Bulletin of the American Mathematical Society

MathSciNet review: 1567568
Full text of review: PDF   This review is available free of charge.
Book Information:

Author: M. C. Fitting
Title: Fundamentals of generalized recursion theory
Additional book information: Studies in Logic and The Foundations of Mathematics, Volume 105, North-Holland Publishing Company, Amsterdam, 1981, xx + 302 pp., $63.75 U.S./Dfl. 150.00.

S. I. Adian, William W. Boone, and Graham Higman (eds.), Word problems. II, Studies in Logic and the Foundations of Mathematics, vol. 95, North-Holland Publishing Co., Amsterdam-New York, 1980. The Oxford book; Outgrowth of a Conference on Decision Problems in Algebra held in Oxford, Summer 1976. MR 579933

A. Baker, Recent advances in transcendence theory, Proceedings of the International Conference on Number Theory (Moscow, 1971), 1973, pp. 67–69, 264 (English, with Russian summary). MR 0332669

A. Baker and H. Davenport, The equations $3x^{2}-2=y^{2}$ and $8x^{2}-7=z^{2}$, Quart. J. Math. Oxford Ser. (2) 20 (1969), 129–137. MR 248079, DOI 10.1093/qmath/20.1.129

Jon Barwise (ed.), The syntax and semantics of infinitary languages, Lecture Notes in Mathematics, No. 72, Springer-Verlag, Berlin-New York, 1968. MR 0234827

Handbook of mathematical logic, Studies in Logic and the Foundations of Mathematics, vol. 90, North-Holland Publishing Co., Amsterdam, 1977. With the cooperation of H. J. Keisler, K. Kunen, Y. N. Moschovakis and A. S. Troelstra. MR 457132

Errett Bishop, Foundations of constructive analysis, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1967. MR 0221878

E. Bombieri and J. Vaaler, On Siegel’s lemma, Invent. Math. 73 (1983), no. 1, 11–32. MR 707346, DOI 10.1007/BF01393823

Georg Cantor, Gesammelte Abhandlungen mathematischen und philosophischen Inhalts. Mit erläuternden Anmerkungen sowie mit Ergänzungen aus dem Briefwechsel Cantor-Dedekind, Georg Olms Verlagsbuchhandlung, Hildesheim, 1962 (German). Herausgegeben von Ernst Zermelo; Nebst einem Lebenslauf Cantors von Adolf Fraenkel. MR 0148517

J. W. S. Cassels, Diophantine equations with special reference to elliptic curves, J. London Math. Soc. 41 (1966), 193–291. MR 199150, DOI 10.1112/jlms/s1-41.1.193

[C2] J. W. S. Cassels, Review of: Introduction to Number Theory by Hua Loo Keng (translated by Peter Shiu), Math. Intelligencer 5 no 2 (1983), 57.

Dirk van Dalen, D. Lascar, and T. J. Smiley (eds.), Logic Colloquium ’80, Studies in Logic and the Foundations of Mathematics, vol. 108, North-Holland Publishing Co., Amsterdam-New York, 1982. Papers intended for the European Summer Meeting of the Association for Symbolic Logic to have been held in Prague, August 24–30, 1980. MR 673782

Ju. L. Eršov, Theorie der Numerierungen. II, Z. Math. Logik Grundlagen Math. 21 (1975), no. 6, 473–584 (German). MR 389562, DOI 10.1002/malq.19750210164

R. O. Gandy and C. E. M. Yates (eds.), Logic Colloquium ’69, Studies in Logic and the Foundations of Mathematics, Vol. 61, North-Holland Publishing Co., Amsterdam-London, 1971. MR 0272598

Kurt Gödel, Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I, Monatsh. Math. Phys. 38 (1931), no. 1, 173–198 (German). MR 1549910, DOI 10.1007/BF01700692

Kurt Gödel, The Consistency of the Continuum Hypothesis, Annals of Mathematics Studies, No. 3, Princeton University Press, Princeton, N. J., 1940. MR 0002514

G. Higman, Subgroups of finitely presented groups, Proc. Roy. Soc. London Ser. A 262 (1961), 455–475. MR 130286, DOI 10.1098/rspa.1961.0132

