Reflection principles for the continuum

Abstract

Let HC denote the set of sets of hereditary cardinality less than 2 ω . We consider reflection principles for HC in analogy with the Levy reflection principle for HC. Let B be a class of complete Boolean algebras. The principle Max(B) says: If R(x 1 , . . . , x n ) is a property which is provably persistent in extensions by elements of B, then R(a 1 , . . . , a n ) holds whenever a 1 , . . . , a n ∈ HC and R(a 1 , . . . , a n ) has a positive IB-value for some IB ∈ B. Suppose C is the class of Cohen algebras. We prove that Con(ZF ) implies Con(ZF C + Max(C)). For a different principle, let CCC be the class of all CCC algebras. We prove that ZF + Levy schema, and ZF C + Max(CCC) are equiconsistent. Max(CCC) implies M A, while Max(C) implies ¬M A. We give applications of these reflection principles to Löwenheim-Skolem theorems of extensions of first order logic. For example, Max(C) implies that the Löwenheim number of the extension of first order logic by the Härtig quantifier is less than 2 ω . * MSC2000 classification numbers 03E50, 03E35 and 03E40. Keywords: Reflections principle, Martin's axiom, Boolean valued models.† The second author has for many years been unable to reach the first author to get his approval of the final manuscript. Therefore the second author takes all resposibility for any errors or shortcomings of the paper.One of the fundamental properties of the universe of sets is the fact that HC = {x | x is hereditarily countable} reflects all Σ 1 -properties, that is, if a ∈ HC and P (a) is a true Σ 1 -property of a then HC |= P (a). If 2 ω > ω 1 , there is an interesting variant of HC:The basic observation underlying this paper is that while HC trivially reflects all Σ 1properties, it may, in a suitable model of set theory, reflect much more. Typically, it may reflect all properties which are Σ 1 with respect to the class of all cardinals.The strongest and perhaps the most interesting reflection principle to be considered is the following: Let B be a class of complete Boolean algebras. The principle Max(B) says: If R(x 1 , . . . , x n ) is a property which is provably persistent in extensions by elements of B, then R(a 1 , . . . , a n ) holds whenever a 1 , . . . , a n ∈ HC and R(a 1 , . . . , a n ) has a positive IB-value for some IB ∈ B.The relevant classes B to be considered here are the class C of all Cohen algebras (for exploding 2 ω ) and the class CCC of all CCC algebras. The main results are: 1. Con(ZF ) ↔ Con(ZF C + Max(C)). 2. Con(ZF + Levy schema) ↔ Con(ZF C + Max(CCC)). Max(CCC) → M A.The principle Max(C) is inconsistent with M A and hence inconsistent with Max(CCC). Thus we have two mutually inconsistent reflection principles which both make HC reflect all properties Σ 1 with respect to the class of all cardinals.Abstract logics relevant in the applications of these reflection principles are logics the satisfaction relation of which is preserved by C or CCC-extensions. An example of such a logic is the logic L(I) with the Härtig-quantifier IxyA(x)B(y) ↔ |{ a | A(a) }| = |{ b | B(b) }|...

Closed maximality principles: implications, separations and combinations

“…The principle was introduced in [SV01] by Jonathan Stavi and Jouko Väänänen, using a slightly different formulation, and focussing mainly on cardinal preserving forcing notions. Inspired by an idea of Christophe Chalons (see [Cha00]), it was then rediscovered by Joel Hamkins and analyzed in a more general context in [Ham03b].…”

“…Namely, given a class Γ of notions of forcing, let's say that a statement is Γ-possible, or Γ-forceable, if it is true in a forcing extension by a forcing notion from Γ. It is Γ-necessary (or Γ-persistent, to use terminology from [SV01]) if it holds in V and in any forcing extension by a forcing notion from Γ. The principle MP Γ now says that every sentence ϕ which is Γ-forceably Γ-necessary (i.e., the sentence "ϕ is Γ-necessary" is Γ-forceable) is true.…”

Combined Maximality Principles up to large cardinals

“…It is a very natural axiom scheme that was first introduced by Stavi and Väänänen in [SV01], where they focused on…”

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

“…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.…”

“…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.…”