A.: The complexity of theorem proving procedures, 3-rd Symp. on Theory of Computing, 1971 pp. 151–158 Google Scholar

Cook, S. A.: Feasibly constructive proofs and the propositonal calculus. In: Proc. 7–th A.C. M. STOC, 1975 pp. 73–89 Google Scholar

Cook, S. A., Reckhow R. A.: The relative efficiency of propositional proof systems,Journ. Symb. Log. Vol. 44, 1979 pp. 36–50

Cook, S. A., Urquhart, A.: Functional interpretations of feasibly constructive arithmetic. In: Proc. ACM Symp. on Theory of Computing, 1989, pp. 107–112 Google Scholar

Craig, W.: Bases for first order theories and subtheories, Journ. Symb. Log. vol. 25, 1960,pp. 97–142 Google Scholar

Csirmaz, L.: On definability in Peano arithmetic, Bull. Acad. Polon. Sci. Vol. 8, 1979 pp. 148–153 Google Scholar

Csirmaz, L., Paris, J. B.: A property of 2-sorted Peano models and program verification, Zeitschr. Math. Log. Grundl. Math. Vol. 30, 1984 pp. 325–334 Google Scholar

Čubarjan, A. A.: The lengths of derivations of formulae in extensions of formal arithmetic, Izv. Akad. Nauk. Armjan. SSSR Ser. Mat. 9, 1974 pp. 409–429 Google Scholar

Čuda, K.: An elimination of the predicate “to be a standard member” in nonstandardmodels of arithmetic, Comm. Math. Univ. Carol. Vol. 23, 1982 pp. 785–803 Google Scholar

Čudnovsky, G. V.: Diophantine predicates, Uspekhi Matem. Nauk. Vol. 25, 1970 No.4 pp. 185–186 Google Scholar

Daguenet, M.: Un modele non standard de l'arithmétique, Comptes Rendus, 1972 No.274 pp. 685–688 Google Scholar

Van Dalen, D., Gordon, C. E.: Independence problems in subsystems of intuitionisticarithmetic, Indag. Math. Vol. 33, 1971 pp. 448–456 Google Scholar

Van Dalen, D., Mulder, H., Krabbe, E. C. W., Visser, A.: Finite Kripke models of HAare locally PA, Notre Dame Journ. Formal Log. Vol. 27, 1986 pp. 528–532 Google Scholar

Davis, M. D.: Arithmetical problems and recursively enumerable predicates, Journ. Symb.Log. Vol. 18, 1953 pp. 33–41 Google Scholar

Davis, M. D.: One equation to rule them all, Trans. N. Y. Acad. Sci. Vol. 30, 1960 pp. 766–775 Google Scholar

Davis, M. D.: Application of recursive function theory to number theory. In: Recursive Function Theory, Amer. Math. Soc, Providence, 1962, pp. 135–138 Google Scholar

Davis, M. D.: Extensions and corollaries of recent work on Hilberts tenth problem, III.Journ. Math. Vol. 7, 1963 pp. 246–250 Google Scholar

Davis, M. D.: Introduction to Gödel's paper: On formally udecidable propositions of the Principia Mathematica and related systems I. In: The Undecidable. Basic papers on Undecidable Propositions, Unsolvable problems and Computable functions, Raven Press, New York, 1965 pp. 4

Davis, M. D.: Diophantine equations and recursively enumerable sets. In: Automata Theory, New York, Academic Press, 1966, pp. 146–152

Davis, M. D.: An explicit Diophantine definition of the exponencial function, Commun.Pure and Applied Math. Vol. 24, 1971 pp. 137–145 Google Scholar

Davis, M. D.: On the number of solutions of Diphantine equations. In: Proc. Amer. Math.Soc. Vol. 33, 1972 pp. 552–554 Google Scholar

Davis, M. D.: Hilbert tenth problem is unsolvable, Amer. Math. Mon. Vol. 80, 1973 pp.233–269 Google Scholar

Davis, M. D.: Speed-up theorems and Diophantine equations. In: Courant Computer Science Symposium 7: Computational Complexity, New York, Algorithmics Press, 1973

Davis, M. D.: What is a computation? In: Mathematics Today, Heidelberg, Springer-Verlag, 1978

Davis, M. D. (ED): The undecidable. Basic papers on undecidable propositions, unsolvableproblems and computable functions, Raven Press, New York, 1965, 440 p.

Davis, M. D., Matiyasevič, I., Robinson, J.: Hilbert's tenth problem. Diophantine equations: positive aspects of a negative solution. In: Mathematical Developmants Arising From Hilbert's Problems, Amer. Math. Soc. Proc. Symp. Pured Math. 28, 1976, pp.223–378 Google Scholar

Davis, M. D., Putnam, H.: eduction of Hilbert's tenth problem, Journ. Symb. Log. Vol.23, 1958 pp. 183–187 Google Scholar

Davis, M. D., Putnam, H.: Diophantine sets over polynomial rings, III. Journ. Math. Vol.7, 1963 pp. 251–256 Google Scholar

Davis, M. D., Putnam, H., Robinson, J.: The decision problem for exponential Diophantine equations, Ann. Math. Vol. 74, 1961 pp. 425–436 Google Scholar

Dawes, A. M., Florence, J. B.: Independent Gödel sentences and independent sets, Journ. Symb. Log. Vol. 40, 1975 pp. 159–166 Google Scholar

Dawes, A. M.: First order hierarchies in general models and in models of Peano arithmetic,Dissertation, Toronto, 1972

Dawes, A. M.: End extensions which are models of a given theory, Zeitschr. Math. Log.Grundl. Math. Vol. 23, 1977 pp. 463–467 Google Scholar

Dawson, J.: The Gödel incompleteness theorem from a length-of-proof perspective, Amer.Math. Mon. Vol. 86, 1979 pp. 740–748 Google Scholar

Dedekind, R.: Was sind und was sollen die, Zahlen, Braunschweig, F. Vieweg und, Sohn,1939, 58 pp.

Denef, J.: Hilbert's tenth problem for quadratic rings, Proc. Amer. Math. Soc. Vol. 48, 1975 pp. 214–220 Google Scholar

Denef, J.: Diophantine sets over Z(T). In: Proc. Amer. Math. Soc. Vol. 69, 1978 pp.148–150 Google Scholar

Denef, J.: The Diophantine problem for polynomial rings and fields of rational functions, Trans. Amer. Math. Soc, 1978 No.242 pp. 391–399 Google Scholar

Denef, J.: The Diophantine problem for polynomial rings of positive characteristic. In:Logic Colloquium '78, Amsterdam, North-Holland, 1979, pp. 131–145

Denef, J.: Diophantine sets over algebraic number rings II, Trans. Amer. Math. Soc, 1980 No.257 pp. 227–236 Google Scholar

Denef, J., Lipshitz, L.: Diophantine sets over some rings of algebraic integers, Journ.London Math. Soc. Vol. (2) 18, 1978 pp. 385–391 Google Scholar

Denis, R.: The arithmetics as theories of two orders. In: Orders: descriptions and roles(L'Arbresle, 1982), North-Holland, 1984 pp. 287–311

Denisov, S. D.: Models of noncontradictory formulas and the Ershov hierarchy, Algebra i Logika Vol. 11, 1972 pp. 648–655 Google Scholar

Derevyankina, E. A.: Some questions on the completeness of arithmetic, Sibirski Mat.Zh. Vol. 15, 1974 pp. 299–317 Google Scholar

Diaconescu, R., Kirby, L. A. S.: Models of arithmetic and categories with finitenessconditions, Annals Pure Appl. Logic Vol. 35, 1987 pp. 123–148 Google Scholar

Dickman, M.: Types remarquables et extensions de modèles dans l'arithmétique de Peano I, Astérisque Vol. 73, 1980 pp. 59–117 Google Scholar

Dimitracopoulos, C: Matiyasevič theorem and fragments of arithmetic, Dissertation Manchester, 1980

Dimitracopoulos, C: A generalization of a theorem of H. Friedman, Zeitschr. Math. Log.Grundl. Math. Vol. 31, 1985 pp. 221–225 Google Scholar

Dimitracopoulos, C, Graifman, H.: Fragments of Peano's arithmetic and the MRDP theorem. In: Logic and Algorithmic An International Symposium Held in Honour of Ernst Specker, Enseign Math. Univ Genève: Genève, 1982 pp. 187–206 Google Scholar

Dimitracopoulos, C, Paris, J. B.: Truth definitions for ∆ 0formulae. In: Logic and Algorithmic. An International Symposium Held in Honour of Ernst Specker, Monogr.Enseign. Math. Univ Geneve, 1982 pp. 317–329 Google Scholar

