Adding a lot of random reals by adding a few

TL;DR

This paper investigates how adding a small number of random reals to a model can result in a larger number of random reals being added to an extension, revealing complex interactions in set-theoretic models.

Contribution

It introduces a framework for understanding how adding a few random reals can lead to the addition of many more in extended models of ZFC.

Findings

Adding $ ext{κ}$-many random reals over $V_1$ can add $ ext{λ}$-many over $V$ with $ ext{λ} > ext{κ}$

The study characterizes conditions under which small additions produce large extensions

Provides new insights into the behavior of random reals in models of set theory

Abstract

We study pairs $(V, V_1)$ of models of $ZFC$ such that adding $\kappa$-many random reals over $V_1$ adds $\lambda$-many random reals over $V$, for some $\lambda > \kappa.$

Equations

$$[s]=\{x\in 2^{I\times \omega }:x\upharpoonright \mathrm{dom}(s)=s\},$$

$$S\sim T⟺S△T\text{ is null, }$$

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

$$[S]\le [T]⟺{\mu }_I(S\setminus T)=0.$$

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

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

$$\forall i\in I,\forall n<\omega ,\exists x\in T({\mu }_I(T\cap [x\upharpoonright \{(i,m):m<n\}])>0\text{ and }\forall m<n,x(i,m)=r_i(m)).$$

$$\bar{b}=[T\cap [x\upharpoonright \{(i,m):m<n\}]]$$

$$G=\{[T]\in \mathbb{R}(I):⟨r_i:i\in I⟩\text{ extends }[T]\}$$

$$c(j_{1},j_{2},l_{1},l_{2},n_{1},n_{2})=\parallel\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{s}}}_{\alpha_{j_{1},l_{1}}}(n_{1})\neq\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{s}}}_{\alpha_{j_{2},l_{2}}}(n_{2})\parallel.$$

$$T_b=\bigcap _{k<\omega }(2^{I\times \omega }\setminus [t_k])=\bigcap _{k<\omega }(\bigcup _{t\in {\Omega }_k}[t]).$$

$${\mu }_I(2^{I\times \omega }\setminus T_b)=\sum _{k<\omega }{2}^{-|t_k|}<1.$$

$$\{({\alpha }_{j_i,l_i},n_i):(j_1,j_2,l_1,l_2,n_1,n_2)\in X_k\text{ and }i=1,2\}\setminus \mathrm{dom}(t_k)$$

$$Y_k=\mathrm{dom}(t_k)\cup \{({\alpha }_{j_i,l_i},n_i):(j_1,j_2,l_1,l_2,n_1,n_2)\in X_k\text{ and }i=1,2\}.$$

$${\zeta }_k=|\{({\alpha }_{j_i,l_i},n_i):(j_1,j_2,l_1,l_2,n_1,n_2)\in X_k\text{ and }i=1,2\}\cap \mathrm{dom}(t_k)|.$$

$${\mu }_I(\bigcup _{t\in {\Omega }_k}(\bigcup _{t^{\prime }\in {\Lambda }_{k,t}}[t^{\prime }]))=\sum _{t\in {\Omega }_k}(\sum _{t^{\prime }\in {\Lambda }_{k,t}}{\mu }_I([t^{\prime }]))=(2^{|t_k|}-1)(2^{2{\eta }_k}-2^{{\xi }_k})2^{-(|t_k|+2{\eta }_k)}.$$

$${\mu }_I(\bar{T})=1-{\mu }_I(2^{I\times \omega }\setminus \bar{T})>0.$$

$$\bar{S}=\{y\in S^{\prime \prime }:y({ı}_{{\bar{j}}_1,{\bar{l}}_1},{ı}_{{\bar{j}}_1,{\bar{l}}_1},0)=y({ı}_{{\bar{j}}_2,{\bar{l}}_2},{ı}_{{\bar{j}}_2,{\bar{l}}_2},0)=1\}.$$

$$\bar{p}\text{~{}}\Vdash\text{``}\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{k}}}(\imath_{\bar{j}_{1},\bar{l}_{1}})=\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{k}}}(\imath_{\bar{j}_{2},\bar{l}_{2}})=0\text{''}.$$

$$b\nleq\parallel\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{s}}}_{\alpha_{\bar{j}_{1},\bar{l}_{1}}}(\bar{n}_{1})\neq\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{s}}}_{\alpha_{\bar{j}_{2},\bar{l}_{2}}}(\bar{n}_{2})\parallel.$$