D. Hilbert and P. Bernays, Grundlagen der Mathematik. II, Die Grundlehren der mathematischen Wissenschaften, Band 50, Springer-Verlag, Berlin-New York, 1970 (German). Zweite Auflage. MR 0272596

Leo Harrington and Saharon Shelah, The undecidability of the recursively enumerable degrees, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 1, 79–80. MR 634436, DOI 10.1090/S0273-0979-1982-14970-9

R. Björn Jensen, The fine structure of the constructible hierarchy, Ann. Math. Logic 4 (1972), 229–308; erratum, ibid. 4 (1972), 443. With a section by Jack Silver. MR 309729, DOI 10.1016/0003-4843(72)90001-0

S. C. Kleene, Hierarchies of number-theoretic predicates, Bull. Amer. Math. Soc. 61 (1955), 193–213. MR 70593, DOI 10.1090/S0002-9904-1955-09896-3

S. C. Kleene, Recursive functionals and quantifiers of finite types. I, Trans. Amer. Math. Soc. 91 (1959), 1–52. MR 102480, DOI 10.1090/S0002-9947-1959-0102480-9

Wilbur Knorr, “La croix des mathématiciens”: the Euclidean theory of irrational lines, Bull. Amer. Math. Soc. (N.S.) 9 (1983), no. 1, 41–69. MR 699316, DOI 10.1090/S0273-0979-1983-15155-8

Georg Kreisel, Mathematical logic: tool and object lesson for science, Synthese 62 (1985), no. 2, 139–151. The present state of the problem of the foundations of mathematics (Florence, 1981). MR 788502, DOI 10.1007/BF00486043

G. Longo and E. Moggi, The hereditary partial effective functionals and recursion theory in higher types, J. Symbolic Logic 49 (1984), no. 4, 1319–1332. MR 771797, DOI 10.2307/2274281

Donald A. Martin, Borel determinacy, Ann. of Math. (2) 102 (1975), no. 2, 363–371. MR 403976, DOI 10.2307/1971035

Marian Boykan Pour-El and Ian Richards, The wave equation with computable initial data such that its unique solution is not computable, Adv. in Math. 39 (1981), no. 3, 215–239. MR 614161, DOI 10.1016/0001-8708(81)90001-3

T. L. Saaty (ed.), Lectures on modern mathematics. Vol. I, John Wiley & Sons, Inc., New York-London, 1963. MR 0177842

J. R. Shoenfield, Measurable cardinals, Logic Colloquium ’69 (Proc. Summer School and Colloq., Manchester, 1969) North-Holland, Amsterdam, 1971, pp. 19–49. MR 0282830

Richard A. Shore, The theory of the degrees below ${\bf 0}^{\prime }$, J. London Math. Soc. (2) 24 (1981), no. 1, 1–14. MR 623666, DOI 10.1112/jlms/s2-24.1.1

Wacław Sierpiński, Le dernier théorème de Fermat pour les nombres ordinaux, Fund. Math. 37 (1950), 201–205 (French). MR 40372, DOI 10.4064/fm-37-1-201-205

Steve Smale, The fundamental theorem of algebra and complexity theory, Bull. Amer. Math. Soc. (N.S.) 4 (1981), no. 1, 1–36. MR 590817, DOI 10.1090/S0273-0979-1981-14858-8

C. E. M. Yates, Prioric games and minimal degrees below ${\bf 0}^{(1)}$, Fund. Math. 82 (1974/75), 217–237. MR 406784, DOI 10.4064/fm-82-3-217-237

[Z] K. Zuse, Der Computer―mein Lebenswerk, Verlag für moderne Industrie, Munich, 1970.

[ABH] S. I. Adian, W. W. Boone and G. Higman (eds.), Word problems II, North-Holland, 1980. MR 0579933




[B] A. Baker, Recent advances in transcendence theory, Trudy Math. Inst. Steklov 132 (1973), 67-69. MR 332669




[BD] A. Baker and H. Davenport, The equations 3x³ - 2 = y² and 8x² - 7 = z², Quarterly J. Math. Oxford Ser. (2) 20 (1969), 129-137. MR 248079




[B1] J. Barwise (ed.), Syntax and semantics of infinitary languages, Lecture Notes, vol. 72, Springer-Verlag, 1968. MR 234827




[B2] J. Barwise (ed.), Handbook of mathematical logic, North-Hollaǹd, 1977. MR 457132