Dimitracopoulos, C, Paris, J. B.: A note on the undefinability of cuts, Journ. Symb.Log. Vol. 48, 1983 pp. 564–569 Google Scholar

Dimitracopoulos, C, Paris, J. B.: The pigeon hole principle and fragments of arithmetic, Zeitschr. Math. Log. Grundl. Math. Vol. 32, 1986 pp. 73–80 Google Scholar

Dimitracopoulos, C, Paris, J. B.: The pigeonhole principle and fragments of arithmetic.(abstract), Journ. Symb. Log. Vol. 51, 1986 pp. 485

Dimitracopoulos, C, Paris, J. B.: A note on a theorem of H. Friedman, Zeitschr. Math.Log. Grundl. Math. Vol. 34, 1988 pp. 13–17 Google Scholar

Dowd, M.: Propositional representation of arithmetic proofs (dissertation), Univ. of Toronto, 1979

Dowd, M.: Model theoretic aspects of PNP (preprint), 1985

Van Den, Dries L.: Some model theory and number theory for models of weak systemsof arithmetic. In: Model Theory of Algebra and Arithmetic (Lect. Notes Math. 834),Springer-Verlag, 1980 pp. 346–362

Ehrenfeucht, A.: Polynomial functions with exponentiation are well ordered, Algebra Universalis Vol. 3, 1973 pp. 261–262) Google Scholar

Ehrenfeucht, A.: Discernible elements in models for Peano arithmetic, Journ. Symb.Log. Vol. 38, 1973 pp. 291–292 Google Scholar

Ehrenfeucht, A., Feferman, S.: Represent ability of recursively enumerable sets in formaltheories, Arch. Math. Log. Vol. 5, 1959 pp. 37–41 Google Scholar

Ehrenfeucht, A., Jensen, D. C: Some problems in elementary arithmetics, Fund. Math.Vol. 92, 1976 pp. 223–245 Google Scholar

Ehrenfeucht, A., Kreisel, G.: Strong models for arithmetic, Bull. Acad. Polon. Sci. Vol.14, 1966 pp. 107–110 Google Scholar

Ehrenfeucht, A., Mycielski, J.: Abbreviating proofs by adding new axioms, Bull. Amer.Math. Soc. Vol. 77, 1971 pp. 366–367 Google Scholar

Ellentuck, E.: A model of arithmetic, Bull. Acad. Polon. Sci. Vol. 22, 1974 pp. 353–355 Google Scholar

Enderton, H. B.: On provable recursive functions, Notre Dame Journ. Formal Log. Vol.9, 1968 pp. 86–88 Google Scholar

Erdos, P., Mills, G.: Some bounds for the Ramsey-Paris-Harrington numbers, Journ.Combin. Theory Ser. A Vol. 30, 1981 pp. 53–70 Google Scholar

Esenine-Volpine, A. S.: Le programme ultra-intuitionniste des fondements des mathematiques. In: Infinitistic Methods, Proceedings of the Symposium on Foundations of Mathematics, PWN: Warsaw, 1961 pp. 201–223 Google Scholar

Farkas, E. J.,Szabo, M. E.: On the programs-as-formulas interpretation of parallelprograms in Peano arithmetic, Annals Pure Appl. Logic Vol. 37, 1988 pp. 111–127 Google Scholar

Feferman, S.: Formal consistency proofs and interpretabilily of theories. In: Summaries of Talks Presented at the Summer Institute for Symbolic Logic; 1957 Ithaca, Institute for Defense Analyses, Com. Res. Division, 1957 pp. 71–77 Google Scholar

Feferman, S.: Arithmetization of metamathematics in a general setting, Fund. Math. Vol.49, 1960 pp. 35–92 Google Scholar

Feferman, S.: Transfinite recursive progressions of axiomatic theories, Journ. Symb. Log.Vol. 27, 1962 pp. 259–316 Google Scholar

Feferman, S.: Autonomous transfinite progressions and the extent of predicative mathematics. In: Proceedings of 3rd International Congress for Logic, Methodology and Philosophy of Science (Stud. Logic Found. Math.), North Holland: Amsterdam, 1968 pp. 121–135

Feferman, S.: Theories of finite type related to mathematical practice. In: Handbook of Mathematical Logic, Amsterdam, North-Holland, 1977, pp. 913–971

Feferman, S.: Inductively presented systems and the formalization of meta-mathematics.In: Logic Colloquium '80 (Stud. Logic Found. Math. 108), North Holland: Amsterdam,1982 pp. 95–128

Feferman, S.: Intensionality in mathematics, Journ. Philosophical Logic Vol. 14,1985 pp.41–56 Google Scholar

Feferman, S., Kreisel, G., Orey, S.: 1–consistency and faithful interpretations, Arch.Math. Log. Vol. 6, 1962 pp. 52–63 Google Scholar

Feferman, S., Spector, C: In completeness along paths in progressions of theories, Journ.Symb. Log. Vol. 27, 1962 pp. 383–390 Google Scholar

Feferman, S., Kreisel, G., Orey, S.: 1–consistency and faithful interpretations, Arch. Math. Log. Vol. 6, 1962 No.6 pp. 52–63 Google Scholar

Fenstad, J. E.: On the completeness of some transfinite recursive progressions of axiomatictheories, Journ. Symb. Log. Vol. 33, 1968 pp. 69–76 Google Scholar

Fenstad, J. E.: Non-standard models for arithmetic and analysis. In: Proceedings of the15th Scandinavian Congress (Lect. Notes Math. 118), Springer-Verlag, 1970 pp. 30–47

Fischer, M. J., Rabin, M. O.: Super-exponential complexity of Presburger arithmetic. In:Complexity of computation, SIAM-Amer. Math. Soc. Proceeding 7, Providence, 1974, pp. 27–44 Google Scholar

Fischer, P. C: Theory of provably recursive functions. Trans. Amer. Math. Soc. Vol. 117,1965 pp. 494–520 Google Scholar

Flagg, R. C, Friedman, H. M.: Epistemic and intuitionisticformal systems, Annals Pure Appl. Logic Vol. 32, 1986 pp. 53–60 Google Scholar

Flannagan, T. B.: A note on a proof of Shepherdson, Arch. Math. Log. Vol. 23, 1983 pp. 55–60 Google Scholar

Friedman, H. M.: Iterated inductive definitions and Σ 1 2 -AC In: Intuitionism and proof theory, Amsterdam, North-Holland, 1970, pp. 435–442

Friedman, H. M.: Countable models of set theories. In: Cambridge Summer School in Mathematical Logic, (Lect. Notes Math. 337), Springer-Verlag, 1973, pp. 539–573

Friedman, H. M.: Provable equality in primitive recursive arithmetic with and withoutinduction, Pac. Journ. Math. Vol. 57, 1975 pp. 379–392 Google Scholar

Friedman, H. M.: One hundred and two problems in mathematical logic, Journ. Symb.Log., Vol. 40, 1975, pp. 113–129 Google Scholar

Friedman, H. M.: Set theoretic foundations for constructive analysis, Ann. Math. Vol.105, 1977 pp. 1–28 Google Scholar

Friedman, H. M.: Classically and intuitionistically provably recursive functions. In: Higher Set Theory (Lect. Notes Math. 669), Springer-Verlag, 1978 pp. 21–27

Friedman, H. M.: Simpler combinatorial theorem independent of set theory, Preprint, Ohio State Univ., 1979, 6 p.

Friedman, H. M.: On the consistency, completeness, and correctness problems, Preprint, Ohio State Univ. 1979, 10 p.

Friedman, H. M.: A strong conservative extension of Peano arithmetic. In: The Kleene Symposium (Stud. Logic Found. Math. 101), North Holland: Amsterdam, 1980 pp.113–122

Friedman, H. M.: Translatability and relative consistency II. Preprint, Ohio State Univ. 1980, 6 p. (see [Smoryński 82, Nonstandard models])

Friedman, H. M., McAloon, K., Simpson, S.: A finite combinatorial principle which is equivalent to the 1-consistency of predicative analysis. In: Patras Logic Symp, 1982 pp.197–230

Friedman, H. M., Sheard, M.: An axiomatic approach to self-referential truth, Annals Pure Appl. Log. Vol. 33, 1987 pp. 1–21 Google Scholar

Friedman, H. M., Sheard, M.: The disjunction and existence properties for axiomaticsystems of truth, Annals Pure Appl. Log. Vol. 40, 1988 pp. 1–10 Google Scholar

Friedman, H. M., Simpson, S. G., Smith, R. L.: Countable algebra and set existenceaxioms, Ann. Pure Appl. Log. Vol. 25(2), 1983 pp. 141–182 Google Scholar

Friedman, H. M., Simpson, S. G., Smith, R. L.: Addendum to “Countable algebra andset existence axioms”, Annals Pure Appl. Log. Vol. 28, 1985 pp. 319–320 Google Scholar

Friedrichsdorf, U.: Einige Bemerkungen zur Peano Arithmetik, Zeitschr. Mat. Log.Grundl. Math. Vol. 22, 1976 pp. 431–436 Google Scholar

Gaifman, H.: Uniform extension operators for models and their applications. In: Sets, Models and Recursion Theory, Amsterdam, North-Holland, 1967, pp. 122–155

Gaifman, H.: On local arithmetical functions, and their applications for constructingtypes of Peano's arithmetic. In: Mathematical Logic and Foundations of Set Theory, Amsterdam, North-Holland, 1970, pp. 105–121

Gaifman, H.: A note on models and submodels of arithmetic. In: Conference in Mathematical Logic, London ’70, Springer-Verlag, 1972 pp. 128–144

Gaifman, H.: Models and types of Peano's arithmetic, Arch. Math. Log. Vol. 9, 1976 pp.223–306 Google Scholar

Gaifman, H., Dimitracopoulos, C: Fragments of Peano's arithmetic and the MRDP theorem. In: Logic and algorithmic (Zurich, 1980) (Monograph. Enseign. Math, 30), Geneve, Univ. Geneve, 1982 pp. 187–206

Gandy, R. O.: Note on a paper of Kemeny's, Math. Ann, 1958 No.136 pp. 466 Google Scholar

Garey, M. R., Johnson, D. S.: Computers and Intractability: A Guide to the theory of NP-completeness, Freeman, San Francisco, 1979, 338 pp.

Gandy, R. O.: Limitations to mathematical knowledge. In: Logic Colloquium '80 (Studiesin Logic and the Foundations of Mathematics 108), North-Holland, 1982 pp. 129–146

Garro, I.: Independence proofs in arithmetic theories with very weak induction, Univ Math Inst: Bonn, 1973, 71 p.

Garro, I.: Nonstandard models for a fragment of the arithmetic and their decision problem, Zeitschr. Math. Log. Grundl. Math. Vol. 33, 1987 pp. 481–483 Google Scholar

Geiser, J. R.: A formalization of Essenin-Volpin's proof theoretical studies by means ofnonstandard analysis, Journ. Symb. Log. Vol. 39, 1974 pp. 81–87 Google Scholar

Gentzen, G.: Untersuchungen über das logische Schliessen I. Math. Zeitschr. 39, 1935 pp. 176–210 (Russian translation in Matematiceskaja teoria logiceskogo vyvoda, Nauka 1967, 9–76)Google Scholar

Gentzen, G.: Untersuchungen über das logische Schliessen II. Math. Zeitschr. Vol. 39,1935 pp. 405–443 (Russian translation in Matematiceskaja teoria logiceskogo vyvoda, Nauka 1967, 9–76)Google Scholar

Gentzen, G.: Die Widerspruchsfreiheit der reinen Zahlen theorie. Math. Ann. Vol. 112, 1936 pp. 493–565 (Russian translation in Matematiceskaja teoria logiceskogo vyvoda,Nauka 1967, 77–153)Google Scholar

Gentzen, G.: Neue Fassung des Widerspruchsfreiheitsbeweises für die reine Zahlentheorie. Forschungen zur Logik und zur Grundlegung der exakten Wissenschaften Vol. 4, 1938 pp. 19–44 (Russian translation in Matematiceskaja teoria logiceskogo vyvoda, Nauka 1967, 154–190)Google Scholar

Gentzen, G.: Zusammenfassung von mehreren vollständigen Induktionen zu einer einzigen, Arch. Math. Log. Vol. 2, 1954 pp. 1–3 Google Scholar

Gentzen, G.: The collected papers of Gerhard Gentzen, North Holland: Amsterdam, 1969, 338 p.

Gentzen, G.: Der erste Widerspruchsfreiheitsbeweis für die klassiche Zahlentheorie, Arch.Math. Log. Vol. 16, 1974 pp. 97–118 Google Scholar

Gentzen, G.: Über das Verhaltnis zwischen intuitionistischer und klassicher Arithmetik, Arch. Math. Log. Vol. 16, 1974 pp. 119–132 Google Scholar

Germano, G.: Incompleteness theorem via weak definability of truth: a short proof, Notre Dame Journ. Formal Log. Vol. 14, 1973 pp. 377–380 Google Scholar

Ginsburg, S., Spanier, E.: Semigroups, Presburger formulas and languages, Pac. Journ.Math. Vol. 16, 1966 pp. 285–296 Google Scholar

Gödel, K.: Die Vollständigkeit der Axiome des logischen Funktionenkalküls, Monatsheftefür Math, und Phys. vol 37, 1930, pp. 349–360.Google Scholar

Gödel, K.: Einige metamathematische Resultate über Entscheidungsdefinitheit und Widerspruchsfreiheit. In: Math. Naturw. Klasse, Anzeiger, 67, Akad. Wiss. Wien, 1930 pp. 214–215 Google Scholar

Gödel, K.: Remarks contributed to a “Diskussion zur Grundlegung der Mathematik”, Erkenntnis Vol. 2, 1931 pp. 147–148 Google Scholar

Gödel, K.: Eine Interpretation des intuitionistischen Aussagenkalküls, Ergebnisse einesmath. Kolloquiums Vol. 4, 1931 pp. 39–40 Google Scholar

Gödel, K.: Über formal unentscheidbare Satze der Principia Mathematica und verwandter Systeme I Monatshefte Math. Phys. Vol. 38, 1931 pp. 173–198 Google Scholar

Gödel, K.: Über Vollständigkeit und Widerspruchsfreiheit, Ergebnisse eines Mathematischen Kolloquiums Vol. 3, 1932 pp. 12–13 Google Scholar

Gödel, K.: Zu intuitionistischen Arithmetik und Zahlentheorie. In: Ergebnisse eines Math.Kolloquiums, 1933 pp. 34–38 Google Scholar

Gödel, K.: On undecidable propositions of formal mathematical systems (mimeographedlecture notes by Kleene, S. C. and Rosser, J. B.). Princeton 1934 (reprinted in Davis1965 and Gödel 1986)

Gödel, K.: Über die Länge von Beweisen. In: Ergeb. eines math. Kolloq. Vol. 7, 1936 pp.23–24 Google Scholar

Gödel, K.: The consistency of the axiom of choice and of the generalized continuum hypothesis. Proc Nat Acad Sci USA Vol. 24, 1938 pp. 556–557 Google Scholar

Gödel, K.: Über eine bisher noch nicht benutzte Erweiterung des finiten Standpunktes, Dialectica Vol. 12, 1958 pp. 280–287 Google Scholar

Gödel, K.: On undecidable propositions of formal mathematical systems. In: The Undecidable, N. Y, Raven Press, Hewlett, 1965, pp. 41–71

Gödel, K.: Collected works, vol. I (edited by Feferman, S. et al.). Oxford Univ. Press New York and Oxford, 1986, xvi + 474 pp.

Gödel, K.: Collected works, vol. II (edited by Feferman, S. et al.). Oxford Univ. Press New York and Oxford, 1990, xvi + 407 pp.

Gold, B.: Relatively diophantine correct models of arithmetic, Notre Dame Journ. Formal Log. Vol. 28, 1987 pp. 291–296 Google Scholar

Goldblatt, R.: Arithmetic necessity, provability and intuitionistic logic, Theoria Vol. 44,1978 pp. 38–46 Google Scholar

Goldfarb, W. D.: On the effective ω-rule, Zeitschr. Math. Log. Grundl. Math. Vol. 21,1975 pp. 409–412 Google Scholar

Goldfarb, W. D.: Ordinal bounds for κ-consistency, Journ. Symb. Log. Vol. 39, 1978 pp.693–699 Google Scholar

Goldfarb, W. D., Scanlon, T. M.: The u-consistency of number theory via Herbrand's theorem, Journ. Symb. Log. Vol. 39, 1974 pp. 678–692 Google Scholar

Goldfrei, D. C, Macintyre, A., Simmons, H.: The forcingcompanions of number theories, Israel Journ. Math. Vol. 14, 1973 pp. 318–337 Google Scholar

Goodman, N. D.: Flagg readability in arithmetic, Journ. Symb. Log. Vol. 51, 1986 pp.387–392 Google Scholar

Goodstein, R. L.: On the restricted ordinal theorem, Journ. Symb. Log. Vol. 9, 1944 pp.33–41 Google Scholar

Goodstein, R. L.: Transfinite ordinals in recursive number theory, Journ. Symb. Log. Vol.12, 1947 pp. 123–129 Google Scholar

Goodstein, R. L.: Logic-free formalism of recursive arithmetic, Math. Scand. Vol. 2, 1954,pp. 247–261 Google Scholar

Gordeev, L.: Proof-theoretical analysis: weak systems of functions and classes, Annals Pure Appl. Logic Vol. 38, 1988 pp. 1–121 Google Scholar

Gorjačev, S. V.: On interpret ability of some extensions of arithmetic, Matemeticeskie Zametki Vol. 40, 1986 pp. 561–571 Google Scholar

Grattan-Guiness, I.: In memorium Kurt Gödel: His 1931 correspondence with Zermeloon his incompletability theorem, Hist. Mathematica Vol. 6, 1979 pp. 294–304 Google Scholar

Grilliot, T. J.: Disturbing arithmetic, Journ. Symb. Log. Vol. 50, 1985 pp. 375–379 Google Scholar

Grzegorczyk, A.: Some classes of recursive functions. In: Rozprawy Matematiczne, IV, Warszawa, 1953 (Russian translation in Problemy matematiceskoj logiki, Mir 1970,9–49)Google Scholar

Grzegorczyk, A.: Some proofs of undecidability of arithmetic, Fund. Math. Vol. 43, 1956 pp. 166–177 Google Scholar

Grzegorczyk, A.: An unfinitizability proof by means of restricted reduced power, Fund.Math. Vol. 73, 1971 pp. 37–49 Google Scholar

Grzegorczyk, A.: Zarys arytmetyki teoretycznej. (Second edition), Panstwowe Wydawnictwo Naukowe, 1983, 315 p.Google Scholar

Grzegorczyk, A., Mostowski, A., Ryll-Nardzewski, C: The classical and the ω-complete arithmetic, Journ. Symb. Log. Vol. 23, 1958 pp. 188–206 Google Scholar

Guaspari, D.: Partially conservative extensions of arithmetic, Transactions Amer. Math.Soc. Vol. 254, 1979 pp. 47–68 Google Scholar

Guaspari, D.: Sentences implying their own provability, Journ. Symb. Log. Vol. 48, 1983 pp. 777–789 Google Scholar

Guaspari, D., Solovay, R. M.: Rosser sentences, Ann. Math. Log. Vol. 16, 1979 pp. 81–99 Google Scholar

Hájek, P.: Syntactic models of axiomatic theories, Bull. Acad. Polon. Sci. Vol. 13, 1965 pp. 273–278 Google Scholar

Hájek, P.: Generalized interpret ability in terms of models, Čas. pĕst. mat. Vol. 91, 1966 pp. 352–357 Google Scholar

Hájek, P.: On interpret ability in set theories, Comm. Math. Univ. Carol. Vol. 12, 1971 pp. 73–79 Google Scholar

Hájek, P.: On interpret ability in set theories II, Comm. Math. Univ. Carol. Vol. 13, 1972 pp. 445–455 Google Scholar

Hájek, P.: Experimental logics and Pi 0 3 theories, Journ. Symb. Log. Vol. 42, 1977 pp.515–522 Google Scholar

Hájek, P.: Arithmetical complexity of some problems in computer science. In: Math.Foundations of Computer Science (Lect. Notes Comp. Sci. 53), Springer-Verlag, 1977 pp. 282–297

Hájek, P.: Arithmetical hierarchy and complexity of Computations, Theor. Comp. Sci. Vol. 8, 1979 pp. 227–237 Google Scholar

Hájek, P.: On partially conservative extensions of arithmetic. In: Logic Colloquim 78, North-Holland Pub, 1979 pp. 225–234

Hájek, P.: Making dynamic logic first order. In: Mathematical foundations of Computer Science, (Lect. Notes Comp. Sci. 118), Springer-Verlag, 1981 pp. 287–295

Hájek, P.: On interpret ability in theories containing arithmetic II, Comm. Math. Univ.Carol. Vol. 22, 1981 pp. 617–688 Google Scholar

Hájek, P.: Completion closed algebras and models of Peano arithmetic, Comm. Math. Univ. Carol. Vol. 22, 1981 pp. 585–594 Google Scholar

Hájek, P.: Arithmetical interpretations of dynamic logic, Journ. Symb. Log. Vol. 48, 1983 pp. 704–713 Google Scholar

Hájek, P.: On a new notion of partial conservativity. In: Computation and proof theory,Proc. Logic Colloq, Aachen 1983, Part II, (Lect. Notes Math. 1104), Springer-Verlag, 1984 pp. 217–232

Hájek, P.: Some coriservativeness results for non-standard dynamic logic. In: Demetrovics(ed.), Proc. Conf. on Algebra, Combinatorics and Logic in Comp. Sci, North-Holland, 1986 pp. 443–449

Hájek, P.: A simple dynamic logic, Theor. Comp. Sci. Vol. 46, 1986 pp. 239–259 Google Scholar

Hájek, P.: Partial conservativity revisited, Comm. Math. Univ. Carol. Vol. 28, 1987 pp.679–690 Google Scholar

Hájek, P.: Partial conservativity revisited, (abstract) Journ. Symb. Log. Vol. 52, 1987 pp.1066 Google Scholar

Hájek, P.: Interpret ability and fragments of arithmetic. In: Arithmetic, Proof Theory and Computational Complexity, eds. Clote, P. and Krajíček, J., Oxford Univ. Press 1992, pp.185–196

Hájek, P., Hájková, M.: On interpretability in theories containing arithmetic, Fund.Math. Vol. 76, 1972 pp. 131–137 Google Scholar

Hájek, P., Kučera, A.: On recursion theory in IΣ 1) Journ. Symb. Log. Vol. 54, 1989, pp.576–589 Google Scholar

Hájek, P., Montagna, F.: ILM is the logic of П1 conservativity, Arch. Math. Log. Vol.30, 1990, pp. 113–123 Google Scholar

Hajek, P., Montagna, F., Pudlák, P.: Abbreviating proofs using metamathematicalrules. In: Arithmetic, Proof Theory and Computational Complexity, eds. Clote, P. and Krajíček, J., Oxford Univ. Press 1992, pp. 197–221

Hájek, P., Paris, J. B.: Combinatorial principles concerning approximation of functions, Arch. Math. Log. Vol. 26, 1986 pp. 13–28 Google Scholar

Hájek, P., Pudlák, P.: Two orderings of the class of all countable models of arithmetic. In: Model theory of algebra and arithmetic, (Lect. Notes Math. 843), Springer-Verlag, 1980 pp. 174–185

Hájková, M.: The lattice of binumerations of arithmetic, Comm. Math. Univ. Carol. Vol.12, 1971 pp. 81n,281n.Google Scholar

Handley, W. G.: Some machine characterizations of classes close to ∆N 0 , Ph. D. dissertation, Manchester University, Manchester, 1986 pp. iv+106 ppGoogle Scholar

Handley, W. G., Paris, J. B., Wilkie, A. J.: Characterizing some low arithmetic classes. In: Theory of algorithms (Pecs), Coll. Math. Soc. Janos Bolyai 44, 1984 pp. 353–364 Google Scholar

Hardy, G. H., Wright, E. M.: An introduction to the theory of numbers, (3. ed.), Oxford Univ. Press, 1954

Harnik, V.: ω1-like recursively saturated models of Presburger's arithmetic, Journ. Symb.Log. Vol. 51, 1986 pp. 421–429 Google Scholar

Harrington, L. A., Paris, J. B.: A mathematical incompleteness in Peano arithmetic. In: Handbook of Mathematical Logic (Stud Logic Found Math 90), North Holland: Amsterdam, 1977 pp. 1133–1142

Hart, J.: ℰ 0-arithmetic, Zeitschr. Math. Log. Grundl. Math. Vol. 15, 1969 pp. 237 Google Scholar

Hartmanis, J., Hopcroft, J. E.: Independence results in computer science, SIGACTNews 8, 1976 No. 4 pp. 13–24 Google Scholar

Hartmanis, J., Lewis, P. M., Stearns, R. E.: Hierarchies of memory limited computations, IEEE Conference Record on Switching Circuit Theory and Logical Design, Ann.Arbor, 1965 pp. 179–190 Google Scholar

Hartmanis, J., Stearns, R. E. On the computational complexity of algorithms, Trans.Amer. Math. Soc., Vol. 117, 1965 pp. 285–306 Google Scholar

Hauschild, K.: Über die Charakterisierbarkeit der Zahlenreihe in gewissen Nichtstandard-modellen der Arithmetik, Zeitschr. Math. Log. Grundl. Math. Vol. 9, 1963 pp. 113–116 Google Scholar

Hauschild, K.: Nichtaxiomatisierbarkeit von Satzmengen durch Ausdriicke spezieller Gestalt, Fund. Math. Vol. 72, 1971 pp. 245–253 Google Scholar

Van Heijenoort, J. (ED.): From Frege to Gödel, A Source Book in Mathematical Logic,1879–1931. Harvard Univ. Press 1967, 660 p.

