The Necessary Maximality Principle for c. c. c. forcing is equiconsistent with a weakly compact cardinal

Abstract

The Necessary Maximality Principle for c. c. c. forcing, denoted 2 MP ccc (R), asserts that any set theoretic statement with a real parameter in a c. c. c. extension that could become true in a further c. c. c. extension and remain true in all subsequent c. c. c. extensions, is already true in the minimal extension containing the real. We show that this principle is equiconsistent with the existence of a weakly compact cardinal.The principle is one of a family of principles considered in [3], which builds on ideas of [2] and contains a rediscovery in parts of earlier independent work in [8] 1) . The family begins with the Maximality Principle MP, the scheme asserting the truth of any statement that holds in some forcing extension V P and all subsequent extensions V P * Q (these are the forceably necessary statements). The boldface form MP(R) allows real parameters in the scheme, and the Necessary Maximality Principle 2 MP(R) asserts MP(R) in all forcing extensions, using the parameters available in those extensions. The main results of [3] show that MP is equiconsistent with ZFC, while MP(R) is equiconsistent with the Lévy scheme "ORD is Mahlo" and 2 MP(R) is far stronger. Philip Welch proved that 2 MP(R) implies Projective Determinacy, and the second author of this paper improved the conclusion to AD L(R) . He also provided an upper bound by proving the consistency of 2 MP(R) from the theory AD R + "Θ is regular".In this article, we focus on the principles obtained by restricting attention to the class of c. c. c. forcing notions. The parameter-free version MP ccc asserts the truth of any statement holding in some c. c. c. extension V P and all subsequent c. c. c. extensions V P * Q . This is equiconsistent with ZFC by [3, Corollary 32].The principle MP ccc implies a spectacular failure of the Continuum Hypothesis. The reason is that with c. c. c. forcing one may add as many Cohen reals as desired, and once they are added, of course, the value of the continuum 2 ω remains inflated in all subsequent c. In particular, Stavi and Väänänen first considered versions of the Maximality Principle for c. c. c. and other classes of forcing in 1977, finally publishing this work in [8]. The published version of [8], unfortunately, makes problematic claims about the forceability of the Maximality Principle for c. c. c. and other forcing in connection with Theorems 28, 32, 33, 34, 35 and 39, arising from the invalid use of a Tarskian truth predicate in the consistency constructions. Arguments of [3] show that these claims fail in any model of ZFC + V = L whose definable ordinals are unbounded. One should instead understand the [8] construction in the context of V δ ≺ V , as in [3] and this article, and Väänänen reports that his 1977 notes used this setup. The CON(ZF) constructions of [8, Theorems 33, 35], however, seem to require the more elaborate methods of [6], the dissertation of a student of the first author.

Closed maximality principles: implications, separations and combinations

“…A proof that this works can be found in [HW05]. 2 I shall also need the following fact, which is again easily verified:…”

Cohen forcing and inner models

“…This behavior suggests a connection with term forcing (or termspace forcing ‐ the nomenclature varies in the literature). The following account of term forcing is taken from [2]. Definition Suppose is a partial order and is a ‐name for a partial order.…”

“…Add(δ, λ δ ) →Q δ is a projection map. 2 Now fix p ∈ P δ+1 . Let π δ+1 ( p) = π δ ( p δ) q, whereq ∈ dom(Q δ ) is selected so that 1 κ∈I ∩δ Add(κ,λ κ ) ḟ ( p(δ)) =q (fullness of the nameQ δ guarantees that such aq will exist in dom(Q δ )).…”

Moving Up and Down in the Generic Multiverse

“…§ 1.4). Various other aspects of the modal logic of forcing are considered in [15,12,3,4,16,19,2,1,10]. The paper [8] presented at ICLA 2009 gives an overview of the status of research and creates a connection between the modal logic of forcing and "set-theoretic geology", i.e., going down from a universe to its ground models.…”