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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, HI96822, USAE-mail: khan@math.hawaii.edu
JOSEPH S. MILLER
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF WISCONSIN MADISON, WI53706-1388, USAE-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
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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