Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-25T02:28:16.444Z Has data issue: false hasContentIssue false
  • English
  • Français

NONSTANDARD MODELS IN RECURSION THEORY AND REVERSE MATHEMATICS

Published online by Cambridge University Press:  26 June 2014

C. T. CHONG ,
WEI LI  and
YUE YANG
C. T. CHONG
Affiliation:
DEPARTMENT OF MATHEMATICS NATIONAL UNIVERSITY OF SINGAPORE 10 LOWER KENT RIDGE ROAD SINGAPORE 119076, SINGAPORE
WEI LI*
Affiliation:
DEPARTMENT OF MATHEMATICS NATIONAL UNIVERSITY OF SINGAPORE 10 LOWER KENT RIDGE ROAD SINGAPORE 119076, SINGAPORE
YUE YANG
Affiliation:
DEPARTMENT OF MATHEMATICS NATIONAL UNIVERSITY OF SINGAPORE 10 LOWER KENT RIDGE ROAD SINGAPORE 119076, SINGAPORE
*
*Current address of Wei Li: KURT GÖDEL RESEARCH CENTER FOR MATHEMATICAL LOGIC UNIVERSITY OF VIENNA, AUSTRIA E-mail:[email protected]E-mail:[email protected]E-mail:[email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We give a survey of the study of nonstandard models in recursion theory and reverse mathematics. We discuss the key notions and techniques in effective computability in nonstandard models, and their applications to problems concerning combinatorial principles in subsystems of second order arithmetic. Particular attention is given to principles related to Ramsey’s Theorem for Pairs.

Keywords

Nonstandard modelsfragments of Peano arithmeticreverse mathematicsreverse recursion theorycombinatorial principlesRamsey’s Theorem for Pairs
Type
Articles
Information
Bulletin of Symbolic Logic , Volume 20 , Issue 2 , June 2014 , pp. 170 - 200
Copyright
Copyright © The Association for Symbolic Logic 2014 

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

Klaus, Ambos-Spies, Jockusch, Carl G., Shore, Richard A., and Soare, Robert I., An algebraic decomposition of the recursively enumerable degrees and the coincidence of several degree classes with the promptly simple class. Transactions of the American Mathematical Society, vol. 281 (1984), no. 1, pp. 109–128.Google Scholar
Chang, C. C. and Jerome Keisler, H., Model Theory, Studies in Logic and the Foundations of Mathematics, vol. 73, North–Holland Publishing Company, Amsterdam, 1973.Google Scholar
Cholak, Peter, Jockusch, Carl G., and Slaman, Theodore A., On the strength of Ramsey’s theorem for pairs. Journal of Symbolic Logic, vol. 66 (2001), no. 1, pp. 1–55.Google Scholar
Cholak, Peter, Jockusch, Carl G., and Slaman, Theodore A., Corrigendum to: “On the strength of Ramsey’s theorem for pairs”. Journal of Symbolic Logic, vol. 74 (2009), no. 4, pp. 1438–1439.Google Scholar
Chong, C. T., Techniques of Admissible Recursion Theory, Lecture Notes in Mathematics, vol. 1106, Springer–Verlag, Berlin, Heidelberg, 1984.Google Scholar
Chong, C. T., Qian, Lei, Slaman, Theodore A., and Yang, Yue, ${\rm{\Sigma }}_2 $induction and infinite injury priority arguments, Part III: Prompt sets, minimal pair and Shoenfield’s conjecture. Israel Journal of Mathematics, vol. 121 (2001), no. 1, pp. 1–28.Google Scholar
Chong, C. T., Lempp, Steffen, and Yang, Yue, On the role of the collection principle for ${\rm{\Sigma }}_2^0 $-formulas in second-order reverse mathematics. Proceedings of the American Mathematical Society, vol. 138 (2010), no. 3, pp. 1093–1100.CrossRefGoogle Scholar
Chong, C. T. and Mourad, K. J., ${\rm{\Sigma }}_n $definable sets without ${\rm{\Sigma }}_n $induction. Transactions of the American Mathematical Society, vol. 334 (1992), no. 1, pp. 349–363.Google Scholar
Chong, C. T. and Mourad, K. J., The degree of a ${\rm{\Sigma }}_n $cut. Annals of Pure and Applied Logic, vol. 48 (1980), no. 3, pp. 227–235.Google Scholar
Chong, C. T., Shore, Richard A., and Yang, Yue, Interpreting arithmetic in the r.e. degrees under Σ4induction. Reverse Mathematics 2001 (Stephen G. Simpson, editor), Association for Symbolic Logic, 2005, pp. 120–146.CrossRefGoogle Scholar
Chong, C. T., Slaman, Theodore A., and Yang, Yue, ${\rm{\Pi }}_1^1 $-conservation of combinatorial principles weaker than Ramsey’s theorem for pairs. Advances in Mathematics, vol. 230 (2012), no. 3, pp. 1060–1077.CrossRefGoogle Scholar
Chong, C. T., Slaman, Theodore A., and Yang, Yue, The metamathematics of stable Ramsey’s theorem for pairs. Journal of the American Mathematical Society, vol. 27 (2014), no. 3, pp. 863–892.Google Scholar
Chong, C. T., Slaman, Theodore A., and Yang, Yue, The inductive strength of Ramsey’s theorem for pairs. in preparation.Google Scholar
Chong, C. T. and Yang, Yue, Σ2induction and infinite injury priority arguments, part II: Tame Σ2coding and the jump operator. Annals of Pure and Applied Logic, vol. 87 (1997), no. 2, pp. 103–116.Google Scholar
Chong, C. T. and Yang, Yue, Recursion theory in weak fragments of Peano arithmetic: A study of cuts. Proceedings of the Sixth Asian Logic Conference (Beijing, China) (Chong, C. T., Feng, Q., Ding, D., Huang, Q., and Yasugi, M., editors), World Scientific, 1998, pp. 47–65.Google Scholar
Chong, C. T. and Yang, Yue, Σ2induction and infinite injury priority arguments, part I: Maximal sets and the jump operator. Journal of Symbolic Logic, vol. 63 (1998), no. 3, pp. 797–814.Google Scholar
Chong, C. T. and Yang, Yue, The jump of a Σn-cut. Journal of the London Mathematical Society, second series, vol. 75 (2007), no. 3, pp. 690–704.Google Scholar
Chubb, Jennifer, Hirst, Jeffry L., and McNicholl, Timothy H., Reverse mathematics, computability, and partitions of trees. Journal of Symbolic Logic, vol. 74 (2009), no. 1, pp. 201–215.Google Scholar
Barry Cooper, S., Degrees of Unsolvability, Ph.D. thesis, Leicester University, 1971.Google Scholar
Corduan, Jared, Groszek, Marcia J., and Mileti, Joseph R., Reverse mathematics and Ramsey’s property for trees. Journal of Symbolic Logic, vol. 75 (2010), no. 3, pp. 945–954.Google Scholar
Conidis, Chris J. and Slaman, Theodore A., Random reals, the rainbow Ramsey theorem, and arithmetic conservation. Journal of Symbolic Logic, vol. 78 (2013), no. 1, pp. 195–206. Abstract.CrossRefGoogle Scholar
Csima, Barbara F. and Mileti, Joseph R., The strength of the rainbow Ramsey theorem. Journal of Symbolic Logic, vol. 74 (2009), no. 4, pp. 1310–1324.Google Scholar
Downey, Rod, Hirschfeldt, Denis R., Lempp, Steffen, and Solomon, Reed, A ${\rm{\Delta }}_2^0 $set with no infinite low subset in either it or its complement. Journal of Symbolic Logic, vol. 66 (2001), no. 3, pp. 1371–1381.Google Scholar
Friedberg, Richard M., Three theorems on recursive enumeration. I. Decomposition. II. Maximal set. III. Enumeration without duplication. Journal of Symbolic Logic, vol. 23 (1958), no. 3, pp. 309–316.Google Scholar
Groszek, Marcia J., Mytilinaios, Michael E., and Slaman, Theodore A., The Sacks density theorem and Σ2 -bounding. Journal of Symbolic Logic, vol. 61 (1996), no. 2, pp. 450–467.Google Scholar
Groszek, Marcia J. and Slaman, Theodore A., On Turing reducibility. 1994, preprint.Google Scholar
Hirschfeldt, Denis and Shore, Richard A., Combinatorial principles weaker than Ramsey’s theorem for pairs. Journal of Symbolic Logic, vol. 72 (2007), no. 1, pp. 171–206.Google Scholar
Hirst, Jeffry L., Combinatorics in Subsystems of Second Order Arithmetic, Ph.D. thesis, Pennsylvania State University, 1987.Google Scholar
Jensen, Ronald B., The fine structure of the constructible hierarchy. Annals of Mathematical Logic, vol. 4 (1972), pp. 229–308.Google Scholar
Jockusch, Carl G., Ramsey’s theorem and recession theory. Journal of Symbolic Logic, vol. 37 (1972), no. 2, pp. 268–280.Google Scholar
Jockusch, Carl G. Jr. and Stephan, Frank, A cohesive set which is not high. Mathematical Logic Quarterly, vol. 39 (1993), pp. 515–530.CrossRefGoogle Scholar
Kaye, Richard, Models of Peano arithmetic, Oxford Logic Guides, vol. 15, the Clarendon Press Oxford University Press, 1991.Google Scholar
Kirby, Laurie A. and Paris, Jeff B., Σn-collection schemas in arithmetic. Logic Colloquium ’77 (J. Barwise, D. Kaplan, H. J. Keisler, P. Suppes, and A. S. Troelstra, editors), North–Holland Publishing Company, Amsterdam, 1978, pp. 199–209.CrossRefGoogle Scholar
Kreisel, Georg, Some reasons for generalizing recursion theory. Logic Colloquium ’69 (Gandy, R. O. and Yates, C. E. M., editors), North–Holland Publishing Company, Amsterdam, 1971, pp. 139–198.Google Scholar
Kummer, Martin, An easy priority-free proof of a theorem of Friedberg. Theoretical Computer Science, vol. 74 (1990), no. 2, pp. 249–251.Google Scholar
Lachlan, Alistair H., A recursively enumerable degree which will not split over all lesser ones. Annals of Mathematical Logic, vol. 9 (1975), no. 4, pp. 307–365.Google Scholar
Lachlan, Alistair H., Bounding minimal pairs. Journal of Symbolic Logic, vol. 44 (1979), no. 4, pp. 626–642.Google Scholar
Lerman, Manuel, On suborderings of the α-recursively enumerable α-degrees. Annals of Mathematical Logic, vol. 4 (1972), pp. 369–392.CrossRefGoogle Scholar
Lerman, Manuel and Simpson, Stephen G., Maximal sets in α-recursion theory. Israel Journal of Mathematics, vol. 4 (1973), pp. 236–247.Google Scholar
Lerman, Manuel, Solomon, Reed, and Towsner, Henry, Separating principles below Ramsey’s Theorem for Pairs. Journal of Mathematical Logic, vol. 13 (2013), no. 2, 1350007, 44 pp.Google Scholar
Li, Wei, Friedberg numbering in fragments of Peano arithmetic and α-recursion theory. Journal of Symbolic Logic, vol. 78 (2013), no. 4, pp. 1135–1163.Google Scholar
Li, Wei, ${\rm{\Delta }}_2 $degrees without Σ1induction. Israel Journal of Mathematics, to appear.Google Scholar
Liu, Jiayi, $RT_2^2 $does not prove WKL 0. Journal of Symbolic Logic, vol. 77 (2012), no. 2,pp. 609–620.Google Scholar
Maass, Wolfgang A., Recursively enumerable generic sets. Journal of Symbolic Logic, vol. 47 (1982), no. 4, pp. 809–823.Google Scholar
McAloon, Kenneth, Completeness theorems, incompleteness theorems and models of arithmetic. Transactions of the American Mathematical Society, vol. 239 (1978), pp. 253–277.Google Scholar
Mourad, K. J., Recursion Theoretic Statements Equivalent to Induction Axioms for Arithmetic, Ph.D. thesis, University of Chicago, 1988.Google Scholar
Mytilinaios, Michael E., Finite injury andΣ1 -induction. Journal of Symbolic Logic, vol. 54 (1989), no. 1, pp. 38–49.Google Scholar
Mytilinaios, Michael E., and Slaman, Theodore A., collection and the infinite injury priority method. Journal of Symbolic Logic, vol. 53 (1988), no. 1, pp. 212–221.Google Scholar
Robinson, Abraham, Nonstandard Analysis (revised edition), Princeton Landmarks in Mathematics, Princeton University Press, 1996.Google Scholar
Sacks, Gerald E., Higher Recursion Theory, Perspectives in Logic, vol. 2, Springer–Verlag, Berlin, Heidelberg, 1990.Google Scholar
Sacks, Gerald E. and Simpson, Stephen G., The α-finite injury method. Annals of Pure and Applied Logic, vol. 4 (1972), pp. 343–367.Google Scholar
Seetapun, David and Slaman, Theodore A., On the strength of Ramsey’s theorem. Notre Dame Journal of Formal Logic, vol. 36 (1995), no. 4, pp. 570–582.Google Scholar
Shore, Richard A., Splitting an α-recursively enumerable set. Transactions of the American Mathematical Society, vol. 204 (1975), pp. 65–78.Google Scholar
Shore, Richard A., On the jump of an α-recursively enumerable set. Transactions of the American Mathematical Society, vol. 217 (1976), pp. 351–363.Google Scholar
Shore, Richard A. and Slaman, Theodore A., Working below a high recursively enumerable degree. Journal of Symbolic Logic, vol. 58 (1993), no. 3, pp. 824–859.Google Scholar
Simpson, Stephen G., Subsystems of Second Order Arithmetic, Perspectives in Logic, Springer–Verlag, Berlin, 1999.Google Scholar
Skolem, Th., Über die Nicht-charakterisierbarkeit der Zahlenreihe mittels endlich oder abzählbar unendlich vieler Aussagen mit ausschliesslich Zahlenvariablen. Fundamenta Mathematicae, vol. 23 (1934), no. 1, pp. 150–161.Google Scholar
Skolem, Th., Peano axioms and models of arithmetic. Mathematical Interpretations of Formal Systems (Skolem, Th., Hasenjaeger, G., Kreisel, G., Robinson, A., Wang, Hao, Henkin, L., and Łoś, J., editors), North–Holland Publishing Company, Amsterdam, 1955, pp. 1–14.Google Scholar
Slaman, Theodore A., The density of infima in the recursively enumerable degrees. Annals of Pure and Applied Logic, vol. 52 (1991), no. 1–2, pp. 1–25.CrossRefGoogle Scholar
Slaman, Theodore A., Σ1-bounding and ${\rm{\Delta }}_n $ -induction. Proceedings of the American Mathematical Society, vol. 132 (2004), no. 8, pp. 2449–2456.Google Scholar
Slaman, Theodore A. and Hugh Woodin, W., Σ1-collection and the finite injury method. Mathematical Logic and Applications (Juichi Shinoda, Theodore A. Slaman, and T. Tugué, editors), Springer–Verlag, Heidelberg, 1989, pp. 178–188.Google Scholar
Yang, Yue, Σ2induction and cuppable degrees. Recursion Theory and Complexity: Proceedings of the Kazan ’97 Workshop (Kazan, Russia) (Marat M. Arslanov and Steffen Lempp, editors), de Gruyter, Berlin, 1999, pp. 215–228.Google Scholar