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November 2020 Projective Games on the Reals
Juan P. Aguilera, Sandra Müller
Notre Dame J. Formal Logic 61(4): 573-589 (November 2020). DOI: 10.1215/00294527-2020-0027


Let Mn(R) denote the minimal active iterable extender model which has n Woodin cardinals and contains all reals, if it exists, in which case we denote by Mn(R) the class-sized model obtained by iterating the topmost measure of Mn(R) class-many times. We characterize the sets of reals which are Σ1-definable from R over Mn(R), under the assumption that projective games on reals are determined:

1. for even n, Σ1Mn(R)=RΠn+11;

2. for odd n, Σ1Mn(R)=RΣn+11.

This generalizes a theorem of Martin and Steel for L(R), that is, the case n=0. As consequences of the proof, we see that determinacy of all projective games with moves in R is equivalent to the statement that Mn(R) exists for all nN, and that determinacy of all projective games of length ω2 with moves in N is equivalent to the statement that Mn(R) exists and satisfies AD for all nN.


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Juan P. Aguilera. Sandra Müller. "Projective Games on the Reals." Notre Dame J. Formal Logic 61 (4) 573 - 589, November 2020.


Received: 8 June 2019; Accepted: 10 July 2020; Published: November 2020
First available in Project Euclid: 29 December 2020

Digital Object Identifier: 10.1215/00294527-2020-0027

Primary: 03E45
Secondary: 03E15, 03E55, 03E60

Rights: Copyright © 2020 University of Notre Dame


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Vol.61 • No. 4 • November 2020
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