Cohen Extensions Research Articles

Filter‐Menger set of reals in Cohen extensions

AbstractWe prove that for every ultrafilter on there exists a filter on which is ‐Menger and . We show that in the Cohen model there exists such which are tall by using a construction of Nyikos's [10]. These answer a question of Das [2, Problem 7]. We prove that there is a Menger filter of character that is not Hurewicz in the ‐Cohen model where is uncountable regular. This shows that the positive answer to a question of Hernández‐Gutiérrez and Szeptycki [3, Question 2.8] is consistent with . We also study the filter generated by the set of mutually Cohen reals in the ‐Cohen model. We prove that and and every ‐dominating family in the ground model is ‐unbounded in extension. Two questions are posed.

Gap‐2 morass‐definable η1‐orderings

AbstractWe prove that in the Cohen extension adding ℵ3 generic reals to a model of containing a simplified (ω1, 2)‐morass, gap‐2 morass‐definable η1‐orderings with cardinality ℵ3 are order‐isomorphic. Hence it is consistent that and that morass‐definable η1‐orderings with cardinality of the continuum are order‐isomorphic. We prove that there are ultrapowers of over ω that are gap‐2 morass‐definable. The constructions use a simplified gap‐2 morass, and commutativity with morass‐maps and morass‐embeddings, to extend a transfinite back‐and‐forth construction of order‐type ω1 to an order‐preserving bijection between objects of cardinality ℵ3.

Small MAD families whose Isbell--Mr\'owka space has pseudocompact hyperspace

Given a countable transitive model $M$ for ZFC+CH, we prove that one can produce a maximal almost disjoint family in $M$ whose Vietoris Hyperspace of its Isbell-Mr\'owka space is pseudocompact on every Cohen extension of $M$. We also show that a classical example of $\omega_1$-sized maximal almost disjoint family obtained by a forcing iteration of length $\omega_1$ in a model of non CH is such that the Vietoris Hyperspace of its Isbell-Mr\'owka space is pseudocompact.

Selective covering properties of product spaces, II: $\gamma $ spaces

We study productive properties of gamma spaces, and their relation to other, classic and modern, selective covering properties. Among other things, we prove the following results: 1. Solving a problem of F. Jordan, we show that for every unbounded tower set of reals X of cardinality aleph_1, the space Cp(X) is productively FU. In particular, the set X is productively gamma. 2. Solving problems of Scheepers and Weiss, and proving a conjecture of Babinkostova-Scheepers, we prove that, assuming CH, there are gamma spaces whose product is not even Menger. 3. Solving a problem of Scheepers-Tall, we show that the properties gamma and Gerlits--Nagy (*) are preserved by Cohen forcing. Moreover, every Hurewicz space that Remains Hurewicz in a Cohen extension must be Rothberger (and thus (*)). We apply our results to solve a large number of additional problems, and use Arhangel'skii duality to obtain results concerning local properties of function spaces and countable topological groups.

Order-isomorphic [formula omitted]-orderings in Cohen extensions

In this paper we prove that, in the Cohen extension (adding ℵ 2 -generic reals) of a model M of ZFC+CH containing a simplified ( ω 1 , 1 ) -morass, η 1 -orderings without endpoints having cardinality of the continuum, and satisfying specified technical conditions, are order-isomorphic. Furthermore, any order-isomorphism in M between countable subsets of the η 1 -orderings can be extended to an order-isomorphism between the η 1 -orderings in the Cohen extension of M . We use the simplified ( ω 1 , 1 ) -morass, and commutativity conditions with morass maps on terms in the forcing language, to extend countable partial functions on terms in the forcing language that are forced in all generic extensions to be order-preserving injections. This technique provides for the construction of functions in Cohen extensions adding ℵ 2 generic reals for which the only known arguments require transfinite constructions of order type no greater than ω 1 in models of ZFC+CH. The specific example presented in this paper is an extension of Tarski’s classic result that in models of ZFC+CH, η 1 -orderings are order-isomorphic.

Potential continuity of colorings

We say that a coloring \({c: [\kappa]^n\to 2}\) is continuous if it is continuous with respect to some second countable topology on κ. A coloring c is potentially continuous if it is continuous in some \({\aleph_1}\)-preserving extension of the set-theoretic universe. Given an arbitrary coloring \({c:[\kappa]^n\to 2}\), we define a forcing notion \({\mathbb P_c}\) that forces c to be continuous. However, this forcing might collapse cardinals. It turns out that \({\mathbb P_c}\) is c.c.c. if and only if c is potentially continuous. This gives a combinatorial characterization of potential continuity. On the other hand, we show that adding \({\aleph_1}\) Cohen reals to any model of set theory introduces a coloring \({c:[\aleph_1]^2 \to 2}\) which is potentially continuous but not continuous. \({\aleph_1}\) has no uncountable c-homogeneous subset in the Cohen extension, but such a set can be introduced by forcing. The potential continuity of c can be destroyed by some c.c.c. forcing.

The power set of ω Elementary submodels and weakenings of CH

We dene a new principle, SEP, which is true in all Cohen extensions of models of CH, and explore the relationship between SEP and other such principles. SEP is implied by each of CH , the weak Freeze{Nation property ofP(!), and the (@1;@0)-ideal property. SEP implies the principle C s(!2), but does not follow from C s(!2), or even C s (!2).

Notes on monadic logic. part A. Monadic theory of the real line

The second-order theory of the continuum in the Cohen extension of a set-theoretic universe is interpreted in the monadic theory of the real line and may be interpreted in the monadic topology of Cantor’s discontinuum as well.

On ideals of subsets of the plane and on Cohen reals

AbstractLet be any proper ideal of subsets of the real line R which contains all finite subsets of R. We define an ideal * ∣ as follows: X ∈ * ∣ if there exists a Borel set B ⊂ R × R such that X ⊂ B and for any x ∈ R we have {y ∈ R: 〈x, y〉 ∈ B} ∈ . We show that there exists a family ⊂ * ∣ of power ω1 such that ⋃ ∉ * ∣ .In the last section we investigate properties of ideals of Lebesgue measure zero sets and meager sets in Cohen extensions of models of set theory.

Thomas Jech and Karel Prikry. On ideals of sets and the power set operation. Bulletin of the American Mathematical Society, vol. 82 (1976), pp. 593–595. - F. Galvin, T. Jech, and M. Magidor. An ideal game. The journal of symbolic logic, vol. 43 (1978), pp. 284–292. - T. Jech, M. Magidor, W. Mitchell, and K. Prikry. Precipitous ideals. The journal

Thomas Jech and Karel Prikry. On ideals of sets and the power set operation. Bulletin of the American Mathematical Society, vol. 82 (1976), pp. 593–595. - F. Galvin, T. Jech, and M. Magidor. An ideal game. The journal of symbolic logic, vol. 43 (1978), pp. 284–292. - T. Jech, M. Magidor, W. Mitchell, and K. Prikry. Precipitous ideals. The journal

On a condition for Cohen extensions which preserve precipitous ideals

T. Jech and K. Prikry introduced the concept of precipitous ideals as a counterpart of measurable cardinals for small cardinals (Jech and Prikry [1]). To get a precipitous ideal on ℵ1( W. Mitchell used the Cohen extension by the standard Levy collapse that makes κ = ℵ1, where κ is a measurable cardinal in the ground model (Jech, Magidor, Mitchell and Prikry [2]). His proof essentially used the fact that , where is the notion of forcing of Levy collapse and j is the elementary embedding obtained by a normal ultrafilter on κ.On the other hand, we know that has the κ-chain condition. In this paper, we show that the κ-chain condition for notions of forcing plays an essential role for preserving normal precipitous ideals. Namely,Theorem 1. Let κ be a regular uncountable cardinal and I a normal ideal on κ. Let be a notion of forcing with the κ-chain condition. Then, I is precipitous iff ⊩“the ideal on κ generated by I is precipitous”.As an application of Theorem 1, we have the following theorem.Theorem 2. Let κ be a regular uncountable cardinal, and Γ be a κ-saturated normal ideal on κ. Then, {a < κ; the ideal of thin sets on a is precipitous} has either Γ-measure one or Γ-measure zero, and the ideal of thin sets on κ is precipitous iff {α< κ; the ideal of thin sets on α is precipitous) has Γ-measure one.