[Bi] E. Bishop, Foundations of constructive analysis, McGraw-Hill, 1967. MR 221878




[BV] E. Bombieri and J. Vaalen, On Siegel's Lemma, Invent. Math. 73 (1983), 11-32, addendum 75 (1984), 377. MR 707346




[Ca] G. Cantor, Gesammelte Abhandlungen (E. Zermelo, ed.), Springer-Verlag, 1932. MR 148517




[C1] J. W. S. Cassels, Diophantine equations with special reference to elliptic curves, J. London Math. Soc. 41 (1966), 193-291. MR 199150




[C2] J. W. S. Cassels, Review of: Introduction to Number Theory by Hua Loo Keng (translated by Peter Shiu), Math. Intelligencer 5 no 2 (1983), 57.




[DLS] D. van Dalen, D. Lascar and T. J. Smiley (eds.), Logic Colloquium '80, North-Holland, 1983. MR 673782




[E] J. L. Ersov, Theorie der Numerierungen II, III, Z. Math. Logik Grundlag. Math. 21 (1975) 473-584; und 23 (1977), 289-371. MR 389562




[GY] R. O. Gandy and C. E. M. Yates (eds.), Logic Colloquium '69, North-Holland, 1971. MR 272598




[G1] K. Gödel, Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I, Mh. Math. Physik 38 (1931), 173-198. MR 1549910




[G2] K. Gödel, The consistency of the continuum hypothesis, Ann. Math. Studies no. 3, 1940. MR 2514




[H] G. Higman, Subgroups of finitely presented groups, Proc. Roy. Soc. Ser. A 262 (1961), 455-475. MR 130286




[HB] D. Hilbert and P. Bernays, Grundlagen der Mathematik, vol. 2, Springer-Verlag, Berlin and New York, 1970. MR 272596




[HS] L. Harrington and S. Shelah, The undecidability of the recursively enumerable degrees, Bull. Amer. Math. Soc. (N.S). 6 (1982), 79-80. MR 634436




[J] R. B. Jensen, The fine structure of the constructible hierarchy, Ann. of Math. Logic 4 (1972), 229-308. MR 309729




[K1] S. C. Kleene, Hierarchies of number-theoretic predicates, Bull. Amer. Math. Soc. 61 (1955), 193-213; reviewed MR 17 (1956), 4. MR 70593




[K2] S. C. Kleene, Recursive functional and quantifiers of finite type, Trans. Amer. Math. Soc. 91 (1959), 1-52. MR 102480




[Kn] W. Knorr, La croix des mathématiciens, Bull. Amer. Math. Soc. (N.S.) 9 (1983), 41-69. MR 699316




[Kr] G. Kreisel, Mathematical logic: tool and object lesson for science, Synthese 62 (1985), 139-152. MR 788502




[LM] G. Longo and E. Moggi, The hereditary partial effective functional and recursion theory in higher types, J. Symbolic Logic 49 (1984), 1319-1332. MR 771797




[M] D. A. Martin, Borel determinacy, Ann. of Math. (2) 102 (1975), 363-371. MR 403976




[PR] M. B. Pour-El and I. Richards, The wave equation with computable initial data such that its unique solution is not computable, Advances in Math. 39 (1981), 215-239; reviewed J. Symbolic Logic 47 (1982), 900-902. MR 614161




[S] T. L. Saaty (ed.), Lectures on modern mathematics. III, Wiley, New York, 1965. MR 177844




[Sh] J. R. Shoenfield, Degrees of undecidability, North-Holland, 1971. MR 282830




[Sho] R. A. Shore, The theory of the degrees below 0', J. London Math. Soc. (2) 24 (1981), 1-14. MR 623666




[Si] W. Sierpiński, Le dernier théorème de Fermat pour les nombres ordinaux, Fund. Math. 37 (1950), 201-205. MR 40372




[Sm] S. Smale, The fundamental theorem of algebra and complexity theory, Bull. Amer. Math. Soc. (N.S.) 4 (1981), 1-36. MR 590817




[Y] C. E. M. Yates, Prioric games and minimal degrees below 0', Fund. Math. 82 (1974), 217-237. MR 406784




[Z] K. Zuse, Der Computer―mein Lebenswerk, Verlag für moderne Industrie, Munich, 1970.