Ranked structures and arithmetic transfinite recursion
Authors:
Noam Greenberg and Antonio Montalbán
Journal:
Trans. Amer. Math. Soc. 360 (2008), 1265-1307
MSC (2000):
Primary 03F35, 03D45; Secondary 03C57, 03B30
DOI:
https://doi.org/10.1090/S0002-9947-07-04285-7
Published electronically:
October 23, 2007
MathSciNet review:
2357696
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: $\mathsf {ATR}_0$ is the natural subsystem of second-order arithmetic in which one can develop a decent theory of ordinals. We investigate classes of structures which are in a sense the “well-founded part" of a larger, simpler class, for example, superatomic Boolean algebras (within the class of all Boolean algebras). The other classes we study are: well-founded trees, reduced Abelian $p$-groups, and countable, compact topological spaces. Using computable reductions between these classes, we show that Arithmetic Transfinite Recursion is the natural system for working with them: natural statements (such as comparability of structures in the class) are equivalent to $\mathsf {ATR}_0$. The reductions themselves are also objects of interest.
- C. J. Ash and J. Knight, Computable structures and the hyperarithmetical hierarchy, Studies in Logic and the Foundations of Mathematics, vol. 144, North-Holland Publishing Co., Amsterdam, 2000. MR 1767842
- Ewan J. Barker, Back and forth relations for reduced abelian $p$-groups, Ann. Pure Appl. Logic 75 (1995), no. 3, 223–249. MR 1355134, DOI https://doi.org/10.1016/0168-0072%2894%2900045-5
- Jon Barwise and Paul Eklof, Infinitary properties of abelian torsion groups, Ann. Math. Logic 2 (1970/71), no. 1, 25–68. MR 279180, DOI https://doi.org/10.1016/0003-4843%2870%2990006-9
- Wesley Calvert, The isomorphism problem for classes of computable fields, Arch. Math. Logic 43 (2004), no. 3, 327–336. MR 2052886, DOI https://doi.org/10.1007/s00153-004-0219-1
- U. Kalvert, D. Kammins, Dz. F. Naĭt, and S. Miller, Comparison of classes of finite structures, Algebra Logika 43 (2004), no. 6, 666–701, 759 (Russian, with Russian summary); English transl., Algebra Logic 43 (2004), no. 6, 374–392. MR 2135387, DOI https://doi.org/10.1023/B%3AALLO.0000048827.30718.2c
- Riccardo Camerlo and Su Gao, The completeness of the isomorphism relation for countable Boolean algebras, Trans. Amer. Math. Soc. 353 (2001), no. 2, 491–518. MR 1804507, DOI https://doi.org/10.1090/S0002-9947-00-02659-3
- Peter Cholak, Alberto Marcone, and Reed Solomon, Reverse mathematics and the equivalence of definitions for well and better quasi-orders, J. Symbolic Logic 69 (2004), no. 3, 683–712. MR 2078917, DOI https://doi.org/10.2178/jsl/1096901762
- Harvey M. Friedman and Jeffry L. Hirst, Weak comparability of well orderings and reverse mathematics, Ann. Pure Appl. Logic 47 (1990), no. 1, 11–29. MR 1050559, DOI https://doi.org/10.1016/0168-0072%2890%2990014-S
- Harvey M. Friedman and Jeffry L. Hirst, Reverse mathematics and homeomorphic embeddings, Ann. Pure Appl. Logic 54 (1991), no. 3, 229–253. MR 1133006, DOI https://doi.org/10.1016/0168-0072%2891%2990048-Q
- Harvey Friedman, Metamathematics of comparability, Manuscript dated October 2001.
- ---, Metamathematics of Ulm theory, Manuscript dated November 2001.
- Harvey M. Friedman, Stephen G. Simpson, and Rick L. Smith, Countable algebra and set existence axioms, Ann. Pure Appl. Logic 25 (1983), no. 2, 141–181. MR 725732, DOI https://doi.org/10.1016/0168-0072%2883%2990012-X
- Su Gao, The homeomorphism problem for countable topological spaces, Topology Appl. 139 (2004), no. 1-3, 97–112. MR 2051099, DOI https://doi.org/10.1016/j.topol.2003.09.005
- S. S. Goncharov and Dzh. Naĭt, Computable structure and antistructure theorems, Algebra Logika 41 (2002), no. 6, 639–681, 757 (Russian, with Russian summary); English transl., Algebra Logic 41 (2002), no. 6, 351–373. MR 1967769, DOI https://doi.org/10.1023/A%3A1021758312697
- Jeffry L. Hirst, Reverse mathematics and ordinal exponentiation, Ann. Pure Appl. Logic 66 (1994), no. 1, 1–18. MR 1263322, DOI https://doi.org/10.1016/0168-0072%2894%2990076-0
- Jeffry L. Hirst, Reverse mathematics and rank functions for directed graphs, Arch. Math. Logic 39 (2000), no. 8, 569–579. MR 1797809, DOI https://doi.org/10.1007/s001530050165
- Jeffry L. Hirst, A survey of the reverse mathematics of ordinal arithmetic, Reverse mathematics 2001, Lect. Notes Log., vol. 21, Assoc. Symbol. Logic, La Jolla, CA, 2005, pp. 222–234. MR 2185437
- Greg Hjorth and Alexander S. Kechris, Analytic equivalence relations and Ulm-type classifications, J. Symbolic Logic 60 (1995), no. 4, 1273–1300. MR 1367210, DOI https://doi.org/10.2307/2275888
- Greg Hjorth and Alexander S. Kechris, Recent developments in the theory of Borel reducibility, Fund. Math. 170 (2001), no. 1-2, 21–52. Dedicated to the memory of Jerzy Łoś. MR 1881047, DOI https://doi.org/10.4064/fm170-1-2
- Irving Kaplansky, Infinite abelian groups, Revised edition, The University of Michigan Press, Ann Arbor, Mich., 1969. MR 0233887
- Sabine Koppelberg, Handbook of Boolean algebras. Vol. 1, North-Holland Publishing Co., Amsterdam, 1989. Edited by J. Donald Monk and Robert Bonnet. MR 991565
- Antonio Montalbán, Beyond the arithmetic, Ph.D. thesis, Cornell University, Ithaca, New York, 2005.
- Antonio Montalbán, Up to equimorphism, hyperarithmetic is recursive, J. Symbolic Logic 70 (2005), no. 2, 360–378. MR 2140035, DOI https://doi.org/10.2178/jsl/1120224717
- James R. Munkres, Topology, second ed., Prentice-Hall Inc., Englewood Cliffs, N.J., 2000.
- Gerald E. Sacks, Higher recursion theory, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1990. MR 1080970
- V. L. Selivanov, The fine hierarchy and definable index sets, Algebra i Logika 30 (1991), no. 6, 705–725, 771 (Russian, with Russian summary); English transl., Algebra and Logic 30 (1991), no. 6, 463–475 (1992). MR 1213731, DOI https://doi.org/10.1007/BF02018741
- John N. Crossley, Jeffrey B. Remmel, Richard A. Shore, and Moss E. Sweedler (eds.), Logical methods, Progress in Computer Science and Applied Logic, vol. 12, Birkhäuser Boston, Inc., Boston, MA, 1993. Papers from the conference in honor of Anil Nerode’s sixtieth birthday held at Cornell University, Ithaca, New York, June 1–3, 1992. MR 1281145
- Richard A. Shore, Invariants, Boolean algebras and ${\rm ACA}^+_0$, Trans. Amer. Math. Soc. 358 (2006), no. 3, 989–1014. MR 2187642, DOI https://doi.org/10.1090/S0002-9947-05-03802-X
- Stephen G. Simpson, Subsystems of second order arithmetic, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1999. MR 1723993
- Clifford Spector, Recursive well-orderings, J. Symbolic Logic 20 (1955), 151–163. MR 74347, DOI https://doi.org/10.2307/2266902
Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 03F35, 03D45, 03C57, 03B30
Retrieve articles in all journals with MSC (2000): 03F35, 03D45, 03C57, 03B30
Additional Information
Noam Greenberg
Affiliation:
Department of Mathematics, Notre Dame University, Notre Dame, Indiana 46556
Address at time of publication:
School of Mathematics, Statistics and Computer Science, Victoria University, Wellington, New Zealand
MR Author ID:
757288
ORCID:
0000-0003-2917-3848
Email:
erlkoenig@nd.edu, greenberg@mcs.vuw.ac.nz
Antonio Montalbán
Affiliation:
Department of Mathematics, University of Chicago, Chicago, Illinois 60637
Email:
antonio@math.uchicago.edu
Received by editor(s):
August 29, 2005
Published electronically:
October 23, 2007
Additional Notes:
We would like to thank our advisor Richard A. Shore for introducing us to questions that are discussed in this paper and for many useful conversations. Both authors were partially supported by NSF Grant DMS-0100035. This paper is part of the second author’s doctoral thesis.
Article copyright:
© Copyright 2007
American Mathematical Society