Henkin, L.: A problem concerning provability, Journ. Symb. Log. Vol. 17, 1952 pp. 160 Google Scholar

Henson, C. W., Kaufmann, M., Keisler, H. J.: The strength of nonstandard methodsin aritmetic, Journ. Symb. Log. Vol. 49, 1984 pp. 1039–1058 Google Scholar

Herbrand, J.: Non-contradiction des axiomes arithmetiques, C. R. Acad Sci, Paris Vol.188, 1929 pp. 303–304 Google Scholar

Herbrand, J.: Recherches sur la théorie de la démonstration (thesis, Paris), Prace Towarzystwa Naukowego Warszawskiego vol. III no. 33, 1930. (Chap. 5 see van Heienoort1967).Google Scholar

Herbrand, J.: Sur la non-contradiction de l'arithmétique, Journal für die Reine und Angewandte Mathematik Vol. 166, 1931 pp. 1–8 Google Scholar

Hermes, H.: Ein mengentheoretisches Modell des Peanoschen Axiomensystem, Math Unterricht Vol. 13/3, 1967 pp. 5–31 Google Scholar

Hilbert, D., Ackermann, W.: Grundziige der theoretischen Logik. Springer-Verlag, 1928,120 pp.

Hilbert, D., Bernays, P.: Grundlagen der Mathematik I, Springer-Verlag, 1934, 471 p.

Hilbert, D., Bernays, P.: Grundlagen der Mathematik II, Springer-Verlag, 1939, 561 p.

Hirose, K., Iida, S.: A proof of negative answer to Hilbert's 10th problem, Proc. Japan Acad. Vol. 49, 1973 pp. 10–12 Google Scholar

Hirschfeld, J.: Existentially complete and generic structures in arithmetic. In: Dissertation, Yale, 1972 Google Scholar

Hirschfeld, J.: Models of arithmetic and the semiring of recursive functions. In: Victoria Symposium on Non-Standard analysis (Lect. Notes Math. 369), Springer-Verlag, 1974 pp. 99–105

Hirschfeld, J.: Models of arithmetic and recursive functions, Israel J. Math. Vol. 20, 1975 pp. 111–126 Google Scholar

Hirschfeld, J.: Finite forcing and generic filters in arithmetic. In: Model Theory and Algebra, A Memorial Tribute to Abraham Robinson, Springer-Verlag, 1975, pp. 172–199

Hirschfeld, J., Wheeler, W. H.: Forcing, arithmetic, division rings. In: (Lect. Notes Math. 454), Springer-Verlag, 1975, 266 p.

Hodges, W.: Interpreting number theory in nilpotent groups, Arch. Math. Log. Vol. 20,1980 pp. 103–111 Google Scholar

Hodges, W.: Truth in a structure. In: Proc. of the Aristotelian Society (New series) vol.86, 1985/86, 135–151 Google Scholar

Hopcroft, J. E., Paul, W., Valiant, L.: On time versus space and related problems, 16–th Symp. on Foundations of Comp. Sci., 1975 pp. 57–64 Google Scholar

Höring, W.: Absolut unentscheidbare Sätze der Mathematik. In: Logik und Logikkalkül,Alber: Freiburg, 1962 pp. 189–194 Google Scholar

Höward, P. E.: A proof of the theorem of Tennenbaum, Zeitschr. Math. Log. Grundl.Math. Vol. 18, 1972 pp. 111–112 Google Scholar

Huber-Dyson, V., Jones, J. P., Sheperdson, J. C: Some diophantine forms of Gödel'stheorem, Arch. Math. Log. Vol. 22, 1982 pp. 51–60 Google Scholar

Ignjatovic, A.: An ordering of the set of sentences of Peano arithmetic, Publ. Inst. Math,Nouv. Ser. Vol. 38 (52), 1985 pp. 13–15 Google Scholar

Immerman, N.: Nondeterministic space is closed under complement. In: Proc. 3–rd Annual Conf. Structure in Complexity Theory, IEEE, 1988 pp. 112–115 Google Scholar

Isaacson, D.: Arithmetical truth and hidden higher-order concepts. In: Logic Colloquium'85, Amsterdam: North-Holland, 1987 pp. 147–169

Jensen, D.: On local proof restrictions for strong theories, Diss. Math. Vol. 108, 1973 pp. 1–47 Google Scholar

Jensen, D. Ehrenfeucht, A.: Some problem in elementary arithmetics, Fund. Math. Vol.92, 1976 pp. 223–245 Google Scholar

Jensen, R. B.: Unbhängigkeitsbeweise in Teilsystemen der elementaren Zahlentheorie, Math. Ann. Vol. 173, 1967 pp. 102–126 Google Scholar

Jeroslow, R. G.: Non-effectiveness in S. Orey's compactness theorem, Zeitschr. Math.Log. Grundl. Math. Vol. 17, 1971 pp. 285–289 Google Scholar

Jeroslow, R. G.: Consistency statements in formal theories, Fund. Math. Vol. 72, 1971 pp. 17–40 Google Scholar

Jeroslow, R. G.: Redundancies in the Hilbert-Bernays derivability conditions for Gödel'ssecond incompleteness theorem, Journ. Symb. Log. Vol. 38, 1973 pp. 359–367 Google Scholar

Jeroslow, R. G.: Experimental logics and ∆0 0-theories, Journ. Phil. Log. Vol. 4, 1975 pp.253–267 Google Scholar

Jervel, H. R.: Gentzen games. Z Math Logik Vol. 31, pp. 431–439

Jervel, H. R.: Large finite sets, Z Math Logik Vol. 31, pp. 545–549

Jockusch, C. G.: Ramsey's theorem and recursion theory, Journ. Symb. Log. Vol. 37,1972 pp. 268–280 Google Scholar

Jockusch, C. G.: П0 1classes and boolean combinations of recursively enumerable sets,Journ. Symb. Log. Vol. 39, 1974 pp. 95–96 Google Scholar

Jockusch, C. G., Soare, R. I.: П0 1-classes and degrees of theories, Trans. Amer. Math.Soc. Vol. 173, 1972 pp. 33–56 Google Scholar

Jockusch, C. G., Soare, R. I.: Degrees of members of П0 1 classes, Pac. Journ. Math. Vol.40, 1972 pp. 605–616 Google Scholar

Jones, J. P.: Formula for the n-th prime number, Can. Math. Bull. Vol. 18, 1975 pp.433–434 Google Scholar

Jones, J. P.: Diophantine representation of the Fibonacci numbers, Fib. Quart. Vol. 13,1975 pp. 82–88 Google Scholar

Jones, J. P.: Diophantine representation of the Lucas numbers, Fib. Quart. Vol. 14, 1976 pp. 134 Google Scholar

Jones, J. P.: Three universal representation of recursively enumerable sets, Journ. Symb.Log. Vol. 43, 1978 pp. 335–351 Google Scholar

Jones, J. P.: Diophantine representation of Mersenne and Fermat primes, Acta Arithmetica Vol. 35, 1979 pp. 209–221 Google Scholar

Jones, J. P.: Udecidable Diophantine equations, Bull. Amer. Math. Soc. (NS) Vol. 3, 1980 pp. 859–862 Google Scholar

Jones, J. P.: Universal diophantine equation, Journ. Symb. Log. Vol. 47, 1982 pp. 549–571 Google Scholar

Jones, J. P., Levitz, H., Wilkie, A. J.: Classification of quantifier prefixes over exponential diophantine equations, Zeitschr. Math. Log. Grundl. Math. Vol. 32, 1986 pp.399–406 Google Scholar

Jones, J. P., Matiyasevičyu., V.: Register machine proof of the theorem on exponentialdiophantine representation of enumerable sets, Journ. Symb. Log. Vol. 49, 1984 pp. 48–59 Google Scholar

Jones, J. P., Sato, D., Wada, H., Wiens, D.: Diophantine representation of the set ofprime numbers, Amer. Math. Monthly Vol. 83, 1976 pp. 449–464 Google Scholar

Jones, J. P., Shepherdson, J. C: Variants of Robinson's essentially undecidable theory R, Arch. Math. Log. Vol. 23, 1983 pp. 61–64 Google Scholar

De Jongh, D.: A simplification of a completeness proof of Guaspari and Solovay, Studialogica Vol. 46, 1987 pp. 187–192 Google Scholar

De Jongh, D.: Provability logics for relative interpretability (preprint for the Heytingconference Varna), 1988, 19 p.

