Skip to main content
SpringerLink
Log in

Arithmetic Encoding

  • Chapter
Logical Number Theory I

Part of the book series: Universitext ((UTX))

Abstract

The rare graduate student or advanced undergraduate who hasn’t haunted the library and leafed through many an incomprehensible mathematical monograph and has, thus, not yet heard of (say) algebraic geometry or orthogonal polynomials or the unsolvability of Hilbert’s tenth problem might well wonder why a book written for graduate students and advanced undergraduates should begin with something so simple as polynomials. Well, polynomials have had a long history and they form a recurring theme throughout all of mathematics.

Die Mathematiker sind eine Art Franzosen: redet man zu ihnen, so übersetzen sie es in ihre Sprache, und dann ist es sobald ganz etwas anderes.

—J. W. von Goethe

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 69.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 89.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Reading List

§1

  1. I.G. Bashmakova, Diophant und diophantische Gleichungen, UTB-Birkhäuser, Basel, 1974.

    MATH  Google Scholar 

  2. G Boole, A Treatise on the Calculus of Finite Differences, Dover, New York, 1960.

    MATH  Google Scholar 

  3. H Eves, An Introduction to the History of Mathematics, 6th ed., Saunders, Philadelphia, 1990.

    MATH  Google Scholar 

  4. Anthony Hyman, Charles Babbage; Pioneer of the Computer, Princeton, 1982.

    Google Scholar 

  5. P and E Morrison, Charles Babbage and His Calculating Engines, Dover, New York, 1961.

    MATH  Google Scholar 

§2

  1. David Eugene Smith, A Source Book in Mathematics, I, Dover, New York, 1959. (Cf. in particular the entry, “Bernoulli on ‘Bernoulli Numbers’”.)

    Google Scholar 

  2. Dirk J. Struik, A Source Book in Mathematics, 1200–1800, Harvard, Cambridge (Mass.), 1969. (Cf. in particular the entry “Sequences and series” by Jakob Bernoulli.)

    MATH  Google Scholar 

  3. R. Calinger, Classics of Mathematics, Moore Pub. Co., Oak Park (Ill.), 1982.

    Google Scholar 

  4. John Fauvel and Jeremy Gray, The History of Mathematics; A Reader, Macmillan Education, London, 1987.

    Google Scholar 

§3

  1. Georg Cantor, Contributions to the Founding of the Theory of Transfinite Numbers, Dover, New York, 1955.

    Google Scholar 

  2. Richard Dedekind, “The nature and meaning of numbers”, in: Richard Dedekind, Essays in the Theory of Numbers, Dover, New York, 1963.

    Google Scholar 

  3. Galileo Galilei, Dialogues Concerning the Two New Sciences, Dover, New York, 1954.

    Google Scholar 

  4. Jean van Heijenoort, From Frege to Gödel; A Source Book in Mathematical Logic, 1879 – 1931, Harvard, Cambridge (Mass.), 1967.

    MATH  Google Scholar 

  5. Herbert Meschkowski, Das Problem des Unendlichen, Deutscher Taschenbuch Verlag, München, 1974.

    MATH  Google Scholar 

§5

  1. Serge Lang, Algebra, Addison-Wesley, Reading (Mass.), 1965.

    MATH  Google Scholar 

  2. Serge Lang, Complex Analysis, Addison-Wesley, Reading (Mass.), 1977.

    MATH  Google Scholar 

  3. John S. Lew and Arnold L. Rosenberg, “Polynomial indexing of integer lattice points, I, II”, J. Number Theory 10 (1978), 192 – 214, 215 – 243.

    Article  MATH  MathSciNet  Google Scholar 

§6

  1. Leonard Eugene Dickson, History of the Theory of Numbers, II, Chelsea, New York, 1952.

    Google Scholar 

  2. Ho Peng-Yoke, “Ch’in Chiu-shao”, Dictionary of Scientific Biography, Scribner’s, New York, 1971.

    Google Scholar 

§8

  1. Richard Dedekind, “The nature and meaning of numbers”, in: Richard Dedekind, Essays in the Theory of Numbers, Dover, New York, 1963.

    Google Scholar 

  2. Kurt Gödel, “Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I”, Monatsheft f. Mathematik u. Physik 38 (1931), 173 – 198; English translations in: Martin Davis, ed., The Undecidable, Raven Press, Hewlett (NY), 1965; Jean van Heijenoort, ed., From Frege to Gödel; A Source Book in Mathematical Logic, 1879 – 1931, Harvard, Cambridge (Mass.), 1967; and Solomon Feferman et al., eds., Kurt Gödel; Collected Works, I, Oxford, 1986.

    Article  Google Scholar 

  3. Thoralf Skolem, “Begründung der elementären Arithmetik durch die rekurrierende Denkweise ohne Anwendung scheinbarer Veränderlichen mit unendlichem Ausdehnungsbereich”, most accessible in: Thoralf Skolem, Selected Works in Logic, Universitetsforlaget, Oslo, 1970; English translation in: Jean van Heijenoort, ed., From Frege to Gödel; A Source Book in Mathematical Logic, 1879 – 1931, Harvard, Cambridge (Mass.), 1967.

    Google Scholar 

  4. Rosza Péter, Recursive Functions, Academic Press, New York, 1967.

    MATH  Google Scholar 

  5. H.E. Rose, Subrecursion; Functions and Hierarchies, Oxford, 1984.

    Google Scholar 

  6. K. Chandrasekharan, Introduction to Analytic Number Theory, Springer-Verlag, New York, 1968.

    Book  MATH  Google Scholar 

§9

  1. Wilhelm Ackermann, “Zum Hilbertschen Aufbau der reelen Zahlen”, Math. Annalen 99 (1928), 118 – 133; English translation in: Jean van Heijenoort, ed., From Frege to Gödel; A Source Book in Mathematical Logic, 1879 – 1931, Harvard, Cambridge (Mass.), 1967.

    Article  MATH  MathSciNet  Google Scholar 

  2. David Hilbert, “Über das Unendliche”, Math. Annalen 95 (1926), 161 – 190; English translation in: Jean van Heijenoort, ed., From Frege to Gödel; A Source Book in Mathematical Logic, 1879 – 1931, Harvard, Cambridge (Mass.), 1967.

    Article  MathSciNet  Google Scholar 

  3. Rosza Péter, “Konstruktion nichtrekursiver Funktionen”, Math. Annalen 111 (1935), 42 – 60.

    Article  Google Scholar 

  4. Gabriel Sudan, “Sur le nombre transfini ωω”, Bulletin mathematique de la Société roumaine des sciences 30 (1927), 11 – 30; cf.: Christian Calude and Solomon Marcus, “The first example of a recursive function which is not primitive recursive”, Historia Mathematica 6 (1979), 380 – 384.

    MATH  Google Scholar 

  5. Ronald Graham, Bruce Rothschild, and Joel Spencer, Ramsey Theory, J. Wiley and Sons, New York, 1980.

    MATH  Google Scholar 

  6. A.Y. Khinchin, Three Pearls of Number Theory, Graylock Press, Baltimore, 1952.

    MATH  Google Scholar 

  7. Steven F. Bellenot, “The Banach space T and the fast growing hierarchy from logic”, to appear.

    Google Scholar 

§10

  1. Kurt Gödel, “on undecidable propositions of formal mathematical systems”, in: Martin Davis, ed., The Undecidable, Raven Press, Hewlett (NY), 1965; and Solomon Feferman, et al., eds., Kurt Gödel; Collected Works, I, Oxford, 1986.

    Google Scholar 

  2. Thoralf Skolem, “Über die Zurückführbarkeit einiger durch Rekursionen definierter Relationen auf ‘Arithmetische’”, most accessible in: Thoralf Skolem, Selected Works in Logic, Universitetsforlaget, Oslo, 1970.

    Google Scholar 

  3. Raymond Smullyan, Theory of Formal Systems, Princeton, 1961.

    Google Scholar 

§11

  1. Eugene Wigner, “The unreasonable effectiveness of mathematics in the natural sciences”, Commun. in Pure and Applied Math. 13 (1960), 1 – 14; reprinted in: Douglas Campbell and John Higgins, eds., Mathematics; People, Problems, Results, III, Wadsworth, Belmont (Cal.), 1984.

    Article  MATH  Google Scholar 

  2. Alan M. Turing, “On computable numbers with an application to the Entscheidungsproblem”, Proc. London Math. Soc., ser. 2, vol. 42 (1936 – 37), 230 – 265; reprinted in: Martin Davis, ed., The Undecidable, Raven Press, Hewlett (NY), 1965.

    MathSciNet  Google Scholar 

  3. Stephen Kleene, “Origins of recursive function theory”, Annals of the History of Computing 3 (1981), 52 – 67.

    Article  MATH  MathSciNet  Google Scholar 

  4. Heinz-Dieter Ebbinghaus, Jörg Rum, and Wolfgang Thomas, Mathematical Logic, Springer-Verlag, New York, 1984. (Chapter X)

    MATH  Google Scholar 

  5. John Bell and Moshe Machover, A Course in Mathematical Logic, North-Holland, Amsterdam, 1977. (Chapter 6)

    MATH  Google Scholar 

§12

  1. Hartley Rogers, Jr., Theory of Recursive Functions and Effective Computability, McGraw-Hill, New York, 1967.

    MATH  Google Scholar 

  2. Michael Burke and Ron Genise, LOGO and Models of Computation, Addison-Wesley, Menlo Park (Cal.), 1987.

    Google Scholar 

  3. William F. Dowling, “There are no safe virus tests”, Amer. Math. Monthly 96 (1989), 835 – 836.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. 429 S. Warwick, Westmont, IL, 60559, USA

    Craig Smoryński

Authors
  1. Craig Smoryński
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 1991 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Smoryński, C. (1991). Arithmetic Encoding. In: Logical Number Theory I. Universitext. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-75462-3_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-75462-3_1

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-52236-2

  • Online ISBN: 978-3-642-75462-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation