Adding many random reals may add many Cohen reals

TL;DR

The paper investigates how forcing with two copies of random reals over an infinite cardinal can lead to the addition of many Cohen reals, revealing interactions between different forcing notions.

Contribution

It demonstrates that forcing with two products of random reals can add a generic filter for Cohen reals, showing a new interaction between these forcing notions.

Findings

Forcing with R(κ)×R(κ) adds a generic filter for C(κ).

Adding many random reals can result in adding many Cohen reals.

The result holds for any infinite cardinal κ.

Abstract

Let $\kappa$ be an infinite cardinal. Then, forcing with $\mathbb{R}(\kappa)$$\times$$\mathbb{R}(\kappa)$ adds a generic filter for $\mathbb{C}(\kappa);$ where $\mathbb{R}(\kappa)$ and $\mathbb{C}(\kappa)$ are the forcing notions for adding $\kappa$-many random reals and adding $\kappa$-many Cohen reals respectively.

Equations

$$[s_p]=\{x\in 2^{I\times \omega }:x\supseteq p\},$$

$$S\sim T⟺S△T\text{ is null, }$$

$$\mathbb{R}(I)=\mathbb{B}(I)/\sim .$$

$$[S]\le [T]⟺\mu (S\setminus T)=0.$$

$$p_k^{i,n}=[x\in 2^{I\times \omega }:x(i,n)=k].$$

$$\parallel\forall n<\omega,~{}\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{r}}}_{i}(n)=\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{F}}}(i,n)\parallel_{{\mathbb{R}}(I)}=1_{{\mathbb{R}}(I)}=[2^{I\times\omega}].$$

$$(x+y)(\alpha ,n)=x(\alpha ,n)+y(\alpha ,n)(\text{mod}2).$$

Full text

The author’s research has been supported by a grant from IPM (No. 95030417). He also thanks Moti Gitik for his useful comments and suggestions.

1. Introduction

For a cardinal $\kappa>0$ let ${\mathbb{R}}(\kappa)$ be the forcing notion for adding $\kappa$-many random reals and let ${\mathbb{C}}(\kappa)$ be the Cohen forcing for adding $\kappa$-many Cohen reals111See Section 2 for the definition of the forcing notions ${\mathbb{R}}(\kappa)$ and ${\mathbb{C}}(\kappa)$..

It is a well-know fact that forcing with ${\mathbb{R}}(1)\times{\mathbb{R}}(1)$ adds a Cohen real; in fact, if $r_{1},r_{2}$ are the added random reals, then $c=r_{1}+r_{2}$ is Cohen [1]. This in turn implies all reals $c+a,$ where $a\in{\mathbb{R}}^{V},$ are Cohen, and so, we have continuum many Cohen reals over $V$. However, the sequence $\langle c+a:a\in{\mathbb{R}}^{V}\rangle$ fails to be ${\mathbb{C}}((2^{\aleph_{0}})^{V})$-generic over $V$. In fact, there is no sequence $\langle c_{i}:i<\omega_{1}\rangle\in V[r_{1},r_{2}]$ of Cohen reals which is ${\mathbb{C}}(\omega_{1})$-generic over $V$.

In this paper, we extend the above mentioned result by showing that if we force with ${\mathbb{R}}(\kappa)\times{\mathbb{R}}(\kappa)$, then in the resulting extension, we can find a sequence $\langle c_{i}:i<\kappa\rangle$ of reals which is ${\mathbb{C}}(\kappa)$-generic over the ground model:

Theorem 1.1**.**

Let $\kappa$ be an infinite cardinal. Then, forcing with ${\mathbb{R}}(\kappa)\times{\mathbb{R}}(\kappa)$ adds a generic filter for ${\mathbb{C}}(\kappa).$

In Section 2, we briefly review the forcing notions ${\mathbb{C}}(\kappa)$ and ${\mathbb{R}}(\kappa)$. Then in Section 3, we state some results from analysis which are needed for the proof of above theorem and in Section 4, we give a proof of Theorem 1.1.

2. Cohen and Random forcings

In this section we briefly review the forcing notions ${\mathbb{C}}(\kappa)$ and ${\mathbb{R}}(\kappa)$, and present some of their properties.

2.1. Cohen forcing

Let $I$ be a non-empty set. The forcing notion ${\mathbb{C}}(I)$, the Cohen forcing for adding $|I|$-many Cohen reals is defined by

${\mathbb{C}}(I)=\{p:I\times\omega\to 2:|p|<\aleph_{0}\}$,

which is ordered by reverse inclusion.

