Skip to main content Accessibility help
×
Home
Hostname: page-component-59b7f5684b-npccv Total loading time: 0.902 Render date: 2022-10-01T05:19:17.038Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "displayNetworkTab": true, "displayNetworkMapGraph": false, "useSa": true } hasContentIssue true

Reverse Mathematics: The Playground of Logic

Published online by Cambridge University Press:  15 January 2014

Richard A. Shore*
Affiliation:
Department of Mathematics, Cornell University, Ithaca, NY 14853, USA, E-mail: shore@math.cornell.edu

Abstract

This paper is essentially the author's Gödel Lecture at the ASL Logic Colloquium '09 in Sofia extended and supplemented by material from some other papers. After a brief description of traditional reverse mathematics, a computational approach to is presented. There are then discussions of some interactions between reverse mathematics and the major branches of mathematical logic in terms of the techniques they supply as well as theorems for analysis. The emphasis here is on ones that lie outside the usual main systems of reverse mathematics. While retaining the usual base theory and working still within second order arithmetic, theorems are described that range from those far below the usual systems to ones far above.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2010

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

REFERENCES

Afshari, B. and Rathjen, M. [2009], Reverse mathematics and well-ordering principles: A pilot study, Annals of Pure and Applied Logic, vol. 160, pp. 231237.CrossRefGoogle Scholar
Aharoni, R., Magidor, M., and Shore, R. A. [1992], On the strength of Köonig's duality theorem for infinite bipartite graphs, Journal of Combinatorial Theory (B), vol. 54, pp. 257290.CrossRefGoogle Scholar
Blum, L., Cucker, F., Shub, M., and Smale, S. [1998], Complexity and real computation, Springer-Verlag, New York.CrossRefGoogle Scholar
Cholak, P. A., Jockusch, C. G. Jr., and Slaman, T. A. [2001], On the strength of Ramsey's theorem for pairs, The Journal of Symbolic Logic, vol. 66, pp. 155.CrossRefGoogle Scholar
Chong, C. T. [1984], Techniques of admissible recursion theory, Lecture Notes in Mathematics, vol. 1106, Springer-Verlag, Berlin.CrossRefGoogle Scholar
Chong, C. T., Lempp, S., and Yang, Y. [2010], On the role of the collection principle for -formulas in second-order reverse mathematics, Proceedings of the American Mathematical Society, vol. 13, pp. 10931100.CrossRefGoogle Scholar
Downey, R., Hirschfeldt, D. R., Lempp, S., and Solomon, R. [2001], A set with no infinite low subset in either it or its complement, The Journal of Symbolic Logic, vol. 66, pp. 13711381.CrossRefGoogle Scholar
Dzhafarov, D. and Hirst, J. [2009], The polarized Ramsey theorem, Archive for Mathematical Logic, vol. 48, pp. 141157.CrossRefGoogle Scholar
Fenstad, Jens Erik [1980], General recursion theory: An axiomatic approach, Perspectives in Mathematical Logic, Springer-Verlag, Berlin–New York.CrossRefGoogle Scholar
Friedman, H. [1967], Subsystems of set theory and analysis, Ph.D. thesis, M.I.T..Google Scholar
Friedman, H. [1971], Higher set theory and mathematical practice, Annals of Mathematical Logic, vol. 2, pp. 325357.CrossRefGoogle Scholar
Friedman, H. [1975], Some systems of second order arithmetic and their use, Proceedings of the International Congress of Mathematicians, Vancouver 1974, vol. 1, pp. 235242.Google Scholar
Friedman, H., Robertson, N., and Seymour, P. [1987], The metamathematics of the graph minor theorem, Logic and combinatorics (Simpson, S., editor), Contemporary Mathematics, American Mathematical Society, Providence.Google Scholar
Hajek, P. and Pudlak, P. [1998], Metamathematics of first order arithmetic, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 2nd printing.CrossRefGoogle Scholar
Hirschfeldt, D. R., Lange, K., and Shore, R. A. [2012], The homogeneous model theorem, in preparation.Google Scholar
Hirschfeldt, D. R. and Shore, R. A. [2007], Combinatorial principles weaker than Ramsey's theorem for pairs, The Journal of Symbolic Logic, vol. 72, pp. 171206.CrossRefGoogle Scholar
