Book contents
- Frontmatter
- Contents
- Introduction
- Historical remarks on Suslin's problem
- The continuum hypothesis, the generic-multiverse of sets, and the Ω conjecture
- ω-models of finite set theory
- Tennenbaum's theorem for models of arithmetic
- Hierarchies of subsystems of weak arithmetic
- Diophantine correct open induction
- Tennenbaum's theorem and recursive reducts
- History of constructivism in the 20th century
- A very short history of ultrafinitism
- Sue Toledo's notes of her conversations with Gödel in 1972–5
- Stanley Tennenbaum's Socrates
- Tennenbaum's proof of the irrationality of √2
- References
Tennenbaum's theorem and recursive reducts
Published online by Cambridge University Press: 07 October 2011
Book contents
- Frontmatter
- Contents
- Introduction
- Historical remarks on Suslin's problem
- The continuum hypothesis, the generic-multiverse of sets, and the Ω conjecture
- ω-models of finite set theory
- Tennenbaum's theorem for models of arithmetic
- Hierarchies of subsystems of weak arithmetic
- Diophantine correct open induction
- Tennenbaum's theorem and recursive reducts
- History of constructivism in the 20th century
- A very short history of ultrafinitism
- Sue Toledo's notes of her conversations with Gödel in 1972–5
- Stanley Tennenbaum's Socrates
- Tennenbaum's proof of the irrationality of √2
- References
Summary
To honor and celebrate the memory of Stanley Tennenbaum
Stanley Tennenbaum's influential 1959 theorem asserts that there are no recursive nonstandard models of Peano Arithmetic (PA). This theorem first appeared in his abstract [42]; he never published a complete proof. Tennenbaum's Theorem has been a source of inspiration for much additional work on nonrecursive models. Most of this effort has gone into generalizing and strengthening this theorem by trying to find the extent to which PA can be weakened to a subtheory and still have no recursive nonstandard models. Kaye's contribution [12] to this volume has more to say about this direction.
This paper is concerned with another line of investigation motivated by two refinements of Tennenbaum's theorem in which not just the model is nonrecursive, but its additive and multiplicative reducts are each nonrecursive. For the following stronger form of Tennenbaum's Theorem credit should also be given to Kreisel [5] for the additive reduct and to McAloon [26] for the multiplicative reduct.
Tennenbaum's Theorem. If M = (M, +, ·, 0, 1, ≤) is a nonstandard model of PA, then neither its additive reduct (M, +) nor its multiplicative reduct (M, ·) is recursive.
What happens with other reducts? The behavior of the order reduct, as is well known, is quite different from that of the additive and multiplicative reducts. The order type of every countable nonstandard model is ω + (ω* + ω) · η, where ω and η are the order types of the nonnegative integers ℕ and the rationals ℚ, respectively.
- Type
- Chapter
- Information
- Set Theory, Arithmetic, and Foundations of MathematicsTheorems, Philosophies, pp. 112 - 149Publisher: Cambridge University PressPrint publication year: 2011
References
[1] and , On recursively saturated models of arithmetic, Model Theory and Algebra (AMemorial Tribute to Abraham Robinson), Lecture Notes in Mathematics, vol. 498, Springer, Berlin, 1975, pp. 42–55.Google Scholar
[2] and , An introduction to recursively saturated and resplendent models, The Journal of Symbolic Logic, vol. 41 (1976), no. 2, pp. 531–536.Google Scholar
[3] , Théorie élémentaire de la multiplication des entiers naturels, Model Theory and Arithmetic (Paris, 1979–1980), Lecture Notes in Mathematics, vol. 890, Springer, Berlin, 1981, pp. 44–89.Google Scholar
[4] , Computability theory and linear orderings, Handbook of Recursive Mathematics, Vol. 2, Studies in Logic and the Foundations of Mathematics, vol. 139, North-Holland, Amsterdam, 1998, pp. 823–976.Google Scholar
[5] and , Strong models of arithmetic, Bulletin de l'Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques, vol. 14 (1966), pp. 107–110.Google Scholar
[6] , Skolem functions and constructible models, Algebra i Logika, vol. 12 (1973), pp. 644–654, [Translation: Algebra and Logic, vol. 12 (1973), pp. 368–373].Google Scholar
[7] , Abstract of “Arithmetically definable models of formalized arithmetic”, Notices of the American Mathematical Society, vol. 5 (1958), pp. 679–680.Google Scholar
[8] , Pure computable model theory, Handbook of recursive mathematics, Vol. 1, Studies in Logic and the Foundations of Mathematics, vol. 138, North-Holland, Amsterdam, 1998, pp. 3–114.Google Scholar
[9] , Ordering by divisibility in abstract algebras, Proceedings of the London Mathematical Society. Third Series, vol. 2 (1952), pp. 326–336.Google Scholar
[10] and , Some problem in elementary arithmetics, Polska Akademia Nauk. Fundamenta Mathematicae, vol. 92 (1976), no. 3, pp. 223–245.Google Scholar
[11] and , Πo1 classes and degrees of theories, Trans actions of the American Mathematical Society, vol. 173 (1972), pp. 33–56.Google Scholar
[12] , Tennenbaum's theorem for models of arithmetic, this volume.
[13] , Models of Peano Arithmetic, Oxford Logic Guides, vol. 15, The Clarendon Press Oxford University Press, New York, 1991, Oxford Science Publications.Google Scholar
[14] , Open induction, Tennenbaum phenomena, and complexity theory, Arithmetic, Proof Theory, and Computational Complexity (Prague, 1991), Oxford Logic Guides, vol. 23, Oxford University Press, New York, 1993, pp. 222–237.Google Scholar
[16] , Finite axiomatizability of theories in the predicate calculus using additional predicate symbols. Two papers on the predicate calculus, Memoirs of the American Mathematical Society, vol. 1952 (1952), no. 10, pp. 27–68.Google Scholar
[18] and , The Structure of Models of Peano Arithmetic, Oxford Logic Guides, vol. 50, The Clarendon Press Oxford University Press, Oxford, 2006.Google Scholar
[19] , Well-quasi-ordering, the Tree Theorem, and Vazsonyi's conjecture, Transactions of the American Mathematical Society, vol. 95 (1960), pp. 210–225.Google Scholar
[20] , Finitely constrained classes of homogeneous directed graphs, The Journal of Symbolic Logic, vol. 59 (1994), no. 1, pp. 124–139.Google Scholar
[21] , Structure theorem for tournaments omitting N5, Journal of Graph Theory, vol. 42 (2003), no. 3, pp. 165–192.Google Scholar
[22] and , Theories with recursive models, The Journal of Symbolic Logic, vol. 44 (1979), no. 1, pp. 59–76.Google Scholar
[23] , Über die vollständigkeit eines gewissen systems der arithmetik ganzer zahlen, in welchem die addition als einzige operation hervortritt, Comptes Rendus du Ier congrès des Mathématiciens des Pays Slaves, Warszawa, (1929), pp. 92–101.
[24] , The complexity of types in field theory, Logic Year 1979–80 (Proc. Seminars and Conf. Math. Logic, Univ. Connecticut, Storrs, Conn., 1979/80), Lecture Notes in Mathematics, vol. 859, Springer, Berlin, 1981, pp. 143–156.Google Scholar
[25] and , Degrees of recursively saturated models, Transactions of the American Mathematical Society, vol. 282 (1984), no. 2, pp. 539–554.Google Scholar
[26] , On the complexity of models of arithmetic, The Journal of Symbolic Logic, vol. 47 (1982), no. 2, pp. 403–415.Google Scholar
[27] , Basic wqo- and bqo-theory, Graphs and Order (Banff, Alta., 1984), NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, vol. 147, Reidel, Dordrecht, 1985, pp. 487–502.Google Scholar
[28] , A formula with no recursively enumerable model, Polska Akademia Nauk. Fundamenta Mathematicae, vol. 42 (1955), pp. 125–140.Google Scholar
[29] , The completeness of Peano multiplication, Israel Journal of Mathematics, vol. 39 (1981), no. 3, pp. 225–233.Google Scholar
[30] , On well-quasi-ordering finite trees, Mathematical Proceedings of the Cambridge Philosophical Society, vol. 59 (1963), pp. 833–835.Google Scholar
[31] , Every recursively enumerable extension of a theory of linear order has a constructive model, Algebra i Logika, vol. 12 (1973), pp. 211–219, [Translation: Algebra and Logic, vol. 12 (1973), 120—124.].Google Scholar
[32] , Un bel ordre d'abritement et ses rapports avec les bornes d'une multirelation, Comptes Rendus de l'Académie des Sciences - Series A-B, vol. 274 (1972), pp. A1677–A1680.Google Scholar
[33] , Linear Orderings, Pure and Applied Mathematics, vol. 98, Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1982.Google Scholar
[34] , Toward model theory through recursive saturation, The Journal of Symbolic Logic, vol. 43 (1978), no. 2, pp. 183–206.Google Scholar
[35] , Arborescent structures. I. Recursive models, Aspects of E?ective Algebra (Clayton, 1979), Upside Down A Book Co. Yarra Glen, Victoria, 1981, pp. 226–231.Google Scholar
[36] , Arborescent structures. II. Interpretability in the theory of trees, Transactions of the American Mathematical Society, vol. 266 (1981), no. 2, pp. 629–643.Google Scholar
[37] , Recursive models and the divisibility poset, Notre Dame Journal of Formal Logic, vol. 39 (1998), no. 1, pp. 140–148.Google Scholar
[38] and , Critically indecomposable partially ordered sets, graphs, tournaments and other binary relational structures, Discrete Mathematics, vol. 113 (1993), no. 1-3, pp. 191–205.Google Scholar
[39] , über gewisse satzfunktionen in der arithmetik, Skrifter utgitt av Det Norske videnskaps-akademi i Oslo, vol. 7 (1930), pp. 1–28.Google Scholar
[40] , Recursively saturated nonstandard models of arithmetic, The Journal of Symbolic Logic, vol. 46 (1981), no. 2, pp. 259–286.Google Scholar
[41] , Lectures on nonstandard models of arithmetic, Logic Colloquium '82 (Florence, 1982), Studies in Logic and the Foundations of Mathematics, vol. 112, North-Holland, Amsterdam, 1984, pp. 1–70.Google Scholar
[42] , Abstract of “Non-Archimedean models for arithmetic”, Notices of the American Mathematical Society, vol. 6 (1959), p. 270.Google Scholar
- 1
- Cited by