scholarly journals ON FOREMAN’S MAXIMALITY PRINCIPLE

2016 ◽  
Vol 81 (4) ◽  
pp. 1344-1356 ◽  
Author(s):  
MOHAMMAD GOLSHANI ◽  
YAIR HAYUT
Keyword(s):  
Regular Cardinal

AbstractIn this paper, we consider Foreman’s maximality principle, which says that any nontrivial forcing notion either adds a new real or collapses some cardinals. We prove the consistency of some of its consequences. We observe that it is consistent that every c.c.c. forcing adds a real and that for every uncountable regular cardinal κ, every κ-closed forcing of size 2<κ collapses some cardinal.

Generic Σ31 absoluteness

2004 ◽  
Vol 69 (1) ◽  
pp. 73-80 ◽  
Author(s):  
Sy D. Friedman
Keyword(s):  
Regular Cardinal ◽  
Forcing Axioms ◽  
Proper Forcing ◽  
Class Forcing ◽  
The Universe ◽  
Generic Extensions

In this article we study the strength of absoluteness (with real parameters) in various types of generic extensions, correcting and improving some results from [3]. (In particular, see Theorem 3 below.) We shall also make some comments relating this work to the bounded forcing axioms BMM, BPFA and BSPFA.The statement “ absoluteness holds for ccc forcing” means that if a formula with real parameters has a solution in a ccc set-forcing extension of the universe V, then it already has a solution in V. The analogous definition applies when ccc is replaced by other set-forcing notions, or by class-forcing.Theorem 1. [1] absoluteness for ccc has no strength; i.e., if ZFC is consistent then so is ZFC + absoluteness for ccc.The following results concerning (arbitrary) set-forcing and class-forcing can be found in [3].Theorem 2 (Feng-Magidor-Woodin). (a) absoluteness for arbitrary set-forcing is equiconsistent with the existence of a reflecting cardinal, i.e., a regular cardinal κ such that H(κ) is ∑2-elementary in V.(b) absoluteness for class-forcing is inconsistent.We consider next the following set-forcing notions, which lie strictly between ccc and arbitrary set-forcing: proper, semiproper, stationary-preserving and ω1-preserving. We refer the reader to [8] for the definitions of these forcing notions.Using a variant of an argument due to Goldstern-Shelah (see [6]), we show the following. This result corrects Theorem 2 of [3] (whose proof only shows that if absoluteness holds in a certain proper forcing extension, then in L either ω1 is Mahlo or ω2 is inaccessible).

Constructible lattices of c-degrees

1982 ◽  
Vol 47 (4) ◽  
pp. 739-754
Author(s):  
C.P. Farrington
Keyword(s):  
Set Theory ◽  
Regular Cardinal ◽  
Boolean Algebras ◽  
Combinatorial Complexity ◽  
Generic Extension ◽  
Transitive Model ◽  
Perfect Set ◽  
Consistency Results ◽  
Combinatorial Principles ◽  
Souslin Trees

This paper is devoted to the proof of the following theorem.Theorem. Let M be a countable standard transitive model of ZF + V = L, and let ℒ Є M be a wellfounded lattice in M, with top and bottom. Let ∣ℒ∣M = λ, and suppose κ ≥ λ is a regular cardinal in M. Then there is a generic extension N of M such that(i) N and M have the same cardinals, and κN ⊂ M;(ii) the c-degrees of sets of ordinals of N form a pattern isomorphic to ℒ;(iii) if A ⊂ On and A Є N, there is B Є P(κ+)N such that L(A) = L(B).The proof proceeds by forcing with Souslin trees, and relies heavily on techniques developed by Jech. In [5] he uses these techniques to construct simple Boolean algebras in L, and in [6] he uses them to construct a model of set theory whose c-degrees have orderlype 1 + ω*.The proof also draws on ideas of Adamovicz. In [1]–[3] she obtains consistency results concerning the possible patterns of c-degrees of sets of ordinals using perfect set forcing and symmetric models. These methods have the advantage of yielding real degrees, but involve greater combinatorial complexity, in particular the use of ‘sequential representations’ of lattices.The advantage of the approach using Souslin trees is twofold: first, we can make use of ready-made combinatorial principles which hold in L, and secondly, the notion of genericity over a Souslin tree is particularly simple.

