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BPFA and projective well-orderings of the reals

Published online by Cambridge University Press:  12 March 2014

Andrés Eduardo Caicedo  and
Sy-David Friedman
Andrés Eduardo Caicedo
Affiliation:
Boise State University, Department of Mathematics—1910 University Drive, Boise, ID 83725, USA, URL: http://math.boisestate.edu/~caicedo/, E-mail: [email protected]
Sy-David Friedman
Affiliation:
Kurt Gödel Research Center for Mathematical Logic, University of Vienna, Vienna, Austria, URL: http://www.logic.univie.ac.at/~sdf/, E-mail: [email protected]
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Abstract

If the bounded proper forcing axiom BPFA holds and ω1 = ω1L, then there is a lightface Σ31 well-ordering of the reals. The argument combines a well-ordering due to Caicedo-Veličković with an absoluteness result for models of MA in the spirit of “David's trick.” We also present a general coding scheme that allows us to show that BPFA is equiconsistent with R being lightface Σ41 for many “consistently locally certified” relations R on ℝ. This is accomplished through a use of David's trick and a coding through the Σ2 stable ordinals of L.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 76 , Issue 4 , December 2011 , pp. 1126 - 1136
Copyright
Copyright © Association for Symbolic Logic 2011

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