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-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-28T23:33:41.272Z Has data issue: false hasContentIssue false

Introduction

Published online by Cambridge University Press:  05 December 2013

Edited by
Noam Greenberg ,
Denis Hirschfeldt ,
Joel David Hamkins  and
Russell Miller
Noam Greenberg
Affiliation:
Victoria University of Wellington
Denis Hirschfeldt
Affiliation:
University of Chicago
Joel David Hamkins
Affiliation:
College of Staten Island
Russell Miller
Affiliation:
Queens College, City University of New York
Get access

Summary

Chang and Keisler [8] famously defined model theory as the sum of logic and universal algebra. In the same spirit, one might describe computable model theory to be the investigation of the constraints on information content imposed by algebraic structure. The analogue of the interplay between syntactical objects and the algebraic structure they deine is the connection between deinability and complexity. One asks: How complicated are the constructions of model theory and algebra? What kind of information can be coded in structures like groups, ields, graphs, and orders? What mathematical distinctions are unearthed when “boldface” notions such as isomorphism are replaced by their “lightface” analogues such as, say, computable isomorphism?

A special case of the following definition was first rigorously made by Fröhlich and Shepherdson [11], following work of Hermann [17] and van der Waerden [40], which itself built on the constructive tradition of 19th century algebra. It was further developed by Rabin [32, 33] and Mal'cev [27].

Definition. Let ℒ be a computable signature (language), and let ℳ be an ℒ-structure whose universe is the set of natural numbers. The degree of ℳ is the Turing degree of the atomic (equivalently, quantifier-free) diagram of ℳ.

A structure is computable if its degree is 0, the Turing degree of computable sets. Equivalently, a structure ℳ is computable if, uniformly in the symbols of ℒ, the interpretations in ℳ of the constant symbols, function symbols, and relation symbols of ℒ are computable.

Type
Chapter
Information
Effective Mathematics of the Uncountable , pp. 1 - 13
Publisher: Cambridge University Press
Print publication year: 2013