De Jongh, D., Montagna, F.: Provable fixed points, Zeitschr. Math. Log. Grundl. Math. Vol. 34, 1988 pp. 229–250 Google Scholar

De Jongh, D., Montagna, F.: Much shorter proofs, Report 87–13, Dept. Math. Univ.Amsterdam, 1987, 21 p.Google Scholar

De Jongh, D., Montagna, F.: Generic generalized Rosser fixed points, Studia logica Vol.46, 1987 pp. 193–203 Google Scholar

De Jongh, D., Montagna, F.: Rosser orderings and free variables (preprint for Heytingconference Varna), 1988, 10 p.

De Jongh, D., Montagna, F.: Provable fixed points, Zeitschr. Math. Log. Grundl. Math. Vol. 34, 1988 pp. 229–250 Google Scholar

Joseph, D., Young, P.: Independence results in computer science, Journ. Comp. Syst. Vol. 23, 1981 pp. 205–222 Google Scholar

Joseph, D., Young, P.: A survey of some recent results on computational complexity inweak theories of arithmetic. In: Mathematical foundations of computer science (Lect.Notes. Comp. Sci 118), Springer-Verlag, 1981 pp. 46–60

Jumelet, M.: On Solovay's completeness theorem (thesis), Amsterdam, 1988, 28 p.

Kalsbeek, M. B.: An Orey sentence for predicative arithmetic (master's thesis), Amsterdam 1988

Kamo, S.: Nonstandard natural number systems and nonstandard models, Journ. Symb.Log. Vol. 46, 1981 pp. 365–376 Google Scholar

Kanamori, A., Mcaloon, K.: On Gödel incompleteness and finite combinatorics, Ann.Pure Appl. Logic Vol. 33, 1987 pp. 23–41 Google Scholar

Kanovic, M. I.: An estimate of complexity of arithmetic incompleteness, Doklady Akad. Nauk SSSR Vol. 238, 1978 pp. 1283–1286 Google Scholar

Kanovic, M. I.: Nezavisimost invariantnych predlozenij, Doklady Akad. Nauk SSSR, 1984 No. 276 pp. 27–31 Google Scholar

Karp, R. M.: Reducibility among combinatorial problems, in Complexity of Computer Computations, Miller, and Thatcher, eds., Plenum Press, New York 1972 pp. 85–103

Karp, R. M., Lipton, R. J.: Some connections between nonuniform and uniform complexity classes. In: Proc. 12–th Annual ACM Symp. on Theory of Computing, 1980, pp.302–309 Google Scholar

Kaufmann, M.: A rather classless model, Proc. Amer. Math. Soc. Vol. 62, 1977 pp. 330–333 Google Scholar

Kaufmann, M.: On existence Σn -end extensions. In: Logic Year 1979–80. The Universityof Connecticut (Lect. Notes Math. 859), Springer-Verlag, 1981 pp. 92–103

Kaufmann, M.: On expandability of models of arithmetic and set theory to models ofweak second-order theories, Fund. Math. Vol. 122, 1984 pp. 57–60 Google Scholar

Kaufmann, M., Schmerl, J. H.: Saturation and simple extensions of models of Peano-arithmetic, Annals Pure Appl. Logic Vol. 27, 1984 pp. 109–136 Google Scholar

Kaufmann, M., Schmerl, J. H.: Remarks on weak notions of saturation in models of Peano arithmetic, Journ. Symb. Log. Vol. 52, 1987 pp. 129–148 Google Scholar

Kaye, R.: Characterizing some low complexity classes using weak theories of arithmetic,Transfer Report, Manchester University, 1985

Kaye, R.: Parameter free induction. In: Proc. of the 5th Easter conf. on model theory(Humboldt univ. Berlin), 1987 Google Scholar

Kaye, R.: Diophantine and parameter-free induction (thesis), Manchester University, 1987

Kaye, R.: Axiomatizations and quantifier complexity. In: Proc. of the 6th Easter conf. onmodel theory (Humboldt Univ. Berlin), 1988, pp. 65–84 Google Scholar

Kaye, R.: On functions provably E 1in open induction, 1985 (manuscript)

Kaye, R.: Parameter-free universal induction, Zeitschr. f. Math. Logik 35, 1989, pp. 443–456 Google Scholar

Kaye, R.: Diophantine induction, Annals of Pure and Appl. Logic 46, 1990, pp. 1–40 Google Scholar

Kaye, R.: Models of Peano arithmetic, Oxford Logic Guides, Oxford Univ. Press 1991,292 pp.

Kaye, R.: Model-theoretic properties characterizing Peano arithmetic, Journ. Symb. Logic 56, 1991, pp. 949–963 Google Scholar

Kaye, R.: On cofinal extensions of models of fragments of arithmetic, Notre Dame J. of Formal Logic 32, 1991, pp. 399–408 Google Scholar

Kaye, R.: Open induction, Tennenbaum phenomena, and complexity theory. In: Arithmetic, Proof Theory and Computational Complexity, eds. Clote, P. and Krajíček, J., Oxford Univ. Press 1992, pp. 222–237

Kaye, R.: Using Herbrand-type theorems to separate, strong fragments of arithmetic. In:Arithmetic, Proof Theory and Computational Complexity, eds. Clote, P. and Krajíček, J., Oxford Univ. Press 1992, pp. 238–246

Kaye, R., Kossak, R., Kotlarski, H.: Automorphisms of recursively saturated models ofarithmetic, Annals of Pure and Appl. Logic 55, 1991, pp. 67–99 Google Scholar

Kaye, R., Paris, J. B., Dimitracopoulos C: On parameter free induction schemas, Journ. of Symb. Logic 53, 1988, pp. 1082–1097 Google Scholar

Keller, J. P., Richard, D.: Remarques sur les structures additives des modèles del'arithmétique, Comptes Rendus, Sér. A Vol. 287, 1978 pp. 101–104 Google Scholar

Kemeny, J. G.: Undecidable problems of elementary number theory, Math. Ann. Vol. 135,1958 pp. 160–169 Google Scholar

Kent, C. F.: The relation of A to prov ‘A’ in the Lindenbaum sentence algebra, Journ.Symb. Log. Vol. 38, 1973 pp. 295–298 Google Scholar

Kent, C. F.: “Disorder” in lattices of binumerations, Comm. Math. Univ. Carol. Vol. 15,1974 pp. 221–244 Google Scholar

Kent, C. F.: Independence versus logical independence in the countable case. In: Logic Colloquium 73, North Holland: Amsterdam, 1975, pp. 399–408

Ketonen, J., Solovay, R. M.: Rapidly growing Ramsey functions, Ann. Math. Vol. 113,1981 pp. 267–314 Google Scholar

Kino, A.: On provably recursive functions and ordinal recursive functions, Journ. Math.Soc. Japan Vol. 20, 1968 pp. 456–476 Google Scholar

Kirby, L. A. S.: Initial segments of models of arithmetic. In: Dissertation, Manchester,1977

Kirby, L. A.S. : La méthode des indicatrices et le theorème d'incomplétude. In: Models of arithmetic (Sem, Univ. Paris VII, Paris, 1977) (French), Soc. Math. France, Paris,1980 pp. 5–18 Google Scholar

Kirby, L. A.S. : Flipping properties in arithmetic, Journ. Symb. Log. Vol. 47, 1982 pp.416–422 Google Scholar

Kirby, L. A.S. : Ultrafilters and types on models of arithmetic, Annals Pure Appl. Logic Vol. 27, 1984 pp. 215–252 Google Scholar

Kirby, L. A.S., Mc Aloon, K., Murawski, R.: Indicators, recursive saturation andexpandability, Fund. Math. Vol. 114, 1981 pp. 127–139 Google Scholar

Kirby, L. A.S., Paris, J. B.: Initial segments of models of Peano's axioms. In: Set Theory and Hierarchy Theory V. Proceedings of the 3rd Conference on Set Theory and Hierarchy Theory, Springer-Verlag, 1977 pp. 211–226

Kirby, L. A.S., Paris, J. B.: Σn collection schemas in arithmetic. In: Logic Colloquium 77 (Stud Logic Found Math 96), North Holland: Amsterdam, 1978 pp. 199–209

Kirby, L. A.S., Paris, J. B.: Accessible independence results for Peano arithmetic, Bull. London Math. Soc. Vol. 14, 1982 pp. 285–293 Google Scholar

Kiss, P.: Diophantine representation of generalized Fibonacci numbers, Elemente der Math. Vol. 34, 1979 pp. 129–132 Google Scholar

Kleene, S.: Recursive functions and predicates, Trans. Amer. Math. Soc. Vol. 53, 1943 pp. 41–73 Google Scholar

Kleene, S. C: A symmetric form of Gödel's numbers, Indag. Math. Vol. 12, 1950 pp. 244–246 Google Scholar

Kleene, S. C: Finite axiomatizability of theories in the predicate calculus using additional predicate symbols. Memoirs of the Amer. Math. Soc, 1951 pp. 27–66 (Russian translation in Matematiceskaja teoria logiceskogo vyvoda, Nauka 1967, 237–284)Google Scholar

Kleene, S. C: On the forms of predicates in the theory of constructive ordinals I, Amer. Journ. of Math. Vol. 66, 1944, pp. 41–58 Google Scholar

Kleene, S. C: Introduction to metamathematics, North Holland, Amsterdam, 1952, 550 p.