$$\Lambda =\{(j,l):(j,l)\text{ appears in }\Delta \}.$$

$$f_{{\alpha }_1^{\ast }}(n),f_{{\alpha }_{j_1,l_1}}(n),f_{{\alpha }_{j_2,l_2}}(n),f_{{\alpha }_2^{\ast }}(n),$$

$$p_{1}\Vdash``k(\imath_{jl})=\min\{k<\omega:\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{r}}}_{\imath_{jl},\imath_{jl}}(k)=1\}=m^{*}~{}\text{''}.$$

$$S_{p_1}=\{y\in 2^{(\kappa \times \kappa )\times \omega }:\forall (j,l)\in \Lambda [(\forall n<m^{\ast },y({ı}_{jl},{ı}_{jl},n)=0)\text{ and }y({ı}_{jl},{ı}_{jl},m^{\ast })=1]\}.$$

$$p_{1}\Vdash``k(\imath_{jl})=\min\{k<\omega:\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{$\sim$}}{{r}}}_{\imath_{jl},\imath_{jl}}(k)=1\}=m^{*}~{}\text{''},$$

$$S_n=\{y\in S_{p_1}:\exists x\in T_b,\forall (j,l)\in \Lambda ,\forall m<n(y(f_{{\alpha }_{jl}}(m^{\ast }+m),f_{{\alpha }_{jl}}(m^{\ast }+m),m)=x({\alpha }_{{ı}_{jl}},m))\}.$$

$$W_n=\{(f_{{\alpha }_{jl}}(m^{\ast }+m),f_{{\alpha }_{jl}}(m^{\ast }+m),m):(j,l)\in \Lambda ,m<n\}$$

$${\Delta }_n=\{t:W_n\to 2\mid \exists x\in T_b,\forall (j,l)\in \Lambda ,\forall m<n,(y(f_{{\alpha }_{jl}}(m^{\ast }+m),f_{{\alpha }_{jl}}(m^{\ast }+m),m)=x({\alpha }_{{ı}_{jl}},m))\}.$$

$$\forall m<n(y(f_{{\alpha }_{jl}}(m^{\ast }+m),f_{{\alpha }_{jl}}(m^{\ast }+m),m)=x({\alpha }_{{ı}_{jl}},m)).$$

$$I=I\cap \bigcup _{\sigma <\lambda }{S}_{\sigma }=\bigcup _{j<\bar{\omega }}(I\cap S_{{\sigma }_j}),$$

Full text

The second author’s research has been supported by a grant from IPM (No. 96030417).

1. Introduction

In [1] and [2], we studied pairs $(V,V_{1})$ of models of $ZFC$ such that adding $\kappa$-many Cohen reals over $V_{1}$ adds $\lambda$-many Cohen reals over $V$, for some $\lambda>\kappa.$ In this paper we prove similar results for random forcing, by producing pairs $(V,V_{1})$ of models of $ZFC$ such that adding $\kappa$-many random reals over $V_{1}$ adds $\lambda$-many random reals over $V$, where by $\kappa$-random reals over $V$ we mean a sequence $\langle r_{i}:i<\kappa\rangle$ which is ${\mathbb{R}}(\kappa)$-generic over $V$, and ${\mathbb{R}}(\kappa)$ is the usual forcing notion for adding $\kappa$-many random reals (see Section 2). The proofs are more involved than those given in [1] and [2] for Cohen reals. This is because random reals, in contrast to Cohen reals, may depend on $\omega$-many coordinates, instead of finitely many as in the Cohen case. Also the proofs in [1] and [2] were based on the fact that the product of Cohen forcing with itself is essentially the same as Cohen forcing, while this is not true in the case of random forcing.

2. Random real forcing

In this section we briefly review random forcing and refer the reader to [3] for more details. 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 $2^{I\times\omega}$. Note that the sets of the form

[TABLE]

where $s:I\times\omega\to 2$ is a finite partial function, form a basis of open sets of $2^{I\times\omega}.$

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 equivalence classes $[S]$ of Borel sets modulo null sets. The order relation is defined by

[TABLE]

The following fact is standard.

Lemma 2.1**.**

${\mathbb{R}}(I)$* 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 2.2**.**

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 $V$.