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] U., Andrews, A new spectrum of recursive models using an amalgamation construction. J. Symbolic Logic 76 (2011), 883–896.Google Scholar
[2] C.J., Ash, Isomorphic recursive structures. In Handbook of Recursive Mathematics, vol. 1 (Yu.L., Ershov, S.S., Goncharov, A., Nerode, and J.B., Remmel, eds., V. W., Marek, assoc. ed.), Elsevier, 1998, 167–181.
[3] C.J., Ash and J.F., Knight, Computable Structures and the Hyperarithmetical Hierarchy, Amsterdam, 2000.
[4] C., Ash, J., Knight, M., Manasse, and T., Slaman, Generic copies of countable structures, Ann. Pure Appl. Logic 42 (1989), 195–205.Google Scholar
[5] J.T., Baldwin and A.H., Lachlan, On strongly minimal sets, J. Symbolic Logic 36 (1971), 79–96.Google Scholar
[6] L., Blum, M., Shub, and S., Smale, On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines, Bull. Amer. Math. Soc. (N.S.) 21 (1989), 1–16.Google Scholar
[7] V., Brattka, P., Hertling, and K., Weihrauch, A tutorial on computable analysis. In New Computational Paradigms: Changing Conceptions of What is Computable, (S.B., Cooper, B., Lowe, and A., Sorbi, eds.), Springer, 2008, 425–491.
[8] C.C., Chang and H.J., Keisler, Model Theory, third edition; Studies in Logic and the Foundations of Mathematics 73, North-Holland, 1990.
[9] J., Chisholm, Effective model theory vs. recursive model theory, J. Symbolic Logic 55 (1990), 1168–1191.Google Scholar
[10] Yu.L., Ershov, Σ-definability in admissible sets, Sov. Math. Dokl. 32 (1985), 767–770.Google Scholar
[11] A., Fröhlich and J.C., Shepherdson, Effective procedures in field theory, Philos. Trans. Roy. Soc. London. Ser. A. 248 (1956), 407–132.Google Scholar
[12] S.S., Goncharov and V.D., Dzgoev, Autostability of models, Algebra Logic 19 (1980), 45–58 (Russian), 28–37 (English translation).Google Scholar
[13] S.S., Goncharov and A.T., Nurtazin, Constructive models of complete decidable theories, Algebra Logic 12 (1973), 125–142, 243 (Russian), 67–77 (English translation).Google Scholar
[14] J.D., Hamkins, R., Miller, D., Seabold, and S., Warner, Infinite time computable model theory. In New Computational Paradigms: Changing Conceptions of What is Computable (S.B., Cooper, B., Lowe, and A., Sorbi, eds.), Springer, 2008, 521–557.
[15] V.S., Harizanov, Pure computable model theory. In Handbook of Recursive Mathematics, vol. 1 (Yu.L., Ershov, S.S., Goncharov, A., Nerode, and J.B., Remmel, eds., V.W., Marek, assoc. ed.), Elsevier, 1998, 3–114.
[16] L., Harrington, Recursively presentable prime models, J. Symbolic Logic 39 (1974), 305–309.Google Scholar
[17] G., Hermann, Die Frage der endlich vielen Schritte in der Theorie der Polynomideale, Math. Ann. 95 (1926), 736–788.Google Scholar
[18] D.R., Hirschfeldt, R.A., Shore, and T.A., Slaman, The atomic model theorem and type omitting, Trans. Amer. Math. Soc. 361 (2009), 5805–5837.Google Scholar
[19] G., Hjorth and A., Nies, Borel structures and Borel theories. J. Symbolic Logic 76 (2011), 461–476.Google Scholar
[20] C.G., Jockusch Jr. and R.I., Soare, Degrees of orderings not isomorphic to recursive linear orderings, Ann. Pure Appl. Logic 52 (1991), 39–64.Google Scholar
[21] B., Khoussainov and M., Minnes, Three lectures on automatic structures. In Logic Colloquium '07 (F., Delon, U., Kohlenbach, P., Maddy, and F., Stephan, eds.), Lecture Notes in Logic 35, Cambridge University Press, and Association for Symbolic Logic, 2010, 132–176.
[22] J.F., Knight, Degrees coded in jumps of orderings, J. Symbolic Logic 51 (1986), 1034–1042.Google Scholar
[23] J.F., Knight and M., Stob, Computable Boolean algebras, J. Symbolic Logic 65 (2000), 1605–1623.Google Scholar
[24] K., Ko, Computational Complexity of Real Functions, BirkhauserBoston, 1991.
[25] K.Zh., Kudaibergenov, A theory with two strongly constructible models, Algebra Logic 18 (1979), 176–185, 253 (Russian), 111–117 (English translation).Google Scholar
[26] K., Lange and R.I., Soare, Computability of homogeneous models, Notre Dame J. Formal Logic 48 (2007), 143–170.Google Scholar
[27] A.I., Mal'cev, Constructive algebras I, Uspekhi Mat. Nauk 16 (1961), 3–60.Google Scholar
[28] T., Millar, Omitting types, type spectrums, and decidability, J. Symbolic Logic 48 (1983), 171–181.Google Scholar
[29] T.S., Millar, A complete, decidable theory with two decidable models, J. Symbolic Logic 44 (1979), 307–312.Google Scholar
[30] R., Miller, The -spectrum of a linear order, J. Symbolic Logic 66 (2001), 470–486.Google Scholar
[31] M., Pour-El and I., Richards, Computability in Analysis and Physics, Springer-Verlag, 1989.
[32] M.O., Rabin, Computable algebraic systems. In Summer Institute for Symbolic Logic, Cornell University, Institute for Defence Analyses, 1957, 134–138.
[33] M.O., Rabin, Computable algebra, general theory, and theory of computable fields, Trans. Amer. Math. Soc. 95 (1960), 341–360.Google Scholar
[34] J.B., Remmel, Recursively categorical linear orderings, Proc. Amer. Math. Soc. 83 (1981), 387–391.Google Scholar
[35] L.J., Richter, Degrees of Structures, PhD Dissertation, University of Illinois at Urbana-Champaign, 1979.
[36] L.J., Richter, Degrees of structures, J. Symbolic Logic 46 (1981), 723–731.Google Scholar
[37] R.A., Shore, Controlling the dependence degree of a recursively enumerable vector space, J. Symbolic Logic 43 (1978), 13–22.Google Scholar
[38] S.G., Simpson, Subsystems of Second Order Arithmetic, second edition, Perspectives in Logic, Cambridge University Press, and Association for Symbolic Logic, 2009.
[39] T.A., Slaman, Relative to any nonrecursive set, Proc. Amer. Math. Soc. 126 (1998), 2117–2122.Google Scholar
[40] B.L., van der Waerden, Eine Bemerkung über die Unzerlegbarkeit von Polynomen, Math. Ann. 102 (1930), 738–739.Google Scholar
[41] S., Wehner, Enumeration, countable structures and Turing degrees, Proc. Amer. Math. Soc. 126 (1998), 2131–2139.Google Scholar
[42] K., Weihrauch, Computable Analysis: An Introduction, Springer, Berlin, 2000.

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] 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
×