Hirschfeldt, D. R., Shore, R. A., and Slaman, T. A. [2009], The atomic model theorem, Transactions of the American Mathematical Society, vol. 361, pp. 58055837.CrossRefGoogle Scholar
Jockusch, C. G. Jr. and Soare, R. I. [1972], classes and degrees of theories, Transactions of the American Mathematical Society, vol. 173, pp. 3356.Google Scholar
Jullien, P. [1960], Contribution à l'étude des types d'ordre dispersés , Ph.D. thesis, Marseille.Google Scholar
Keisler, H. J. [2006], Nonstandard arithmetic and reverse mathematics, this Bulletin, vol. 12, pp. 100125.Google Scholar
Keisler, H. J. [2011], Nonstandard arithmetic and recursive comprehension, Annals of Pure and Applied Logic, to appear.Google Scholar
Kirby, L. and Paris, J. [1978], Σn collection schemes in arithmetic, Logic colloquium '77 (Macintyre, A., Pacholski, L., and Paris, J., editors), Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, pp. 199209.Google Scholar
Kohlenbach, U. [2005], Higher order reverse mathematics, Reverse mathematics 2001 (Simpson, S., editor), Lecture Notes in Logic, vol. 21, Association for Symbolic Logic and A. K. Peters, Wellesley, MA, pp. 281295.Google Scholar
Kohlenbach, U. [2008], Applied proof theory: Proof interpretations and their use in mathematics, Springer Monographs in Mathematics, Springer, Berlin.Google Scholar
Lerman, M. [1972], On suborderings of the α-recursively enumerable α-degrees, Annals of Mathematical Logic, vol. 4, pp. 369392.CrossRefGoogle Scholar
Lovasz, L. and Plummer, M. D. [1986], Matching theory, Annals of Discrete Mathematics, vol. 29, North-Holland, Amsterdam.Google Scholar
Marcone, A. and Montalbán, A. [2009], On Fraïssé's conjecture for linear orders of finite Hausdorff rank, Annals of Pure and Applied Logic, vol. 160, pp. 355367.CrossRefGoogle Scholar
Marcone, A. and Montalbán, A. [2011], The Veblen functions for computability theorists, to appear.Google Scholar
Martin, D. A. [1974], Analysis ⊢ -determinacy, circulated handwritten notes dated 04 24, 1974.Google Scholar
Martin, D. A. [1974a], -determinacy, circulated handwritten notes dated 03 20, 1974.Google Scholar
Martin, D. A. [1975], Borel determinacy, Annals of Mathematics, vol. 102, pp. 363371.CrossRefGoogle Scholar
Martin, D. A. [n.d.], Determinacy, circulated drafts, about 578 pp.Google Scholar
MedSalem, M. O. and Tanaka, K. [2007], -determinacy, comprehension and induction, The Journal of Symbolic Logic, vol. 72, pp. 452462.CrossRefGoogle Scholar
MedSalem, M. O. and Tanaka, K. [2008], Weak determinacy and iterations of inductive definitions, Computational prospects of infinity, part II: Presented talks (Chong, C., Feng, Q., Slaman, T. A., Woodin, W. H., and Yang, Y., editors), Lecture Note Series, World Scientific, Singapore, pp. 333353.CrossRefGoogle Scholar
Moldestad, J. [1977], Computations in higher types, Lecture Notes in Mathematics, vol. 574, Springer-Verlag, Berlin.CrossRefGoogle Scholar
Montalbán, A. [2006], Indecomposable linear orderings and hyperarithmetic analysis, The Journal of Mathematical Logic, vol. 6, pp. 89120.CrossRefGoogle Scholar
Montalbán, A. and Shore, R. A. [2011], The limits of determinacy in second order arithmetic, to appear.Google Scholar
Mummert, C. and Simpson, S. G. [2005], Reverse mathematics and comprehension, this Bulletin, vol. 11, pp. 526533.Google Scholar
Neeman, I. [2011], The strength of Jullien's indecomposability theorem, to appear.Google Scholar
Neeman, I. [2012], Necessary uses of induction in a reversal, to appear.Google Scholar
Nemoto, T., MedSalem, M. O., and Tanaka, K. [2007], Infinite games in the Cantor space and subsystems of second order arithmetic, MLQ. Mathematical Logic Quarterly, vol. 53, pp. 226236.CrossRefGoogle Scholar
Odifreddi, P. [1989], Classical recursion theory I, North-Holland, Elsevier, Amsterdam.Google Scholar
Odifreddi, P. [1999], Classical recursion theory II, North-Holland, Elsevier, Amsterdam.Google Scholar
Rathjen, M. and Weiermann, A. [2011], Reverse mathematics and well-ordering principles, to appear.CrossRefGoogle Scholar
Rogers, H. Jr. [1967], Theory of recursive functions and effective computability, McGraw-Hill, New York.Google Scholar
Rosenstein, J. [1982], Linear orderings, Academic Press, New York–London.Google Scholar
Sacks, G. E. [1990], Higher recursion theory, Perspectives in Mathematical Logic, Springer-Verlag, Berlin.CrossRefGoogle Scholar
Schwichtenberg, H. and Wainer, S. [2011], Proofs and computations, Perspectives in Logic, Association for Symbolic Logic and Cambridge University Press, New York, to appear.CrossRefGoogle Scholar
Shore, R. A. [1972], Priority arguments in α-recursion theory, Ph.D. thesis, M.I.T..Google Scholar
Shore, R. A. [1993], On the strength of Fraïssé's conjecture, Logical methods (Crossley, J. N. C., Remmel, J., Shore, R. A., and Sweedler, M., editors), Birkhäuser, Boston, pp. 782813.CrossRefGoogle Scholar
Shore, R. A. [2007], Direct and local definitions of the Turing jump, Journal of Mathematical Logic, vol. 7, pp. 229262.CrossRefGoogle Scholar
Shore, R. A. [2011], Reverse mathematics, countable and uncountable: a computational approach, to appear.Google Scholar
Shore, R. A. and Slaman, T. A. [1999], Defining the Turing jump, Mathematical Research Letters, vol. 6, pp. 711722.CrossRefGoogle Scholar
Simpson, S. G. [1985], Nonprovability of certain combinatorial properties of finite trees, Harvey Friedman's research on the foundations of mathematics (Harrington, L. A., Morley, M. D., Scedrov, A., and Simpson, S. G., editors), Studies in Logic and the Foundations of Mathematics, vol. 117, North-Holland, Amsterdam, pp. 87118.CrossRefGoogle Scholar
Simpson, S. G. [1994], On the strength of Köonig's duality theorem for countable bipartite graphs, The Journal of Symbolic Logic, vol. 59, pp. 113123.CrossRefGoogle Scholar
Simpson, S. G. [2009], Subsystems of second order arithmetic, 2nd ed., Perspectives in Logic, Association for Symbolic Logic and Cambridge University Press, New York.CrossRefGoogle Scholar
Slaman, T. A. [1991], Degree structures, Proceedings of the International Congress of Mathematicians, Kyoto 1990, Springer-Verlag, Tokyo, pp. 303316.Google Scholar
Slaman, T. A. [2008], Global properties of the Turing degrees and the Turing jump, Computational prospects of infinity. Part I: Tutorials (Chong, C. T., Feng, Q, Slaman, T. A., Woodin, W. H., and Yang, Y., editors), Lecture Notes Series, vol. 14, World Scientific, pp. 83101.Google Scholar
Soare, R. I. [1987], Recursively enumerable sets and degrees, Springer-Verlag, Berlin.CrossRefGoogle Scholar
Steel, J. [1976], Determinateness and subsystems of analysis, Ph.D. thesis, University of California, Berkeley.Google Scholar
Steel, J. [1978], Forcing with tagged trees, Annals of Mathematical Logic, vol. 15, pp. 5574.CrossRefGoogle Scholar
Tanaka, K. [1990], Weak axioms of determinacy and subsystems of analysis I: games, Zeitschrift für mathematische Logik und Grundlagen der Mathmatik, vol. 36, pp. 481491.CrossRefGoogle Scholar
Tanaka, K. [1991], Weak axioms of determinacy and subsystems of analysis II ( games), Annals of Pure and Applied Logic, vol. 52, pp. 181193.CrossRefGoogle Scholar
Welch, P. [2009], Weak systems of determinacy and arithmetical quasi-inductive definitions, preprint.Google Scholar
Yokoyama, K. [2007], Non-standard analysis in ACA0 and Riemann mapping theorem, MLQ. Mathematical Logic Quarterly, vol. 53, pp. 132146.CrossRefGoogle Scholar
Yokoyama, K. [2009], Standard and non-standard analysis in second order arithmetic, D.S. thesis, Tohoku University, available as Tohoku Mathematical Publications vol. 34 (2009).Google Scholar
10
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@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.

Reverse Mathematics: The Playground of Logic
Available formats
×

Save article to Dropbox

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

Reverse Mathematics: The Playground of Logic
Available formats
×

Save article to Google Drive

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

Reverse Mathematics: The Playground of Logic
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *