On computable self-embeddings of computable linear orderings
Rodney G. Downey, Bart Kastermans, and Steffen Lempp
Source: J. Symbolic Logic Volume 74, Issue 4 (2009), 1352-1366.
Abstract
We solve a longstanding question of Rosenstein, and make progress toward solving a long-standing open problem in the area of computable linear orderings by showing that every computable η-like linear ordering without an infinite strongly η-like interval has a computable copy without nontrivial computable self-embedding. The precise characterization of those computable linear orderings which have computable copies without nontrivial computable self-embedding remains open.
Primary Subjects: Primary 03D45, Secondary 03C57
Keywords: computable linear ordering; self-embedding
Full-text: Access denied (no subscription detected)
We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.
If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.jsl/1254748695
Digital Object Identifier: doi:10.2178/jsl/1254748695
Zentralblatt MATH identifier:
1105.03036
References
Christopher J. Ash, Carl G. Jockusch, Jr., and Julia F. Knight, Jumps of orderings, Transactions of the American Mathematical Society, vol. 319 (1990), pp. 573--599.
Mathematical Reviews (MathSciNet):
MR955487
Zentralblatt MATH:
0705.03022
Digital Object Identifier: doi:10.2307/2001255
JSTOR: links.jstor.org
Rodney G. Downey, Every recursive Boolean algebra is isomorphic to one with incomplete atoms, Annals of Pure and Applied Logic, vol. 60 (1993), pp. 193--206.
Mathematical Reviews (MathSciNet):
MR1216669
Zentralblatt MATH:
0796.03049
Digital Object Identifier: doi:10.1016/0168-0072(93)90075-O
--------, Computability theory and linear orderings, Handbook of recursive mathematics, vol. 2, North-Holland, Amsterdam, 1998, pp. 823--976.
Mathematical Reviews (MathSciNet):
MR1673590
Zentralblatt MATH:
0941.03045
Rodney G. Downey, Carl G. Jockusch, Jr., and Joseph S. Miller, On self-embeddings of computable linear orderings, Annals of Pure and Applied Logic, vol. 138 (2006), pp. 52--76.
Mathematical Reviews (MathSciNet):
MR2183808
Zentralblatt MATH:
1105.03036
Digital Object Identifier: doi:10.1016/j.apal.2005.06.008
Rodney G. Downey and Steffen Lempp, The proof-theoretic strength of the Dushnik--Miller Theorem for countable linear orders, Recursion theory and complexity (Kazan, 1997), de Gruyter, Berlin, 1999, pp. 55--57.
Mathematical Reviews (MathSciNet):
MR1724931
Zentralblatt MATH:
0951.03053
Rodney G. Downey, Steffen Lempp, and Guohua Wu, On the complexity of the successivity relation in computable linear orderings, to appear.
Rodney G. Downey and Michael F. Moses, On choice sets and strongly nontrivial self-embeddings of recursive linear orders, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 35 (1989), pp. 237--246.
Mathematical Reviews (MathSciNet):
MR1000966
Ben Dushnik and E. W. Miller, Concerning similarity transformations of linearly ordered sets, Bulletin of the American Mathematical Society, vol. 46 (1940), pp. 322--326.
Mathematical Reviews (MathSciNet):
MR1919
Digital Object Identifier: doi:10.1090/S0002-9904-1940-07213-1
Project Euclid: euclid.bams/1183502563
Lawrence Feiner, Hierarchies of Boolean algebras, Journal of Symbolic Logic, vol. 35 (1970), pp. 365--374.
Mathematical Reviews (MathSciNet):
MR282805
Digital Object Identifier: doi:10.2307/2270692
JSTOR: links.jstor.org
Joseph Harrison, Recursive pseudo-well-orderings, Transactions of the American Mathematical Society, vol. 131 (1968), pp. 526--543.
Mathematical Reviews (MathSciNet):
MR244049
Zentralblatt MATH:
0186.01101
Digital Object Identifier: doi:10.2307/1994961
JSTOR: links.jstor.org
Carl G. Jockusch, Jr. and Robert I. Soare, Degrees of orderings not isomorphic to recursive linear orderings, Annals of Pure and Applied Logic, vol. 52 (1991), pp. 39--64, ``International Symposium on Mathematical Logic and its Applications'' (Nagoya, 1988).
Mathematical Reviews (MathSciNet):
MR1104053
Zentralblatt MATH:
0734.03026
Digital Object Identifier: doi:10.1016/0168-0072(91)90038-N
Steffen Lempp, Lecture notes on priority arguments, preprint available at family http://www.math.wisc.edu/\~lempp/papers/prio.pdf.
Antonio Montalbán, Beyond the arithmetic, Ph.D. thesis, Cornell University, 2005.
--------, Countably complementable linear orderings, Order, vol. 23 (2006), pp. 321--331.
Mathematical Reviews (MathSciNet):
MR2309696
Linda Jean C. Richter, Degrees of unsolvability of models, Ph.D. thesis, University of Illinois at Urbana--Champaign, 1977.
Joseph G. Rosenstein, Linear orderings, Academic Press, New York-London, 1982.
Mathematical Reviews (MathSciNet):
MR662564
--------, Recursive linear orderings, Orders: description and roles (l'Arbresle, 1982), North-Holland, Amsterdam, 1984, pp. 465--475.
Mathematical Reviews (MathSciNet):
MR779865
Robert I. Soare, Recursively enumerable sets and degrees, Springer-Verlag, Berlin, New York, 1987.
Mathematical Reviews (MathSciNet):
MR882921
Richard Watnick, A generalization of Tennenbaum's theorem on effectively finite recursive linear orderings, Journal of Symbolic Logic, vol. 49 (1984), pp. 563--569.
Mathematical Reviews (MathSciNet):
MR745385
Zentralblatt MATH:
0585.03015
Digital Object Identifier: doi:10.2307/2274189
JSTOR: links.jstor.org