Given $b=[T]\in{\mathbb{R}}(I)$ and $|I|$-random reals $\langle r_{i}:i\in I\rangle$ over $V$, we say $\langle r_{i}:i\in I\rangle$ extends $b$ if

[TABLE]

This simply says that if $i$ and $n$ are given, then we can extend $b$ to some

[TABLE]

such that $\bar{b}$ decides $r_{i}\upharpoonright n.$ In fact, $\bar{b}\Vdash$“\forall m<n,~{}\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{\sim}}{{r}}}_{i}(m)=x(i,m)”. Note that if $\langle r_{i}:i<\kappa\rangle$ is a sequence of $|I|$-random reals generated by $G$, then

[TABLE]

The next lemma follows from Lemma 2.1.

Lemma 2.3**.**

The sequence $\langle r_{i}:i<\kappa\rangle$ is ${\mathbb{R}}(\kappa)$-generic over $V$ iff for each countable set $I\subseteq\kappa,I\in V,$ the sequence $\langle r_{i}:i\in I\rangle$ is ${\mathbb{R}}(I)$-generic over $V$.

3. The first general fact about adding many random reals

In this section we prove the following theorem, which is an analogue of Theorem 2.1 from [1], and use it to get some consequences.

Theorem 3.1**.**

Let $V_{1}$ be an extension of $V$. Suppose that in $V_{1}:$

1. $(a)$

$\kappa<\lambda$* are infinite cardinals,* 2. $(b)$

$\lambda$* is regular,* 3. $(c)$

there exists an increasing sequence $\langle\kappa_{n}:n<\omega\rangle$ cofinal in $\kappa.$ In particular $cf(\kappa)=\omega,$ 4. $(d)$

there exists an increasing (mod finite) sequence $\langle f_{\alpha}:\alpha<\lambda\rangle$ of functions in the product $\displaystyle\prod_{n<\omega}(\kappa_{n+1}\setminus\kappa_{n}),$ 5. $(e)$

there exists a club $C\subseteq\lambda$ which avoids points of countable $V$-cofinality.

Then adding $\kappa$-many random reals over $V_{1}$ produces $\lambda$-many random reals over $V$.

Proof.

There are two cases to consider: $(1):$ $\lambda=\kappa^{+}$ and $(2):$ $\lambda>\kappa^{+}.$ We give a proof for the first case, as the second case can be proved similarly, using ideas from [1, Theorem 2.1] (combined with the proof of the first case given below). We may assume, for clarity of exposition, that $\min(C)=0$.

Thus assume that $\lambda=\kappa^{+},$ and force to add $\kappa$-many random reals over $V_{1}$. We denote them by $\langle r_{\imath,\tau}:\imath,\tau<\kappa\rangle$. Also let $\langle f_{\alpha}:\alpha<\kappa^{+}\rangle\in V_{1}$ be an increasing (mod finite) sequence in $\displaystyle\prod_{n<\omega}(\kappa_{n+1}\setminus\kappa_{n})$. We define a sequence $\langle s_{\alpha}:\alpha<\kappa^{+}\rangle$ of reals as follows:

Assume $\alpha<\kappa^{+}$. Let $\alpha^{*}$ and $\alpha^{**}$ be two consecutive points of $C$ so that $\alpha^{*}\leq\alpha<\alpha^{**}$. Let $\langle\alpha_{\imath}:\imath<\kappa\rangle$ be some fixed enumeration of the interval $[\alpha^{*},\alpha^{**})$ with $\alpha_{0}=\alpha^{*}$. Then for some $\imath<\kappa$, $\alpha=\alpha_{\imath}$. Let $k(\imath)=\min\{k<\omega:r_{\imath,\imath}(k)=1\}$. Set

$\forall n<\omega$, $s_{\alpha}(n)=r_{f_{\alpha}(k(\imath)+n),f_{\alpha}(k(\imath)+n)}(0)$.

The following lemma completes the proof.

Lemma 3.2**.**

$\langle s_{\alpha}:\alpha<\kappa^{+}\rangle$* is a sequence of $\kappa^{+}$-many random reals over $V$.*

Proof.

