Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-745bb68f8f-b95js Total loading time: 0 Render date: 2025-01-28T21:54:33.529Z Has data issue: false hasContentIssue false

Tennenbaum's theorem for models of arithmetic

Published online by Cambridge University Press:  07 October 2011

By
Richard Kaye
Edited by
Juliette Kennedy  and
Roman Kossak
Juliette Kennedy
Affiliation:
University of Helsinki
Roman Kossak
Affiliation:
City University of New York
Get access

Summary

Abstract. This paper discusses Tennenbaum's Theorem in its original context of models of arithmetic, which states that there are no recursive nonstandard models of Peano Arithmetic. We focus on three separate areas: the historical background to the theorem; an understanding of the theorem and its relationship with the Gödel–Rosser Theorem; and extensions of Tennenbaum's theorem to diophantine problems in models of arthmetic, especially problems concerning which diophantine equations have roots in some model of a given theory of arithmetic.

§ 1.Some historical background. The theorem known as “Tennenbaum's Theorem” was given by Stanley Tennenbaum in a paper at the April meeting in Monterey, California, 1959, and published as a one-page abstract in the Notices of the American Mathematical Society [28]. It is easily stated as saying that there is no nonstandard recursive model of Peano Arithmetic, and is an attractive and rightly often-quoted result.

This paper celebrates Tennenbaum's Theorem; we state the result fully and give a proof of it andother related results later. This introduction is in the main historical. The goals of the latter parts of this paper are: to set out the connections between Tennenbaum's Theorem for models of arithmetic and the Gödel–Rosser Theorem and recursively inseparable sets; and to investigate stronger versions of Tennenbaum's Theorem and their relationship to some diophantine problems in systems of arithmetic.

Tennenbaum's theorem was discovered in a period of foundational studies, associated particularly with Mostowski, where it still seemed conceivable that useful independence results for arithmetic could be achieved by a “handson” approach to building nonstandard models of arithmetic.

