Hostname: page-component-77c89778f8-gvh9x Total loading time: 0 Render date: 2024-07-24T04:06:01.122Z Has data issue: false hasContentIssue false
  • English
  • Français

Finding paths through narrow and wide trees

Published online by Cambridge University Press:  12 March 2014

Stephen Binns  and
Bjørn Kjos-Hanssen
Stephen Binns
Affiliation:
Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia, E-mail: binns@kfupm.edu.sa
Bjørn Kjos-Hanssen
Affiliation:
Department of Mathematics, University of Hawai‘i at Mānoa, Honolulu, Hi 96822, USA, E-mail: bjoern@math.hawaii.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider two axioms of second-order arithmetic. These axioms assert, in two different ways, that infinite but narrow binary trees always have infinite paths. We show that both axioms are strictly weaker than Weak König's Lemma, and incomparable in strength to the dual statement (WWKL) that wide binary trees have paths.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 74 , Issue 1 , March 2009 , pp. 349 - 360
Copyright
Copyright © Association for Symbolic Logic 2009

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

[1]Ambos-Spies, Klaus, Kjos-Hanssen, Bjorn, Lempp, Steffen, and Slaman, Theodore A., Comparing DNR and WWKL, this Journal, vol. 69 (2004), pp. 1089–1104.Google Scholar
[2]Binns, Stephen, The Medvedev and Muchnik lattices of classes, Ph.D. thesis, The Pennsylvania State University, 2003.CrossRefGoogle Scholar
[3]Binns, Stephen, classes with complex elements, this Journal, to appear.Google Scholar
[4]Brown, Douglas K., Giusto, Mariagnese, and Simpson, Stephen G., Vitali's theorem and WWKL, Archive for Mathematical Logic, vol. 41 (2002), pp. 191–206.CrossRefGoogle Scholar
[5]Cholak, Peter, Greenberg, Noam, and Miller, Joseph S., Uniform almost everywhere domination, this Journal, vol. 71 (2006), pp. 1057–1072.Google Scholar
[6]Ebbinghaus, H.-D., Müller, G.H., and Sacks, G.E. (editors), Recursion Theory Week, Lecture Notes in Mathematics, vol. 1141, Springer-Verlag, 1985.CrossRefGoogle Scholar
[7]Jockusch, Carl G. Jr., and Soare, Robert I., classes and degrees of theories, Transactions of the American Mathematical Society, vol. 173 (1972), pp. 35–56.Google Scholar
[8]Kjos-Hanssen, Bjørn, Merkle, Wolfgang, and Stephan, Frank, Kolmogorov complexity and the recursion theorem, Lecture Notes in Computer Science, vol. 3884, 2006.Google Scholar
[9]Kučera, Antonin, Measure, classes and complete extensions of PA, 1985, in [6], pp. 245–259.Google Scholar
[10]Lachlan, A.H., On the lattice of recursively enumerable sets, Transactions of the American Mathematical Society, vol. 130 (1968), pp. 1–27.CrossRefGoogle Scholar
[11]Li, Ming and Vitányi, Paul, An introduction to Kolmogorov complexity and its applications, 2 ed., Graduate Texts in Computer Science, Springer-Verlag, 1997.CrossRefGoogle Scholar
[12]Odifreddi, P., Classical recursion theory, vol. 1, North-Holland, 1950.Google Scholar
[13]Ohashi, Kempachiro, A stronger form of a theorem of Friedberg, Notre Dame Journal of Formal Logic, vol. V (1964), no. 1, pp. 10–12.Google Scholar
[14]Robinson, Robert W., Simplicity of recursively enumerable sets, this Journal, vol. 32 (1967), pp. 162–172.Google Scholar
[15]Simpson, Stephen G., Subsystems of second order arithmetic, Springer-Verlag, 1999.CrossRefGoogle Scholar
[16]Soare, Robert I., Recursively Enumerable Sets and Degrees, Perspectives in Mathematical Logic, Springer-Verlag, 1987.CrossRefGoogle Scholar