scholarly journals Cardinal sequences and Cohen real extensions

2004 ◽  
Vol 181 (1) ◽  
pp. 75-88 ◽  
Author(s):  
István Juhász ◽  
Saharon Shelah ◽  
Lajos Soukup ◽  
Zoltán Szentmiklóssy
Keyword(s):  
Cohen Real

THE SOLIDITY AND NONSOLIDITY OF INITIAL SEGMENTS OF THE CORE MODEL

2018 ◽  
Vol 83 (3) ◽  
pp. 920-938
Author(s):  
GUNTER FUCHS ◽  
RALF SCHINDLER
Keyword(s):  
Core Model ◽  
Solid Core ◽  
The Core ◽  
Inner Model ◽  
Cohen Real ◽  
Image Position ◽  
Woodin Cardinal

AbstractIt is shown that $K|{\omega _1}$ need not be solid in the sense previously introduced by the authors: it is consistent that there is no inner model with a Woodin cardinal yet there is an inner model W and a Cohen real x over W such that $K|{\omega _1}\,\, \in \,\,W[x] \setminus W$. However, if ${0^{\rm{\P}}}$ does not exist and $\kappa \ge {\omega _2}$ is a cardinal, then $K|\kappa$ is solid. We draw the conclusion that solidity is not forcing absolute in general, and that under the assumption of $\neg {0^{\rm{\P}}}$, the core model is contained in the solid core, previously introduced by the authors.It is also shown, assuming ${0^{\rm{\P}}}$ does not exist, that if there is a forcing that preserves ${\omega _1}$, forces that every real has a sharp, and increases $\delta _2^1$, then ${\omega _1}$ is measurable in K.

scholarly journals Some Banach spaces added by a Cohen real

2015 ◽  
Vol 195 ◽  
pp. 246-255
Author(s):  
Mirna Džamonja
Keyword(s):  
Banach Spaces ◽  
Cohen Real

scholarly journals Small forcings and Cohen reals

1997 ◽  
Vol 62 (1) ◽  
pp. 280-284
Author(s):  
Jindřich Zapletal
Keyword(s):  
Uniform Density ◽  
Cohen Real

AbstractWe show that all posets of uniform density ℵ1 may have to add a Cohen real and develop some forcing machinery for obtaining this sort of result.

scholarly journals The Bristol model: An abyss called a Cohen real

2018 ◽  
Vol 18 (02) ◽  
pp. 1850008 ◽  
Author(s):  
Asaf Karagila
Keyword(s):  
Positive Answer ◽  
Cohen Real ◽  
One Step ◽  
Work Done

We construct a model [Formula: see text] of [Formula: see text] which lies between [Formula: see text] and [Formula: see text] for a Cohen real [Formula: see text] and does not have the form [Formula: see text] for any set [Formula: see text]. This is loosely based on the unwritten work done in a Bristol workshop about Woodin’s HOD Conjecture in 2011. The construction given here allows for a finer analysis of the needed assumptions on the ground models, thus taking us one step closer to understanding models of [Formula: see text], and the HOD Conjecture and its relatives. This model also provides a positive answer to a question of Grigorieff about intermediate models of [Formula: see text], and we use it to show the failure of Kinna–Wagner Principles in [Formula: see text].

scholarly journals Orthogonal families of real sequences

1998 ◽  
Vol 63 (1) ◽  
pp. 29-49
Author(s):  
Arnold W. Miller ◽  
Juris Steprans
Keyword(s):  
Hilbert Space ◽  
Inner Product ◽  
Partial Functions ◽  
Perfect Set ◽  
Mad Families ◽  
Real Model ◽  
Cohen Real ◽  
Image Position ◽  
Orthogonal Set ◽  
Orthogonal Sequences

For x, y ϵ ℝω define the inner productwhich may not be finite or even exist. We say that x and y are orthogonal if (x, y) converges and equals 0.Define lp to be the set of all x ϵ ℝω such thatFor Hilbert space, l2, any family of pairwise orthogonal sequences must be countable. For a good introduction to Hilbert space, see Retherford [4].Theorem 1. There exists a pairwise orthogonal family F of size continuum such that F is a subset of lp for every p > 2.It was already known that there exists a family of continuum many pairwise orthogonal elements of ℝω. A family F ⊆ ℝω∖0 of pairwise orthogonal sequences is orthogonally complete or a maximal orthogonal family iff the only element of ℝω orthogonal to every element of F is 0, the constant 0 sequence.It is somewhat surprising that Kunen's perfect set of orthogonal elements is maximal (a fact first asserted by Abian). MAD families, nonprincipal ultrafilters, and many other such maximal objects cannot be even Borel.Theorem 2. There exists a perfect maximal orthogonal family of elements of ℝω.Abian raised the question of what are the possible cardinalities of maximal orthogonal families.Theorem 3. In the Cohen real model there is a maximal orthogonal set in ℝω of cardinality ω1, but there is no maximal orthogonal set of cardinality κ with ω1 < κ < ϲ.By the Cohen real model we mean any model obtained by forcing with finite partial functions from γ to 2, where the ground model satisfies GCH and γω = γ.

scholarly journals Forcing Properties of Ideals of Closed Sets

2011 ◽  
Vol 76 (3) ◽  
pp. 1075-1095 ◽  
Author(s):  
Marcin Sabok ◽  
Jindřich Zapletal
Keyword(s):  
Polish Space ◽  
Borel Function ◽  
Constant Property ◽  
Borel Set ◽  
Borel Sets ◽  
Combinatorial Properties ◽  
Closed Sets ◽  
Cohen Real ◽  
The Ideal

AbstractWith every σ-ideal I on a Polish space we associate the σ-ideal I* generated by the closed sets in I. We study the forcing notions of Borel sets modulo the respective σ -ideals I and I* and find connections between their forcing properties. To this end, we associate to a σ-ideal on a Polish space an ideal on a countable set and show how forcing properties of the forcing depend on combinatorial properties of the ideal.We also study the 1–1 or constant property of σ-ideals, i.e., the property that every Borel function defined on a Borel positive set can be restricted to a positive Borel set on which it either 1–1 or constant. We prove the following dichotomy: if I is a σ-ideal generated by closed sets, then either the forcing P1 adds a Cohen real, or else I has the 1–1 or constant property.

Small filter forcing

1986 ◽  
Vol 51 (3) ◽  
pp. 526-546
Author(s):  
R. Michael Canjar
Keyword(s):  
Regular Cardinal ◽  
Ground Model ◽  
Random Real ◽  
Generic Subset ◽  
Partially Ordered ◽  
Martin's Axiom ◽  
Martin’S Axiom ◽  
Cohen Real ◽  
Basic Facts ◽  
Random Reals

Qλ is the set of nonprincipal filters on ω which are generated by fewer than λ sets, for λ a fixed uncountable, regular cardinal ≤ c. We analyze forcing with Qλ, where Qλ is partially ordered in such a way that a filter F1 is more informative than F2 iff F1 includes F2. Qλ-forcing adjoins an ultrafilter on ω but adds no new reals. We analyze Qλ-forcing from a forcing-theoretic viewpoint. We also analyze the properties of Qλ-generic ultrafilters. These properties are independent of ZFC and depend very much on the ground model. In particular, we study Qλ-forcing over ground models which are Cohen real extensions, random real extensions, and models which satisfy Martin's Axiom.In §2 we give notations and definitions, and review some of the basic facts about forcing and ultrafilters which we will use. In §3 we introduce Qλ-forcing and prove some basic lemmas about it. §4 studies Qc-forcing. §§5, 6, and 7 analyze Qλ-forcing over ground models of Martin's Axiom, ground models which are generated by Cohen reals, and ground models which are generated by random reals, respectively. Qλ-forcing over Cohen real and random real models is isomorphic to the notion of forcing which adjoins a Cohen generic subset of λ; this is proved in §8.

Cohen real model

2017 ◽  
pp. 46-56
Author(s):  
Arnold W. Miller
Keyword(s):  
Real Model ◽  
Cohen Real

scholarly journals Strong measure zero sets without Cohen reals

1993 ◽  
Vol 58 (4) ◽  
pp. 1323-1341 ◽  
Author(s):  
Martin Goldstern ◽  
Haim Judah ◽  
Saharon Shelah
Keyword(s):  
Measure Zero ◽  
Zero Set ◽  
Zero Sets ◽  
Cohen Real ◽  
Strong Measure ◽  
Strong Measure Zero

AbstractIf ZFC is consistent, then each of the following is consistent with :(1) X ⊆ ℝ is of strong measure zero iff ∣X∣ ≤ ℵ1 + there is a generalized Sierpinski set.(2) The union of ℵ many strong measure zero sets is a strong measure zero set + there is a strong measure zero set of size ℵ2 + there is no Cohen real over L.

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