Models of Zermelo Frankel set theory as carriers for the mathematics of physics. II

This paper continues the study of the use of different models of ZF set theory as carriers for the mathematics of quantum mechanics. The basic tool used here is the construction of Cohen extensions of ZFC models by use of Boolean valued ZFC models [C=axiom of choice]. Let M be a standard transitive ZFC model. Inside M, B (ℋM) is the algebra of all bounded linear operators over some Hilbert space ℋM. It is shown that with each state ρ in B (ℋM) and projection operator ο in B (ℋM) one can associate a unique Boolean valued ZFC model Mℬρο. ℬρο is the algebra of all Borel subsets of {0,1}ω, the set of all infinite 0–1 sequences, modulo sets of Pρο =⊗pρο measure zero with pρο({1}) =Trρο in M. Let ΨM and ΦM be respective maps from the sets of state preparation and question measuring procedures into B (ℋM). Let M=M0, the minimal standard transitive ZFC model. It is then shown that with each state preparation procedure s - Dom(ΨM0) and each question measuring procedure q - Dom(ΦM0) and with each infinite repetition (tsq) of doing s and q at times t (0), t (1),..., if the definition of randomness is sufficiently strong, one can associate the Cohen extension M0[ψtsq] of M0 by ψtsq. ψtsq is the random outcome sequence associated with (tsq). A third condition, in addition to the two given in the previous paper, is then given which must be satisfied if a ZFC model M is to serve as a carrier for the mathematics of quantum mechanics. In essence it says that for each pair (tsq) and (wuk) of distinct infinite repetitions of doing s and q and of doing u and k with s, u - Dom(ΨM) and q, k - Dom(ΦM), the two outcome sequences ψtsq and ψwuk are mutually statistically independent. It is then shown that for a strong definition of independence, corresponding to the definition of randomness used previously, no Cohen extension M0[ψtsq] of M0 can serve as the carrier for the mathematics of quantum mechanics.

On cardinal collapsing with reals

We show that every uncountable regular cardinal can become {ie11-1} in a suitable Cohen extension by a real (set of integers) without destroying larger cardinals. The proof is analyzed to obtain results about the powers of Boolean algebras.

R. M. Solovay and S. Tennenbaum. Iterated Cohen extensions and Souslin's problem. Annals of mathematics, ser. 2 vol. 94 (1971), pp. 201–245.

We can characterize the real line, up to order isomorphism, by the following list of properties: R is order complete, order dense, has no first or last elements, and contains a dense subset. (One shows first that the dense is order isomorphic to the rationals, Q, and then that the ordered set is isomorphic to the Dedekind completion of its dense subset.) Souslin raised the question as to whether the countable dense subset condition could be replaced by the following consequence [15]: (*) Every disjoint family of non-empty open intervals is countable.' We use SH (Souslin's Hypothesis) to denote the following proposition: Every order complete order dense linearly ordered set satisfying ( * ) contains a dense subset. We use ZFC to denote Zermelo-Fraenkel set theory (including the axiom of choice). In [16], Tennenbaum constructed models of ZFC in which SH is false. Moreover, in one of these models the continuum hypothesis (CH) is false, while in another one, the generalized continuum hypothesis (GCH) is true. Thus SH is independent of the usual axioms of set theory. (This result is due, independently, to Jech [7].)2

Preserving some properties of large cardinals under mild Cohen extensions

Preserving some properties of large cardinals under mild Cohen extensions

Iterated Cohen Extensions and Souslin's Problem

We can characterize the real line, up to order isomorphism, by the following list of properties: R is order complete, order dense, has no first or last elements, and contains a dense subset. (One shows first that the dense is order isomorphic to the rationals, Q, and then that the ordered set is isomorphic to the Dedekind completion of its dense subset.) Souslin raised the question as to whether the countable dense subset condition could be replaced by the following consequence [15]: (*) Every disjoint family of non-empty open intervals is countable.' We use SH (Souslin's Hypothesis) to denote the following proposition: Every order complete order dense linearly ordered set satisfying ( * ) contains a dense subset. We use ZFC to denote Zermelo-Fraenkel set theory (including the axiom of choice). In [16], Tennenbaum constructed models of ZFC in which SH is false. Moreover, in one of these models the continuum hypothesis (CH) is false, while in another one, the generalized continuum hypothesis (GCH) is true. Thus SH is independent of the usual axioms of set theory. (This result is due, independently, to Jech [7].)2