Increasing u2 by a stationary set preserving forcing

2009 ◽  
Vol 74 (1) ◽  
pp. 187-200
Author(s):  
Benjamin Claverie ◽  
Ralf Schindler
Keyword(s):  
Regular Cardinal ◽  
Nonstationary Ideal ◽  
Stationary Set ◽  
Precipitous Ideal ◽  
Countable Structure

AbstractWe show that if I is a precipitous ideal on ω1 and if θ > ω1 is a regular cardinal, then there is a forcing ℙ = ℙ(I, θ) which preserves the stationarity of all I-positive sets such that in Vℙ, ⟨Hθ; ∈, I⟩ is a generic iterate of a countable structure ⟨M; ∈, Ī⟩. This shows that if the nonstationary ideal on ω1 is precipitous and exists, then there is a stationary set preserving forcing which increases . Moreover, if Bounded Martin's Maximum holds and the nonstationary ideal on ω1 is precipitous, then .

Powers of the ideal of Lebesgue measure zero sets

1991 ◽  
Vol 56 (1) ◽  
pp. 103-107
Author(s):  
Maxim R. Burke
Keyword(s):  
Partial Order ◽  
Lebesgue Measure ◽  
Regular Cardinal ◽  
Measure Zero ◽  
Lebesgue Measure Zero ◽  
Zero Sets ◽  
Covering Lemma ◽  
Inner Model ◽  
The Ideal

AbstractWe investigate the cofinality of the partial order κ of functions from a regular cardinal κ into the ideal of Lebesgue measure zero subsets of R. We show that when add () = κ and the covering lemma holds with respect to an inner model of GCH, then cf (κ) = max{cf(κκ), cf([cf()]κ)}. We also give an example to show that the covering assumption cannot be removed.

Some remarks on the partition calculus of ordinals

1999 ◽  
Vol 64 (2) ◽  
pp. 436-442 ◽  
Author(s):  
Péter Komjáth
Keyword(s):  
Regular Cardinal ◽  
Continuum Hypothesis ◽  
Alternative Proof ◽  
Partition Relation ◽  
Negative Relation ◽  
Partition Calculus ◽  
Counter Example ◽  
The Continuum

One of the early partition relation theorems which include ordinals was the observation of Erdös and Rado [7] that if κ = cf(κ) > ω then the Dushnik–Miller theorem can be sharpened to κ→(κ, ω + 1)2. The question on the possible further extension of this result was answered by Hajnal who in [8] proved that the continuum hypothesis implies ω1 ↛ (ω1, ω + 2)2. He actually proved the stronger result ω1 ↛ (ω: 2))2. The consistency of the relation κ↛(κ, (ω: 2))2 was later extensively studied. Baumgartner [1] proved it for every κ which is the successor of a regular cardinal. Laver [9] showed that if κ is Mahlo there is a forcing notion which adds a witness for κ↛ (κ, (ω: 2))2 and preserves Mahloness, ω-Mahloness of κ, etc. We notice in connection with these results that λ→(λ, (ω: 2))2 holds if λ is singular, in fact λ→(λ, (μ: n))2 for n < ω, μ < λ (Theorem 4).In [11] Todorčević proved that if cf(λ) > ω then a ccc forcing can add a counter-example to λ→(λ, ω + 2)2. We give an alternative proof of this (Theorem 5) and extend it to larger cardinals: if GCH holds, cf (λ) > κ = cf (κ) then < κ-closed, κ+-c.c. forcing adds a counter-example to λ→(λ, κ + 2)2 (Theorem 6).Erdös and Hajnal remarked in their problem paper [5] that Galvin had proved ω2→(ω1, ω + 2)2 and he had also asked if ω2→(ω1, ω + 3)2 is true. We show in Theorem 1 that the negative relation is consistent.

A Technique to Generate -Ary Free Lattices from Finitary Ones

1985 ◽  
Vol 37 (2) ◽  
pp. 324-336
Author(s):  
George Grätzer ◽  
David Kelly
Keyword(s):  
Regular Cardinal

Let be an infinite regular cardinal. A poset L is called an -lattice if and only if for all XL satisfying 0 < |X| < m, ∧ X and ∨ X exist.This paper is a part of a sequence of papers, [5], [6], [7], [8], developing the theory of -lattices. For a survey of some of these results, see [9].The -lattice is described in [6]; γ denotes the zero and γ′ the unit of . In particular, formulas for -joins and meets are given. (We repeat the essentials of this description in Section 4.)In [6] we proved the theorem stated below. Our proof was based on characterization of (the free -lattice on P) due to [1]; as a result, our proof was very computational.

scholarly journals Saturated Ideals in Boolean Extensions

1972 ◽  
Vol 48 ◽  
pp. 159-168 ◽  
Author(s):  
Yuzuru Kakuda
Keyword(s):  
Regular Cardinal ◽  
Uncountable Cardinal

0. Introduction. Let k be an uncountable cardinal, and let λ be a regular cardinal less than k. Let I be a λ-saturated non-trivial ideal on k. Prikry, in his thesis, showed that, in certain Boolean extensions, k has a λ-saturated non-trivial ideal on k.