Lemma 2.1**.**

${\mathbb{C}}(I)$* is c.c.c.*

Assume $G$ is ${\mathbb{C}}(I)$-generic over $V$, and set $F=\bigcup G:I\times\omega\to 2.$ For each $i\in I$ set $c_{i}:\omega\to 2$ be defined by $c_{i}(n)=F(i,n).$ Then:

Lemma 2.2**.**

For each $i\in I,c_{i}\in 2^{\omega}$ is a new real and for $i\neq j$ in $I,c_{i}\neq c_{j}$. Further, $V[G]=V[\langle c_{i}:i\in I\rangle].$

The reals $c_{i}$ are called Cohen reals. By $\kappa$-Cohen reals over $V$, we mean a sequence $\langle c_{i}:i<\kappa\rangle$ which is ${\mathbb{C}}(\kappa)$-generic over $V$.

2.2. Random forcing

In this subsection we briefly review random forcing. Suppose $I$ is a non-empty set and consider the product measure space $2^{I\times\omega}$ with the standard product measure $\mu_{I}$ on it. Let $\mathbb{B}(I)$ denote the class of Borel subsets of $2^{I\times\omega}$. Recall that $\mathbb{B}(I)$ is the $\sigma$-algebra generated by the basic open sets

[TABLE]

where $p\in{\mathbb{C}}(I)$. Also $\mu_{I}([s_{p}])=2^{-|p|}$.

For Borel sets $S,T\in\mathbb{B}(I)$ set

[TABLE]

where $S\bigtriangleup T$ denotes the symmetric difference of $S$ and $T$. The relation $\sim$ is easily seen to be an equivalence relation on $\mathbb{B}(I).$ Then ${\mathbb{R}}(I)$, the forcing for adding $|I|$-many random reals, is defined as

[TABLE]

Thus elements of ${\mathbb{R}}(I)$ are equivalent classes $[S]$ of Borel sets modulo null sets. The order relation is defined by

[TABLE]

The following fact is standard.

Lemma 2.3**.**

${\mathbb{R}}(I)$* is c.c.c.*

Using the above lemma, we can easily show that ${\mathbb{R}}(I)$ is in fact a complete Boolean algebra. Let \smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{\sim}}{{F}}} be an ${\mathbb{R}}(I)$-name for a function from $I\times\omega$ to $2$ such that for each $i\in I,n\in\omega$ and $k<2,$ \parallel\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{\sim}}{{F}}}(i,n)=k\parallel_{{\mathbb{R}}(I)}=p_{k}^{i,n}, where

[TABLE]

This defines ${\mathbb{R}}(I)$-names \smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{\sim}}{{r}}}_{i}\in 2^{\omega},i\in I, such that

[TABLE]

Lemma 2.4**.**

Assume $G$ is ${\mathbb{R}}(I)$-generic over $V$, and for each $i\in I$ set r_{i}=\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{\sim}}{{r}}}_{i}[G]. Then each $r_{i}\in 2^{\omega}$ is a new real and for $i\neq j$ in $I,r_{i}\neq r_{j}$. Further, $V[G]=V[\langle r_{i}:i\in I\rangle].$

The reals $r_{i}$ are called random reals. By $\kappa$-random reals over $V$, we mean a sequence $\langle r_{i}:i<\kappa\rangle$ which is ${\mathbb{R}}(\kappa)$-generic over $V$.

3. Some results from analysis

A famous theorem of Steinhaus [2] from 1920 asserts that if $A,B\subseteq\mathbb{R}^{n}$ are measurable sets with positive Lebesgue measure, then $A+B$ has an interior point; see also [3]. Here, we need a version of Steinhaus theorem for the space $2^{\kappa\times\omega}$.

For $S,T\subseteq 2^{\kappa\times\omega}$, set $S+T=\{x+y:x\in S$ and $y\in T\}$, where $x+y:\kappa\times\omega\to 2$ is defined by

[TABLE]

Note that the above addition is continuous.

Lemma 3.1**.**

Suppose $S\subseteq 2^{\kappa\times\omega}$ is Borel and non-null. Then $S-S$ contains an open set around the zero function [math].

Proof.

We follow [3]. Set $\mu=\mu_{\kappa}$ be the product measure on $2^{\kappa\times\omega}$. As $S$ is Borel and non-null, there is a compact subset of $S$ of positive $\mu$-measure, so may suppose that $S$ itself is compact. Let $U\supseteq S$ be an open set with $\mu(U)<2\cdot\mu(S).$ By continuity of addition, we can find an open set $V$ containing the zero function [math] such that $V+S\subseteq U.$

We show that $V\subseteq S-S$. Thus suppose $x\in V.$ Then $(x+S)\cap S\neq\emptyset,$ as otherwise we will have $(x+S)\cup S\subseteq U,$ and hence $\mu(U)\geq 2\cdot\mu(S),$ which is in contradiction with our choice of $U$. Thus let $y_{1},y_{2}\in S$ be such that $x+y_{1}=y_{2}.$ Then $x=y_{2}-y_{1}\in S-S$ as required. ∎

Similarly, we have the following:

Lemma 3.2**.**

Suppose $S,T\subseteq 2^{\kappa\times\omega}$ are Borel and non-null. Then $S+T$ contains an open set.