First, we may assume that $\langle r_{\imath,\tau}:\imath,\tau<\kappa\rangle$ is ${{\mathbb{R}}}(\kappa\times\kappa)$-generic over $V_{1}$. By Lemma 2.3, it suffices to show that for any countable set $I\subseteq\kappa^{+}$, $I\in V$, the sequence $\langle s_{\alpha}:\alpha\in I\rangle$ is ${{\mathbb{R}}}(I)$-generic over $V$. Thus it suffices to prove the following:

for every $p\in{{\mathbb{R}}}(\kappa\times\kappa)$ and every open dense subset $D\in V$

$(*)$           of ${{\mathbb{R}}}(I)$,  there is $\bar{p}\leq p$ such  that \bar{p}\mbox{\rm|\kern-1.30005pt-}``\langle\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{\sim}}{{s}}}_{\alpha}:\alpha\in I\rangle extends

some element of $D$”.

Let $p$ and $D$ be as above. For simplicity suppose that $p=1_{{\mathbb{R}}(\kappa\times\kappa)}=[2^{(\kappa\times\kappa)\times\omega}]$. By $(e)$ there are only finitely many $\alpha^{*}\in C$ such that $I\cap[\alpha^{*},\alpha^{**})\neq\emptyset$, where $\alpha^{**}=\min(C\setminus(\alpha^{*}+1))$. For simplicity suppose that there are exactly two $\alpha^{*}_{1}<\alpha^{*}_{2}$ in $C$ with this property. Let $n^{*}<\omega$ be such that for all $n\geq n^{*}$, $f_{\alpha^{*}_{1}}(n)<f_{\alpha^{*}_{2}}(n)$.

Let $b=[T_{b}]\in D,$ where $T_{b}\subseteq 2^{I\times\omega}.$ For $j\in\{1,2\}$, let $\{\alpha_{j_{l}}:l<k_{j}\leq\omega\}$ be an enumeration of $I\cap[\alpha^{*}_{j},\alpha^{**}_{j}).$ For $j\in\{1,2\}$ and $l<k_{j}$ let $\alpha_{j,l}=\alpha_{\imath_{jl}}$ where $\imath_{jl}<\kappa$ is the index of $\alpha_{j,l}$ in the enumeration of the interval $[\alpha^{*}_{j},\alpha^{**}_{j})$ considered above.

For every $j_{1},j_{2}\in\{1,2\},l_{1}<k_{j_{1}},l_{2}<k_{j_{2}}$ and $n_{1},n_{2}<\omega$ set

[TABLE]

Claim 3.3**.**

The set $\Delta=\{(j_{1},j_{2},l_{1},l_{2},n_{1},n_{2}):b\leq c(j_{1},j_{2},l_{1},l_{2},n_{1},n_{2})\}$ is finite. Also, $(j_{1},j_{2},l_{1},l_{2},n_{1},n_{2})\in\Delta$ implies $(j_{2},j_{1},l_{2},l_{1},n_{2},n_{1})\in\Delta.$

Proof.

Recall that $b=[T_{b}].$ By shrinking $T_{b}$ if necessary, we can assume that $T_{b}$ is closed. Then $2^{I\times\omega}\setminus T_{b}$ is open, so there are finite partial functions $t_{k}:I\times\omega\to 2$ such that $2^{I\times\omega}\setminus T_{b}=\bigcup_{k<\omega}[t_{k}]$ and for $k\neq l,~{}[t_{k}]\cap[t_{l}]=\emptyset$. For each $k$ set $\Omega_{k}=\{t:\operatorname{dom}(t)=\operatorname{dom}(t_{k})$ and $t\neq t_{k}\}$. Then each $\Omega_{k}$ is finite and $2^{I\times\omega}\setminus[t_{k}]=\bigcup_{t\in\Omega_{k}}[t]$. So

[TABLE]

Also, as $\mu_{I}(T_{b})>0,$ we have

[TABLE]

Note that $\mu_{I}(T_{b})=1-\sum_{k<\omega}2^{-|t_{k}|}>0.$ Fix an increasing sequence $\langle\eta_{k}:k<\omega\rangle$ of natural numbers such that

$({\dagger})$                                   $\sum_{k<\omega}2^{-\eta_{k}}<\dfrac{1-\mu_{I}(2^{I\times\omega}\setminus T_{b})}{1+\mu_{I}(2^{I\times\omega}\setminus T_{b})}.$

Assume, towards a contradiction, that the set $\Delta$ is infinite. For each $k<\omega$ let $X_{k}$ be a finite subset of $\Delta$ such that:

1. (1)

$(j_{1},j_{2},l_{1},l_{2},n_{1},n_{2})\in X_{k}\implies$ at least one of $(\alpha_{j_{1},l_{1}},n_{1})$ or $(\alpha_{j_{2},l_{2}},n_{2})$ is not in $\operatorname{dom}(t_{k})$. 2. (2)

The set

[TABLE]

has size $2\eta_{k}$. 3. (3)

$X_{k}$’s, for $k<\omega,$ are pairwise disjoint.