Kleene, S. C: On the forms of predicates on the theory of constructive ordinals II, Amer.Journ. of Math, vol 77, 1955, pp. 405–428 Google Scholar

Kleene, S. C: Mathematical logic, Wiley & Sons: New York, 1967, 398 p.

Klingen, N.: Zur Idealstruktur in Nichtstandardmodellen von Dedekindringen, Journ.Reine Angew. Math Vol. 274/275, 1975 pp. 38–60 Google Scholar

Knight, J. F.: Complete types and the natural numbers, Journ. Symb. Log. Vol. 38, 1973 pp. 413–415 Google Scholar

Knight, J. F.: Types omitted in uncountable models of arithmetic, Journ. Symb. Log.Vol. 40, 1975 pp. 317–320 Google Scholar

Knight, J. F.: Omitting types in set theory and arithmetic, Journ. Symb. Log. Vol. 41,1976 pp. 25–32 Google Scholar

Knight, J. F.: Hanf numbers for omitting types over particular theories, Journ. Symb.Log. Vol. 41, 1976 pp. 583–588 Google Scholar

Knight, J. F.: Models of arithmetic and closed ideals, Journ. Symb. Log. Vol. 47, 1982 pp. 833 Google Scholar

Knight, J. F: Additive structure in uncountable models for a fixed completion of P, Journ.Symb. Log. Vol. 48, 1983 pp. 623–628 Google Scholar

Knight, J. F.: Effective construction of models. In: Logic Colloquium '84 (Studies in Logicand the Foundations of Mathematics 120), North-Holland, 1986 pp. 105–119

Knight, J. F., Lachlan, A. H., Soare, R. I.: Two theorems on degrees of models of truearithmetic, Journ. Symb. Log. Vol. 49, 1984 pp. 425–436 Google Scholar

Knight, J. F., Nadel, M.: Expansions of models and Turing degrees, Journ. Symb. Log.Vol. 47, 1982 pp. 587 Google Scholar

Kochen, S., Kripke, S. A.: Nonstandard models of Peano arithmetic, Enseign. Math., Ser 2 Vol. 28, 1982 pp. 211–231 Google Scholar

König, D.: Über eine Schlußweise aus dem Endlichen ins Unendliche, Acta litterarum acscientiarum Regiae Universitatis Hungaricae Francisco-Josephinae, Sectio scientiarummathematicarum Vol. 3, 1927, pp. 121–130 Google Scholar

Koppel, M.: Some decidable Diophantine problems: positive solution to a problem of Davis, Matiyasevič and Robinson, Proc. Amer. Math. Soc. Vol. 77, 1979 pp. 319–323 Google Scholar

Kossak, R.: An application of definable types of Peano's arithmetic, Bull. Acad. Polon.Sci. Vol. 28, 1980 pp. 213–217 Google Scholar

Kossak, R.: A certain class of models of Peano arithmetic, Journ. Symb. Log. Vol. 48,1983 pp. 311–320 Google Scholar

Kossak, R.: L∞,ω 1-elementary equivalence of ω 1-like models of PA, Fund. Math. Vol. 123,1984 pp. 123–131 Google Scholar

Kossak, R.: A note on satisfaction classes, Notre Dame Journ. Formal Log. Vol. 26, 1985 pp. 1–8 Google Scholar

Kossak, R.: Recursively saturated ω 1-like models of arithmetic, Notre Dame Journ. Formal Log. Vol. 26, 1985 pp. 413–422 Google Scholar

Kossak, R.: Remarks on free sets, Bull. Pol. Acad. Sci. Vol. 34, 1986 pp. 117–122 Google Scholar

Kossak, R.: Models with the ωproperty, Journ. Symb. Log. Vol. 54, 1989 pp. 177–189 Google Scholar

Kossak, R., Kotlarski, H.: Results on automorphisms of recursively saturated models of PA, Fund. Math. Vol. 129, 1988 pp. 1–7 Google Scholar

Kossak, R., Paris, J. B.: Subsets of models of arithmetic. In: Open Days in Model Th. & Set Th.; 1981 Jadwisin, 1981 pp. 159–174 Google Scholar

Kossovsky, N. K.: Diophantine representation of the sequence of solutions to the Pellequation, Zap. naučn. sem. Leningr. otdel. Mat. in-ta V. A. Steklova Vol. 20, 1971 pp.28–35 Google Scholar

Kotlarski, H.: On Skolem ultrapowers and their non-standard variant, Zeitschr. Math.Log. Grundl. Math. Vol. 26, 1980 pp. 227–236 Google Scholar

Kotlarski, H.: On elementary cuts in models of arithmetic, Bull. Acad. Polon. Sci. Vol.29, 1981 pp. 419–423 Google Scholar

Kotlarski, H.: On confinal extensions of models of arithmetic, Journ. Symb. Log. Vol.48, 1983 pp. 253–262 Google Scholar

Kotlarski, H.: On elementary cuts in models of arithmetic, Fund. Math. Vol. 115, 1983 pp. 27–31 Google Scholar

Kotlarski, H.: On elementary cuts in recursively saturated models of Peano arithmetic, Fund. Math. Vol. 120, 1984 pp. 205–222 Google Scholar

Kotlarski, H.: Some remarks on initial segments in models of Peano arithmetic, Journ.Symb. Log. Vol. 49, 1984 pp. 955–960 Google Scholar

Kotlarski, H.: Bounded induction and satisfaction classes. In: Proceedings of the 3rd Easter Conference on Model Theory, Humboldt-Univ. Berlin: Berlin, 1985 pp. 143–167

Kotlarski, H.: The recursively saturated part of models of Peano arithmetic, Zeitschr.Math. Log. Grundl. Math. Vol. 32, 1986 pp. 365–370 Google Scholar

Kotlarski, H.: Bounded induction and satisfaction classes, Zeitschr. Math. Log. Grundl.Math. Vol. 32, 1986 pp. 531–544 Google Scholar

Kotlarski, H.: On the end extension problem for ∆ 0 –PA(S), Zeitschr. f. Math. Logik 35,1989, pp. 391–397 Google Scholar

Kotlarski, H., Krajewski, S., Lachlan, A. H.: Construction of satisfaction classes fornonstandard models, Canad Math Bull Vol. 24, 1981 pp. 283–293 Google Scholar

Kotlarski, H., Ratajczyk, Z.: A weak extension of bounded induction in the languagewith a satisfaction class, Journ. Symb. Log. Vol. 52, 1987 pp. 1070 Google Scholar

Kotlarski, H., Ratajczyk, Z.: Inductive full satisfaction classes, Annals Pure Appl. Logic Vol. 47, 1990, pp. 199–223 Google Scholar

Krajewski, S.: Predicative expansions of axiomatic theories, Zeitschr. Math. Log. Grundl.Math. Vol. 20, 1974 pp. 435–452 Google Scholar

Krajewski, S.: Mutually inconsistent satisfaction classes, Bull. Acad. Polon. Sci. Vol. 22,1974 pp. 983–987 Google Scholar

Krajewski, S.: A remark on automorphisms and nonstandard properties, Bull. Acad.Polon. Sci. Vol. 22, 1974 pp. 989–991 Google Scholar

Krajewski, S.: Non-standard satisfaction classes. In: Set Theory and Hierarchy Theory.A Memorial Tribute to Andrzej Mostowski. Proceedings of the 2nd Conference on Set Theory and Hierarchy Theory (Lect. Notes. Math 537), Springer-Verlag, 1976 pp. 121–144

Krajewski, S.: A note on expansion of models of set theories. In: Set Theory and Hierarchy Theory. Proceedings of the 1st Colloquium in Set Theory and Hierarchy Theory,Politechnical University Wroclaw, 1977 pp. 63–67 Google Scholar

Krajíček, J.: Generalizations of proofs. In: Proc. 8–th Easter Conference on Model Theory,Wendisch-Rietz, 1987 pp. 82–99 Google Scholar

Krajíček, J.: A note on proofs of falsehood, Arch. Math. Log. Vol. 26, 1987 pp. 169–176 Google Scholar

Krajíček, J.: Speed-up for propositional Prege systems via generalizations of proofs, Comm. Math. Univ. Carol, Vol. 30, 1989, pp. 137–140 Google Scholar