Suppose $S,T\subseteq 2^{\kappa\times\omega}$ are Borel and non-null. It follows from Lemma 3.2 that for some $p\in{\mathbb{C}}(\kappa),$ $[s_{p}]\subseteq S+T.$ Thus, by continuity of the addition, we can find $x\in S$ and $y\in T$ such that:

• •

$(x+y)\upharpoonright\operatorname{dom}(p)=p.$

• •

The sets $S\cap[s_{x\upharpoonright\operatorname{dom}(p)}]$ and $T\cap[s_{y\upharpoonright\operatorname{dom}(p)}]$ are Borel and non-null.

4. Proof of Theorem 1.1

In this section, we complete the proof of Theorem 1.1. Thus force with ${\mathbb{R}}(\kappa)\times{\mathbb{R}}(\kappa)$ and let $G\times H$ be generic over $V$. Let $\langle\langle r_{\alpha}:\alpha<\kappa\rangle,\langle s_{\alpha}:\alpha<\kappa\rangle\rangle$ be the sequence of random reals added by $G\times H.$

For $\alpha<\kappa$ set $c_{\alpha}=r_{\alpha}+s_{\alpha}$. The following completes the proof:

Lemma 4.1**.**

The sequence $\langle c_{\alpha}:\alpha<\kappa\rangle$ is a sequence of $\kappa$-Cohen reals over $V$.

Proof.

It suffices to prove the following:

For every $([S],[T])\in{{\mathbb{R}}}(\kappa)\times{\mathbb{R}}(\kappa)$, and every open dense subset $D\in V$

$(*)$          of ${{\mathbb{C}}}(\kappa)$,  there is $([\bar{S}],[\bar{T}])\leq([S],[T])$ such  that ([\bar{S}],[\bar{T}])\mbox{\rm|\kern-1.30005pt-}``\langle\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{\sim}}{{c}}}_{\alpha}:\alpha\in\kappa\rangle

extends some element of $D$”.

Thus fix $([S],[T])\in{{\mathbb{R}}}(\kappa)\times{\mathbb{R}}(\kappa)$ and $D\in V$ as above, where $S,T\subseteq 2^{\kappa\times\omega}$ are Borel and non-null. By Lemma 3.2 and the remarks after it, we can find $p\in{\mathbb{C}}(\kappa)$ and $(x,y)\in S\times T$ such that:

1. (1)

$[s_{p}]\subseteq S+T.$ 2. (2)

$(x+y)\upharpoonright\operatorname{dom}(p)=p.$ 3. (3)

The sets $S\cap[s_{x\upharpoonright\operatorname{dom}(p)}]$ and $T\cap[s_{y\upharpoonright\operatorname{dom}(p)}]$ are Borel and non-null.

Now let $q\in D$ be such that

$([S\cap[s_{x\upharpoonright\operatorname{dom}(p)}]],[T\cap[s_{y\upharpoonright\operatorname{dom}(p)}]])\Vdash$“$q\leq_{{\mathbb{C}}(\kappa)}p$”.

Using continuity of the addition and further application of Lemma 3.2 and the remarks after it, we can find $x^{\prime},y^{\prime}$ such that:

1. (4)

$x^{\prime}\in S\cap[s_{x\upharpoonright\operatorname{dom}(p)}]$ and $y^{\prime}\in T\cap[s_{y\upharpoonright\operatorname{dom}(p)}].$ 2. (5)

$(x^{\prime}+y^{\prime})\upharpoonright\operatorname{dom}(q)=q.$ 3. (6)

The sets $S\cap[s_{x^{\prime}\upharpoonright\operatorname{dom}(q)}]$ and $T\cap[s_{y^{\prime}\upharpoonright\operatorname{dom}(q)}]$ are Borel and non-null.

It is now clear that

$([S\cap[s_{x^{\prime}\upharpoonright\operatorname{dom}(q)}]],[T\cap[s_{y^{\prime}\upharpoonright\operatorname{dom}(q)}]])\Vdash$“\langle\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{\sim}}{{c}}}_{\alpha}:\alpha\in\kappa\rangle extends $q$”.

The result follows. ∎