Set

[TABLE]

For each $t\in\Omega_{k}$ let

$\Lambda_{k,t}=\{t^{\prime}:Y_{k}\to 2:t^{\prime}\supseteq t$ and $\exists(j_{1},j_{2},l_{1},l_{2},n_{1},n_{2})\in X_{k},t^{\prime}(\alpha_{j_{1},l_{1}},n_{1})=t^{\prime}(\alpha_{j_{2},l_{2}},n_{2})\}$.

Note that each $t^{\prime}\in\Lambda_{k,t}$ is well-defined by clause (1) above. Let

[TABLE]

Then note that $2\eta_{k}=2\xi_{k}+\zeta_{k},$ where $\xi_{k}$ is the number of those $(j_{1},j_{2},l_{1},l_{2},n_{1},n_{2})\in X_{k}$ such that both $(\alpha_{j_{1},l_{1}},n_{1})$ and $(\alpha_{j_{2},l_{2}},n_{2})$ are not in $\operatorname{dom}(t_{k})$.

Set $\bar{T}=\bigcap_{k<\omega}(\bigcup_{t\in\Omega_{k}}(\bigcup_{t^{\prime}\in\Lambda_{k,t}}[t^{\prime}])).$ Clearly, $|\Omega_{k}|=2^{|t_{k}|}-1,$ $|\Lambda_{k,t}|=2^{2\eta_{k}}-2^{\xi_{k}}$ and for each $t^{\prime}\in\Lambda_{k,t},|t^{\prime}|=|t|+2\eta_{k}=|t_{k}|+2\eta_{k}$, and so

[TABLE]

It follows that

$\mu_{I}(2^{I\times\omega}\setminus\bar{T})\leq\sum_{k<\omega}(1-(2^{|t_{k}|}-1)(2^{2\eta_{k}}-2^{\xi_{k}})2^{-(|t_{k}|+2\eta_{k})})$

$=\sum_{k<\omega}(2^{\xi_{k}-2\eta_{k}}+2^{-|t_{k}|}-2^{\xi_{k}-|t_{k}|-2\eta_{k}})$

$\leq\sum_{k<\omega}(2^{\xi_{k}-2\eta_{k}}+2^{-|t_{k}|}+2^{\xi_{k}-|t_{k}|-2\eta_{k}})$

$=\sum_{k<\omega}2^{\xi_{k}-2\eta_{k}}+\sum_{k<\omega}2^{-|t_{k}|}+\sum_{k<\omega}2^{\xi_{k}-|t_{k}|-2\eta_{k}}$

$\leq\sum_{k<\omega}2^{\eta_{k}-2\eta_{k}}+\sum_{k<\omega}2^{-|t_{k}|}+\sum_{k<\omega}2^{\eta_{k}-|t_{k}|-2\eta_{k}}$ (as $\xi_{k}\leq\eta_{k}$)

$=\sum_{k<\omega}2^{-|t_{k}|}+\sum_{k<\omega}2^{-\eta_{k}}+\sum_{k<\omega}2^{-|t_{k}|-\eta_{k}}$

$\leq\sum_{k<\omega}2^{-|t_{k}|}+\sum_{k<\omega}2^{-\eta_{k}}+(\sum_{k<\omega}2^{-|t_{k}|})(\sum_{k<\omega}2^{-\eta_{k}})$

$\leq\mu_{I}(2^{I\times\omega}\setminus T_{b})+\sum_{k<\omega}2^{-\eta_{k}}+\mu_{I}(2^{I\times\omega}\setminus T_{b})(\sum_{k<\omega}2^{-\eta_{k}})$

$<1$ (by $({\dagger})$).

