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Transactions of the American Mathematical Society

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Any FIP real computes a 1-generic


Authors: Peter Cholak, Rodney G. Downey and Greg Igusa
Journal: Trans. Amer. Math. Soc. 369 (2017), 5855-5869
MSC (2010): Primary 03D28; Secondary 03F35, 03B30, 03E25
DOI: https://doi.org/10.1090/tran/6997
Published electronically: April 24, 2017
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Abstract: We construct a computable sequence of computable reals $ \langle X_i\rangle $ such that any real that can compute a subsequence that is maximal with respect to the finite intersection property can also compute a Cohen 1-generic. This is extended to establish the same result with 2IP in place of FIP. This is the first example of a classical theorem of mathematics that has been found to be equivalent, both proof theoretically and in terms of its effective content, to computing a 1-generic.


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Additional Information

Peter Cholak
Affiliation: Department of Mathematics, University of Notre Dame, 255 Hurley, Notre Dame, Indiana 46556
Email: Peter.Cholak.1@nd.edu

Rodney G. Downey
Affiliation: School of Mathematics, Statistics and Operations Research, Victoria University, P.O. Box 600, Wellington, New Zealand
Email: Rod.Downey@vuw.ac.nz

Greg Igusa
Affiliation: Department of Mathematics, University of Notre Dame, 255 Hurley, Notre Dame, Indiana 46556
Email: Gregory.Igusa.1@nd.edu

DOI: https://doi.org/10.1090/tran/6997
Received by editor(s): February 12, 2015
Received by editor(s) in revised form: May 20, 2016
Published electronically: April 24, 2017
Additional Notes: The first author was partially supported by a grant from the Simons Foundation #315283
The second author was supported by the Marsden Fund of New Zealand
The third author was partially supported by EMSW21-RTG-0838506
This research was (partially) completed while the authors were visiting the Institute for Mathematical Sciences, National University of Singapore in 2014
Article copyright: © Copyright 2017 American Mathematical Society

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