Krajíček, J.: On the number of steps in proofs, Ann. Pure Appl. Logic, Vol. 41, 1989, pp.153–178 Google Scholar

Krajíček, J.: A theorem on uniform provability of schemes. In: Proc. 6–th Easter Conf.on Model Theory, Wendisch-Rietz, 1988 pp. 85–92 Google Scholar

Krajíček, J.: Exponentiation and second order bounded arithmetic, Ann. Pure Appl.Logic, Vol. 48, 1990, pp. 261–276 Google Scholar

Krajíček, J.: No counter-example interpretation and interactive computation. In: Logicfrom Computer Science, ed. Moschovakis, Y. N., Springer-Verlag 1992, pp. 287–293

Krajíček, J.: Fragments of Bounded Arithmetic and bounded query classes, Transactions AMS, to appear

Krajíček, J., Pudlák, P.: The number of proof lines and the size of proofs in first orderlogic, Arch. Math. Log. Vol. 27, 1988 pp. 69–84 Google Scholar

Krajíčef, J., Pudlák, P.: On the structure of initial segments of models of arithmetic, Arch. Math. Log. Vol. 28, 1989 pp. 91–98 Google Scholar

Krajíček, J., Pudlák, P.: Propositional provability and models of weak arithmetic. In:Proc. Computer Science Logic '89, (Lect. Notes Comp. Sci. 440), Springer-Verlag, 1990,pp. 193–210 Google Scholar

Krajíček, J., Pudlák, P.: Propositional proof systems, the consistency of first ordertheories and the complexity of computations, Journ. Symb. Log Vol. 54, 1989 pp. 1063–1079 Google Scholar

Krajíček, J., Pudlák, P.: Quantified propositional calculi and fragments of boundedarithmetic, Zeitschr. Math. Log. Grundl. Math Vol. 36, 1989 pp. 29–46 Google Scholar

Krajíček, J., Pudlák, P., Takeuti, G.: Bounded Arithmetic and the Polynomial Hierarchy, Annals of Pure and Appl. Logic 52, pp. 143–154

Krajíček, J.,Takeuti, G.: On induction-free provability, Annals of Mathematics and Artificial Intelligence, (to appear)

Krajíček, J.,Takeuti, G.: On bounded Σ 1 1polynomial induction. In: Feasible Mathematics, Buss, Scott eds., Birkhäuser 1990, pp. 259–280 Google Scholar

Kreisel, G.: A survey of proof theory II. In: Proceedings of the Second Scandinavian Logic Symposium, North Holland: Amsterdam pp. 109–170

Kreisel, G.: Informal rigour and completeness proofs. In: Proceedings of the International Colloquium in the Philosophy of Science Vol. 1, North-Holland: Amsterdam, 1967, pp.138–186

Kreisel, G.: Note on arithmetic models for consistent formulae of the predicate calculus I, Fund. Math. Vol. 37, 1950 pp. 265–285 Google Scholar

Kreisel, G.: On the interpretation of non-finitist proofs I, Journ. Symb. Logic Vol. 16,1951 pp. 241–267 Google Scholar

Kreisel, G.: On the concepts of completeness and interpretation of formal systems, Fund.Math. Vol. 39, 1952 pp. 103–127 Google Scholar

Kreisel, G.: Some concepts concerning formal systems of number theory, Zeitschr. Math.Log. Grundl. Math. Vol. 57, 1952 pp. 1–12 Google Scholar

Kreisel, G.: On the interpretation of non-finitist proofs II. Interpretation of numbertheory. Applications, Journ. Symb. Log. Vol. 17, 1952 pp. 43–58 Google Scholar

Kreisel, G.: The diagonal method in formalized arithmetic, Brit. Journ. Phil. Sci. Vol. 3,1953 pp. 364–373 Google Scholar

Kreisel, G.: On a problem of Henkin's, Indag. Math. Vol. 15, 1953 pp. 405–406 Google Scholar

Kreisel, G.: Note on arithmetic models for consistent formulae of the predicate calculus II. In: Actes du lleme Congres International de Philosophic, North Holland: Amsterdam Vol. 14, 1953 pp. 39–49

Kreisel, G.: A variant to Hilbert's theory of the foundations of arithmetic,Br. J. Phil.Sci. Vol. 4, 1953 pp. 107–129 Google Scholar

Kreisel, G.: Remark on complete interpretations by models, Arch. Math. Log. Vol. 2,1954 pp. 4–9 Google Scholar

Kreisel, G.: Models, translations and interpretations. In: Mathematical Interpretationof Formal Systems; 1954 Amsterdam (Stud Logic Found Math 10), North Holland,Amsterdam, 1955 pp. 26–50

Kreisel, G.: Gödel's interpretation of Heyting's arithmetic. In: Summaries of Talks Presented at the Summer Institute for Symbolic Logic, Institute for Defense Analyses,Comm. Res. Division, 1957 pp. 125–133 Google Scholar

Kreisel, G.: On weak completeness of intuitionistic predicate logic, Journ. Symb. Log.Vol. 27, 1962 pp. 139–158 Google Scholar

Kreisel, G.: Foundations of intuitionistic logic. In: Logic, Methodology and Philosophy of Science, Stanford, 1962, pp. 198–210 Google Scholar

Kreisel, G.: Mathematical logic. In: Lect. Modern Math., III, Wiley, New York, 1965, pp.95–195

Kreisel, G.: A survey on proof theory, Journ. Symb. Log. Vol. 33, 1968 pp. 321–388 Google Scholar

Kreisel, G.: Axiomatizations of nonstandard analysis that are conservative extensionsof formal systems for classical standard analysis. In: Applications of Model Theory to Algebra, Analysis and Probability, Holt, Rinehart and Winston, 1969 pp. 93–106

Kreisel, G.: Which number theoretic problems can be solved in recursive progressions onП 1 1-paths through O?, Journ. Symb. Log. Vol. 37, 1972 pp. 311–334 Google Scholar

Kreisel, G.: Observations on a recent generalization of completeness theorems due to Schiitte. In: Proof Theory Symposion, Kiel 1974 (Lect. Notes Math. 500), Springer-Verlag, 1975 pp. 164–181

Kreisel, G.: What have we learnt from Hilbert's second problem? In: Mathematical Developments Arising From Hilbert's Problems, Amer. Math. Soc: Providence, 1976, pp. 93–130 Google Scholar

Kreisel, G.: Wie die Beweistheorie zu ihren Ordinalzahlen kam und kommt, Jber. d. Dt.Math.-Verein. Vol. 78, 1977 pp. 177–223 Google Scholar

Kreisel, G.: Finiteness theorems in arithmetic: an application of Herbrand's theorem forΣ 2formulas. In: Proceedings of the Herbrand Symposium. Logic Colloquium '81 (Stud Logic Found Math 107), North Holland: Amsterdam, 1982 pp. 39–55

Kreisel, G., Levy, A.: Reflections principles and their use for establishing the complexityof axiomatic systems, Zeitschr. Math. Log. Grundl. Math. Vol. 14, 1968 pp. 97–142 Google Scholar

Kreisel, G., Minc, G. E. Simpson, S. G.: The use of abstract languages in elementarymathematics: some pedagogic examples. In: Logic Colloquium, Springer-Verlag, 1975 pp. 38–131

Kreisel, G., Shoenfield, J. R., Wang, H.: Number theoretic concepts and recursivewell-orderings, Arch. Math. Log. Vol. 5, 1959 pp. 42–64 Google Scholar

Kreisel, G., Takeuti, G.: Formally self-referential propositions in cut-free classical analysis and related systems. Diss. Math. Vol. 118, 1974 pp. 1–50 Google Scholar

Kreisel, G., Wang, H.: Some applications of formalized consistency proofs, Fund. Math.Vol. 42 (dod. 45), 1955 pp. 101 n. (334)Google Scholar

Kriauciukas, V.: A Diophantine representation of perfect numbers, Zap. naučn. sem.Leningr. otdel. Mat. in-ta V. A. Steklova Vol. 88, 1979 pp. 78–89 Google Scholar

Kripke, S. A.: “Flexible” predicates of formal number theory, Proc. Amer. Math. Soc. Vol. 13, 1962 pp. 647–650 Google Scholar

Kripke, S. A., Pour-El, M. B.: Deduction preserving recursive isomorphisms of theories, Fund. Math. Vol. 61, 1967 pp. 141–147 Google Scholar

Kucera, A.: Measure, П 0 1-classes and complete extensions of PA. In: Recursion Theory Week (Lect. Notes Math. 1141), Springer-Verlag, 1985 pp. 245–259

Kučera, A.: Alternative, priority-free constructions of some r. e. degrees and other resultson degrees, Journ. Symb. Log. Vol. 52, 1987 pp. 1071 Google Scholar