Hence

[TABLE]

Set $\bar{b}=[\bar{T}],$ Then $\bar{b}\in{\mathbb{R}}(I)$ and $\bar{b}\leq b.$ Also note that:

$\forall x\in\bar{T},\forall k<\omega,\exists(j_{1},j_{2},l_{1},l_{2},n_{1},n_{2})\in X_{k},~{}x(\alpha_{j_{1},l_{1}},n_{1})=x(\alpha_{j_{2},l_{2}},n_{2}).$

Let $S^{\prime}$ consists of those $y\in 2^{(\kappa\times\kappa)\times\omega}$ such that for some $k<\omega$, some $(j_{1},j_{2},l_{1},l_{2},n_{1},n_{2})\in X_{k}$ and some $x\in\bar{T}$

1. (1)

$y(f_{\alpha_{j_{1}l_{1}}}(n_{1}),f_{\alpha_{j_{1}l_{1}}}(n_{1}),n_{1})=x(\alpha_{j_{1}l_{1}},n_{1})$. 2. (2)

$y(f_{\alpha_{j_{2}l_{2}}}(n_{2}),f_{\alpha_{j_{2}l_{2}}}(n_{2}),n_{2})=x(\alpha_{j_{2}l_{2}},n_{2})$. 3. (3)

$x(\alpha_{j_{1}l_{1}},n_{1})=x(\alpha_{j_{2}l_{2}},n_{2}).$

Clearly, $\mu_{\kappa\times\kappa}(S^{\prime})>0$. For each $y\in S^{\prime}$ let $k_{y}$ denote the least $k$ as above. Similarly, let $(j^{y}_{1},j^{y}_{2},l^{y}_{1},l^{y}_{2},n^{y}_{1},n^{y}_{2})$ denote the least $(j_{1},j_{2},l_{1},l_{2},n_{1},n_{2})\in X_{k}$ as above (with respect to some fixed well-ordering of $\Delta$). For some $\bar{k}<\omega$ and $(\bar{j}_{1},\bar{j}_{2},\bar{l}_{1},\bar{l}_{2},\bar{n}_{1},\bar{n}_{2})\in X_{k}$, the set $S^{\prime\prime}=\{y\in S^{\prime}:k_{y}=\bar{k}$ and $(j^{y}_{1},j^{y}_{2},l^{y}_{1},l^{y}_{2},n^{y}_{1},n^{y}_{2})=(\bar{j}_{1},\bar{j}_{2},\bar{l}_{1},\bar{l}_{2},\bar{n}_{1},\bar{n}_{2})\}$ has positive measure. Let

[TABLE]

Then \mu_{\kappa\times\kappa}(\bar{S})=$$1\over 4$$\mu_{\kappa\times\kappa}(S^{\prime\prime})>0 and if $\bar{p}=[\bar{S}],$ then $\bar{p}\in{\mathbb{R}}(\kappa\times\kappa)$ and

[TABLE]

For each $y\in\bar{S}$, if $x$ (with $\bar{k}$ and $\bar{u}$) is a witness as above, then

\bar{p}\Vdash~{}\text{``}\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{\sim}}{{s}}}_{\alpha_{\bar{j}_{1},\bar{l}_{1}}}(\bar{n}_{1})=\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{\sim}}{{r}}}_{f_{\alpha_{\bar{j}_{1},\bar{l}_{1}}}(\bar{n}_{1}),f_{\alpha_{\bar{j}_{1},\bar{l}_{1}}}(\bar{n}_{1})}(0)

$=y(f_{\alpha_{\bar{j}_{1},\bar{l}_{1}}}(\bar{n}_{1}),f_{\alpha_{\bar{j}_{1},\bar{l}_{1}}}(\bar{n}_{1}),\bar{n}_{1})$

$=x(\alpha_{\bar{j}_{1},\bar{l}_{1}},\bar{n}_{1})$ (by $(1)$)

$=x(\alpha_{\bar{j}_{2},\bar{l}_{2}},\bar{n}_{2})$ (by $(3)$)

$=y(f_{\alpha_{\bar{j}_{2},\bar{l}_{2}}}(\bar{n}_{2}),f_{\alpha_{\bar{j}_{2},\bar{l}_{2}}}(\bar{n}_{2}),\bar{n}_{2})$ (by $(2)$)

=\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{\sim}}{{r}}}_{f_{\alpha_{\bar{j}_{2},\bar{l}_{2}}}(\bar{n}_{2}),f_{\alpha_{\bar{j}_{2},\bar{l}_{2}}}(\bar{n}_{2})}(0)

=\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{\sim}}{{s}}}_{\alpha_{\bar{j}_{2},\bar{l}_{2}}}(\bar{n}_{2})\text{''}.