AN ABSTRACT ELEMENTARY CLASS NONAXIOMATIZABLE IN

2019 ◽  
Vol 84 (3) ◽  
pp. 1240-1251
Author(s):  
SIMON HENRY
Keyword(s):  
Regular Cardinal ◽  
Vector Spaces ◽  
Left Adjoint ◽  
Scott Topology ◽  
Elementary Class ◽  
Closed Fields ◽  
Uncountable Cardinal ◽  
Image Position ◽  
Bounded Below ◽  
Abstract Elementary Class

AbstractWe show that for any uncountable cardinal λ, the category of sets of cardinality at least λ and monomorphisms between them cannot appear as the category of points of a topos, in particular is not the category of models of a ${L_{\infty ,\omega }}$-theory. More generally we show that for any regular cardinal $\kappa < \lambda$ it is neither the category of κ-points of a κ-topos, in particular, nor the category of models of a ${L_{\infty ,\kappa }}$-theory.The proof relies on the construction of a categorified version of the Scott topology, which constitute a left adjoint to the functor sending any topos to its category of points and the computation of this left adjoint evaluated on the category of sets of cardinality at least λ and monomorphisms between them. The same techniques also apply to a few other categories. At least to the category of vector spaces of with bounded below dimension and the category of algebraic closed fields of fixed characteristic with bounded below transcendence degree.

scholarly journals Closed maximality principles: implications, separations and combinations

2008 ◽  
Vol 73 (1) ◽  
pp. 276-308 ◽  
Author(s):  
Gunter Fuchs
Keyword(s):  
Equivalence Relation ◽  
Regular Cardinal ◽  
Highly Sensitive ◽  
Souslin Trees ◽  
Maximality Principles ◽  
Automorphism Tower

AbstractI investigate versions of the Maximality Principles for the classes of forcings which are <κ-closed, <κ-directed-closed, or of the form Col(κ, <λ). These principles come in many variants, depending on the parameters which are allowed, I shall write MPΓ (A) for the maximality principle for forcings in Γ, with parameters from A. The main results of this paper are:• The principles have many consequences, such as <κ-closed-generic (Hκ) absoluteness, and imply, e.g., that ◊κ holds. I give an application to the automorphism tower problem, showing that there are Souslin trees which are able to realize any equivalence relation, and hence that there are groups whose automorphism tower is highly sensitive to forcing.• The principles can be separated into a hierarchy which is strict, for many κ.• Some of the principles can be combined, in the sense that they can hold at many different κ simultaneously.The possibilities of combining the principles are limited, though: While it is consistent that MP<κ-closed(Hκ +) holds at all regular κ below any fixed α, the “global” maximality principle, stating that MP<κ-closed (Hκ ∪ {κ} ) holds at every regular κ, is inconsistent. In contrast to this, it is equiconsistent with ZFC that the maximality principle for directed-closed forcings without any parameters holds at every regular cardinal. It is also consistent that every local statement with parameters from Hκ⊦ that's provably <κ-closed-forceably necessary is true, for all regular κ.

Complete Boolean ultraproducts

1987 ◽  
Vol 52 (2) ◽  
pp. 530-542
Author(s):  
R. Michael Canjar
Keyword(s):  
Boolean Algebra ◽  
Regular Cardinal ◽  
Complete Boolean Algebra ◽  
Full Structure ◽  
Truth Value ◽  
Image Position

Throughout this paper, B will always be a Boolean algebra and Γ an ultrafilter on B. We use + and Σ for the Boolean join operation and · and Π for the Boolean meet.κ is always a regular cardinal. C(κ) is the full structure of κ, the structure with universe κ and whose functions and relations consist of all unitary functions and relations on κ. κB is the collection of all B-valued names for elements of κ. We use symbols f, g, h for members of κB. Formally an element f ∈ κB is a mapping κ → B with the properties that Σα∈κf(α) = 1B and that f(α) · f(β) = 0B whenever α ≠ β. We view f(α) as the Boolean-truth value indicating the extent to which the name f is equal to α, and we will hereafter write ∥f = α∥ for f(α). For every α ∈ κ there is a canonical name fα ∈ κB which has the property that ∥fα = α∥ = 1. Hereafter we identify α and fα.If B is a κ+-complete Boolean algebra and Γ is an ultrafilter on B, then we may define the Boolean ultraproduct C(κ)B/Γ in the following manner. If ϕ(x0, x1, …, xn) is a formula of Lκ, the language for C(κ) (which has symbols for all finitary functions and relations on κ), and f0, f1, …, fn−1 are elements of κB then we define the Boolean-truth value of ϕ(f0, f1, …, fn−1) as

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