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A Simple Maximality Principle
by
Joel David Hamkins
City University of New York and Carnegie Mellon University
In this talk I propose a new kind of forcing axiom, the Maximality Principle, which asserts that any sentence j holding in some forcing extension VP and all subsequent extensions VP*Q holds already in V. It follows, in fact, that such sentences must also hold in all forcing extensions of V. In modal terms, therefore, the Maximality Principle is expressed by the scheme (\Diamond[¯]j) --> ([¯]j), and is equivalent to the modal theory S5. In this talk, I prove that the Maximality Principle is relatively consistent with ZFC. A boldface version of the Maximality Principle, obtained by allowing real parameters to appear in j, is equiconsistent with the scheme asserting that V\delta is an elementary substructure of V for an inaccessible cardinal \delta, which in turn is equiconsistent with the scheme asserting that ORD is Mahlo. The strongest principle along these lines is the Necessary Maximality Principle, which asserts that the boldface MP holds in V and all forcing extensions. From this, it follows that 0# exists, that x# exists for every set x, that projective truth is invariant by forcing, that Woodin cardinals are consistent and much more. Many open questions remain.
Date received: March 21, 2001
Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cagb-12.