Type
Chapter
Information
Set Theory, Arithmetic, and Foundations of Mathematics
Theorems, Philosophies
 , pp. 66 - 79
Publisher: Cambridge University Press
Print publication year: 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Alessandro, Berarducci and Margarita, Otero, A recursive nonstandard model of normal open induction, The Journal of Symbolic Logic, vol. 61 (1996), no. 4, pp. 1228–1241.Google Scholar
[2] Patrick, Cegielski, Kenneth, McAloon, and George, Wilmers, Modèles recursivement saturés de l'addition et de la multiplication des entiers naturels, Logic Colloquium '80 (Prague, 1980), Studies in Logic and the Foundations of Mathematics, vol. 108, North-Holland, Amsterdam, 1982, pp. 57–68.Google Scholar
[3] Solomon, Feferman, Arithmetization of metamathematics in a general setting, Polska Akademia Nauk. Fundamenta Mathematicae, vol. 49 (1960/1961), pp. 35–92.Google Scholar
[4] Haim, Gaifman and Constantine, Dimitracopoulos, Fragments of Peano's arithmetic and the MRDP theorem, Logic and Algorithmic (Zurich, 1980), L'Enseignement mathématique. Monograph, vol. 30, Université de Genève, Geneva, 1982, pp. 187–206.Google Scholar
[5] Robin O., Gandy, Note on a paper of Kemeny's, Mathematische Annalen, vol. 136 (1958), p. 466.Google Scholar
[6] Richard, Kaye, Diophantine induction, Annals of Pure and Applied Logic, vol. 46 (1990), no. 1, pp. 1–40.Google Scholar
[7] Richard, Kaye, Models of Peano Arithmetic, Oxford Logic Guides, vol. 15, The Clarendon Press Oxford University Press, New York, 1991, Oxford Science Publications.Google Scholar
[8] Richard, Kaye, Hilbert's tenth problem for weak theories of arithmetic, Annals of Pure and Applied Logic, vol. 61 (1993), no. 1-2, pp. 63–73, Provability, Interpretability and Arithmetic Symposium (Utrecht, 1991).Google Scholar
[9] Richard, Kaye, 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
[10] John G., Kemeny, Models of logical systems, The Journal of Symbolic Logic, vol. 13 (1948), pp. 16–30.Google Scholar
[11] John G., Kemeny, Undecidable problems of elementary number theory, Mathematische Annalen, vol. 135 (1958), pp. 160–169.Google Scholar
[12] Georg, Kreisel, Note on arithmetic models for consistent formulae of the predicate calculus. II, Actes du XIème Congrès International de Philosophie, Bruxelles, 20–26 Août 1953, vol. XIV, North-Holland, Amsterdam, 1953, pp. 39–49.Google Scholar
[13] Yuri V., Matijasevič, The Diophantineness of enumerable sets, Doklady Akademii Nauk SSSR, vol. 191 (1970), pp. 279–282.Google Scholar
[14] Andrzej, Mostowski, On a system of axioms which has no recursively enumerable arith- metic model, Polska Akademia Nauk. Fundamenta Mathematicae, vol. 40 (1953), pp. 56–61.Google Scholar
[15] Andrzej, Mostowski, A formula with no recursively enumerable model, Polska Akademia Nauk. Fundamenta Mathematicae, vol. 42 (1955), pp. 125–140.Google Scholar
[16] Andrzej, Mostowski, The present state of investigations on the foundations of mathematics, Rozprawy Matematyczne, vol. 9 (1955), p. 48, in collaboration with A., Grzegorczyk, S., Jaśkowski, J., Łoś, S., Mazur, H., Rasiowa, and R., Sikorski.Google Scholar
[17] Andrzej, Mostowski, On recursive models of formalised arithmetic, Bulletin de l'Académie Polonaise des Sciences. Série III, vol. 5 (1957), pp. 705–710, LXII.Google Scholar
[18] Rohit, Parikh, Existence and feasibility in arithmetic, The Journal of Symbolic Logic, vol. 36 (1971), pp. 494–508.Google Scholar
[19] Jeff B., Paris, On the structure of models of bounded E1-induction, Československá Akademie Věd. Časopis Pro Pěstování Matematiky, vol. 109 (1984), no. 4, pp. 372–379.Google Scholar
[20] Hilary, Putnam, Arithmetic models for consistent formulae of quantification theory, The Journal of Symbolic Logic, vol. 22 (1957), pp. 110–111.Google Scholar
[21] J. Barkley, Rosser and Hao, Wang, Non-standard models for formal logics, The Journal of Symbolic Logic, vol. 15 (1950), pp. 113–129.Google Scholar
[22] Czesław, Ryll-Nardzewski, The role of the axiom of induction in elementary arithmetic, Polska Akademia Nauk. Fundamenta Mathematicae, vol. 39 (1952), pp. 239–263 (1953).Google Scholar
[23] Dana, Scott, On constructing models for arithmetic, Infinitistic Methods (Proc. Sympos. Foundations of Math., Warsaw, 1959), Pergamon, Oxford, 1961, pp. 235–255.Google Scholar
[24] Dana, Scott, Algebras of sets binumerable in complete extensions of arithmetic, Proc. Sympos. Pure Mathematics, Vol. V, American Mathematical Society, Providence, Rhode Island, 1962, pp. 117–121.Google Scholar
[25] John C., Shepherdson, A non-standard model for a free variable fragment of number theory, Bulletin de l'Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques, vol. 12 (1964), pp. 79–86.Google Scholar
[26] Thoralf A., Skolem, Über die Nichtcharakterisierbarkeit der Zalenreihe mittels endlich oder abzählbar unendlich vielen Assagen mit asschliesslich Zahlvariablen, Polska Akademia Nauk. Fundamenta Mathematicae, vol. 23 (1934), pp. 150–161, English version published in 1955 [27].Google Scholar
[27] Thoralf A., Skolem, Peano's axioms and models of arithmetic, Mathematical Interpretation of Formal Systems, North-Holland, Amsterdam, 1955, pp. 1–14.Google Scholar
[28] Stanley, Tennenbaum, Non-archimedian models for arithmetic, Notices of the American Mathematical Society, vol. 270 (1959), p. 270.Google Scholar
[29] George, Wilmers, Bounded existential induction, The Journal of Symbolic Logic, vol. 50 (1985), no. 1, pp. 72–90.Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×