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FORCING WITH BUSHY TREES

Published online by Cambridge University Press:  21 June 2017

MUSHFEQ KHAN  and
JOSEPH S. MILLER
MUSHFEQ KHAN
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF HAWAI‘I AT MĀNOA HONOLULU, HI 96822, USA E-mail: khan@math.hawaii.edu
JOSEPH S. MILLER
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF WISCONSIN MADISON, WI 53706-1388, USA E-mail: jmiller@math.wisc.edu
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Abstract

We present several results that rely on arguments involving the combinatorics of “bushy trees”. These include the fact that there are arbitrarily slow-growing diagonally noncomputable (DNC) functions that compute no Kurtz random real, as well as an extension of a result of Kumabe in which we establish that there are DNC functions relative to arbitrary oracles that are of minimal Turing degree. Along the way, we survey some of the existing instances of bushy tree arguments in the literature.

Keywords

03D3203D55Diagonally noncomputable functionsbushy tree forcingminimal degreesalgorithmic randomness

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Type
Articles
Information
Bulletin of Symbolic Logic , Volume 23 , Issue 2 , June 2017 , pp. 160 - 180
Copyright
Copyright © The Association for Symbolic Logic 2017 

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References

REFERENCES

Ambos-Spies, K., Kjos-Hanssen, B., Lempp, S., and Slaman, T. A., Comparing DNR and WWKL . The Journal of Symbolic Logic, vol. 69 (2004), no. 4, pp. 1089–1104.Google Scholar
Beros, A. A., A DNC function that computes no effectively bi-immune set . Archive for Mathematical Logic, vol. 54 (2015), no. 5–6, pp. 521–530.CrossRefGoogle Scholar
Bienvenu, L. and Patey, L., Diagonally non-computable functions and fireworks, arXiv e-prints, 2014.Google Scholar
Cai, M., Elements of classical recursion theory: Degree-theoretic properties and combinatorial properties, Ph.D. thesis, Cornell University, 2011.Google Scholar
Dorais, F. G., Hirst, J. L., and Shafer, P., Comparing the strength of diagonally nonrecursive functions in the absence of ${\rm{\Sigma }}_2^0$ induction . The Journal of Symbolic Logic, vol. 80 (2015), no. 4, pp. 1211–1235.Google Scholar
Downey, R. G., Greenberg, N., Jockusch, C. G. Jr., and Milans, K. G., Binary subtrees with few labeled paths . Combinatorica, vol. 31 (2011), no. 3, pp. 285–303.Google Scholar
Downey, R. G. and Hirschfeldt, D. R., Algorithmic Randomness and Complexity, Theory and Applications of Computability, Springer, New York, 2010.Google Scholar
Giusto, M. and Simpson, S. G., Located sets and reverse mathematics . The Journal of Symbolic Logic, vol. 65 (2000), no. 3, pp. 1451–1480.Google Scholar
Greenberg, N. and Miller, J. S., Diagonally non-recursive functions and effective Hausdorff dimension . Bulletin of the London Mathematical Society, vol. 43 (2011), no. 4, pp. 636–654.CrossRefGoogle Scholar
Jockusch, C. G. Jr., Degrees of functions with no fixed points , Logic, Methodology and Philosophy of Science, VIII (Moscow, 1987), Studies in Logic and the Foundations of Mathematics, vol. 126, North-Holland, Amsterdam, 1989, pp. 191–201.Google Scholar
Jockusch, C. G. Jr. and Lewis, A. E. M., Diagonally non-computable functions and bi-immunity . The Journal of Symbolic Logic, vol. 78, (2013), no. 3, pp. 977–988.Google Scholar
Kučera, A., Measure, ${\rm{\Pi }}_1^0$ -classes and complete extensions of PA, Recursion Theory Week (Oberwolfach, 1984), Lecture Notes in Mathematics, vol. 1141, Springer, Berlin, 1985, pp. 245–259.Google Scholar
Kumabe, M., A fixed point free minimal degree, unpublished, 1996.Google Scholar
Kumabe, M. and Lewis, A. E. M., A fixed-point-free minimal degree . Journal of the London Mathematical Society (2), vol. 80 (2009), no. 3, pp. 785–797.Google Scholar
Kurtz, S. A., Randomness and genericity in the degrees of unsolvability, Ph.D. thesis, University of Illinois at Urbana-Champaign; ProQuest LLC, Ann Arbor, MI, 1981.Google Scholar
Sacks, G. E., Some open questions in recursion theory , Recursion Theory Week (Ebbinghaus, H.-D., Müller, G. H., and Sacks, G. E., editors), Lecture Notes in Mathematics, vol. 1141, Springer, Berlin, Heidelberg, 1985, pp. 333–342.Google Scholar