So \bar{b}\nleq\parallel\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{\sim}}{{s}}}_{\alpha_{\bar{j}_{1},\bar{l}_{1}}}(\bar{n}_{1})\neq\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{\sim}}{{s}}}_{\alpha_{\bar{j}_{2},\bar{l}_{2}}}(\bar{n}_{2})\parallel, and since $\bar{b}\leq b,$ we have

[TABLE]

It follows that $(\bar{j}_{1},\bar{j}_{2},\bar{l}_{1},\bar{l}_{2},\bar{n}_{1},\bar{n}_{2})\notin\Delta$, which is a contradiction. The second part of the claim is evident and the claim follows. ∎

Say that $(j,l)$ appears in $\Delta$ if $(j,l)=(j_{1},l_{1})$ for some $(j_{1},j_{2},l_{1},l_{2},n_{1},n_{2})\in\Delta.$ Also set

[TABLE]

Then $|\Lambda|\leq 2|\Delta|$ is finite. Let $m^{*}$, with $n^{*}\leq m^{*}<\omega$, be such that for all $n\geq m^{*}$ all of the values

[TABLE]

are all different, where $(j_{1},l_{1}),(j_{2},l_{2})\in\Lambda.$

Claim 3.4**.**

There exists $p_{1}\leq p$ such that for all $(j,l)\in\Lambda$,

[TABLE]

Proof.

Let $S_{p_{1}}\subseteq 2^{(\kappa\times\kappa)\times\omega}$ be defined by

[TABLE]

Then $\mu_{\kappa\times\kappa}(S_{p_{1}})=2^{-|\Lambda|(m^{*}+1)}>0$, so $p_{1}=[S_{p_{1}}]\in{\mathbb{R}}(\kappa\times\kappa)$. Further, for all $(j,l)\in\Lambda$ and $n<m^{*}$, p_{1}\Vdash``\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{\sim}}{{r}}}_{\imath_{jl},\imath_{jl}}(n)=0”, p_{1}\Vdash``\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{\sim}}{{r}}}_{\imath_{jl},\imath_{jl}}(m^{*})=1” and thus for $(j,l)\in\Lambda$,

[TABLE]

as required ∎

Before we continue, let us make an assumption on $T_{b}.$ For each $n<\omega$ let $\Phi_{n}=\{(\alpha_{\imath_{jl}},m):(j,l)\in\Lambda,m<n\}\subseteq I\times\omega$. Then for a countable subset $T^{\prime}$ of $I\times\omega$, $\{x\upharpoonright\Phi_{n}:x\in T^{\prime}\}=2^{H_{n}}$, for all $n<\omega$. As $[T_{b}]=[T_{b}\cup T^{\prime}],$ let us assume without loss of generality that $T^{\prime}\subseteq T_{b}.$

Set

$J=\{f_{\alpha_{jl}}(m^{*}+m):(j,l)\in\Lambda$ and $m<\omega\}\subseteq\kappa.$

Note that by our choice of $m^{*},$ for all $m$ and all $(j_{1},l_{1}),(j_{2},l_{2})\in\Lambda$, $f_{\alpha_{j_{1}l_{1}}}(m^{*}+m)\neq f_{\alpha_{j_{2}l_{2}}}(m^{*}+m)$. Set

$\bar{S}=\{y\in S_{p_{1}}:\forall n<\omega,\exists x\in T_{b},\forall(j,l)\in\Lambda,\forall m<n$

$(~{}y(f_{\alpha_{jl}}(m^{*}+m),f_{\alpha_{jl}}(m^{*}+m),m)=x(\alpha_{\imath_{jl}},m)~{})\}.$

By the above remarks, $\bar{S}$ is well-defined. We also have $\bar{S}=\bigcap_{n<\omega}S_{n},$ where

[TABLE]

Let

[TABLE]

and

[TABLE]

By our assumption, $T^{\prime}\subseteq T_{b},|\Delta_{n}|=2^{|W_{n}|}$, and hence, $\mu_{\kappa\times\kappa}(\bigcup_{t\in\Delta_{n}}[t])=\sum_{t\in\Delta_{n}}2^{|t|}=2^{|W_{n}|}2^{-|W_{n}|}=1$. We have $S_{n}=S_{p_{1}}\cap\bigcup_{t\in\Delta_{n}}[t]$, so

$\mu_{\kappa\times\kappa}(S_{n})=\mu_{\kappa\times\kappa}(S_{p_{1}})+\mu_{\kappa\times\kappa}(\bigcup_{t\in\Delta_{n}}[t])-\mu_{\kappa\times\kappa}(S_{p_{1}}\cup\bigcup_{t\in\Delta_{n}}[t])=\mu_{\kappa\times\kappa}(S_{p_{1}}).$

It follows that $\mu_{\kappa\times\kappa}(S_{p_{1}}\setminus S)=\mu_{\kappa\times\kappa}(\bigcup_{n<\omega}(S_{p_{1}}\setminus S_{n})\leq\sum_{n<\omega}\mu_{\kappa\times\kappa}(S_{p_{1}}\setminus S_{n})=0$, and so $\mu_{\kappa\times\kappa}(S)=\mu_{\kappa\times\kappa}(S_{p_{1}})>0$. Let $\bar{p}=[\bar{S}]$. Then $\bar{p}\in{\mathbb{R}}(\kappa\times\kappa)$ and $\bar{p}\leq p.$

Claim 3.5**.**

$\bar{p}\Vdash$“\langle\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{\sim}}{{s}}}_{\alpha_{jl}}:(j,l)\in\Lambda\rangle extends $b$”.

Proof.

Suppose $(j,l)\in\Lambda$ and $n<\omega.$ Let $y\in\bar{S}$. Thus we can find $x\in T_{b}$ such that

[TABLE]

But then

\bar{p}\Vdash~{}\text{``}\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{\sim}}{{s}}}_{\alpha}(m)=\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{\sim}}{{s}}}_{\alpha_{jl}}(m)

=\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{\sim}}{{r}}}_{f_{\alpha_{jl}}(m^{*}+m),f_{\alpha_{jl}}(m^{*}+m)}(0)

$=y(f_{\alpha_{jl}}(m^{*}+m),f_{\alpha_{jl}}(m^{*}+m),0)$

$=x(\alpha_{jl},m)$

$=x(\alpha,m)\text{''.}$

The result follows. ∎

We now consider those $(j,l)$’s, $j\in\{1,2\},l<k_{j}$, which do not appear in $\Delta$. Fix such a pair $(j,l)$. Also let $n<\omega.$ Then there is $(j_{1},l_{1})\in\Lambda$ such that for each $m<n,$ $b\nleq c(j,j_{1},l,l_{1},m,m)$, i.e., $b\nVdash$“\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{\sim}}{{s}}}_{\alpha_{j,l}}(m)\neq\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{\sim}}{{s}}}_{\alpha_{j_{1},l_{1}}}(m)”. So there exists $b_{jln}=[T_{jln}]\leq b$ such that $\forall m<n,~{}b_{jln}\Vdash$“\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{\sim}}{{s}}}_{\alpha_{j,l}}(m)=\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{\sim}}{{s}}}_{\alpha_{j_{1},l_{1}}}(m)”.

Note that $\mu_{I}(T_{jln}\setminus T_{b})=0$. Since there are only countably many such tuples $(j,l,n),$

$\mu_{I}(\bigcup_{n<\omega,(j,l)\in\Lambda}T_{jln}\setminus T_{b})=0.$

This implies $[T_{b}]=[T_{b}\cup\bigcup_{n<\omega,(j,l)\in\Lambda}T_{jln}],$ so without loss of generality, each $T_{jln}$ is contained in $T_{b}$ where $n<\omega$ and $(j,l)\in\Lambda.$ Now Claim 3.5 implies the following:

Claim 3.6**.**

$\bar{p}\Vdash$“\langle\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{\sim}}{{s}}}_{\alpha}:\alpha\in I\rangle extends $b$”.

$(*)$ follows, which completes the proof of Lemma 3.2. ∎

Theorem 3.1 follows. ∎

The next theorem follows immediately from Theorem 3.1 and the arguments from [1].

Theorem 3.7**.**

• (a)

Suppose that $V$ satisfies $GCH$, $\kappa=\bigcup_{n<\omega}\kappa_{n}$ and $\bigcup_{n<\omega}o(\kappa_{n})=\kappa$ (where $o(\kappa_{n})$ is the Mitchell order of $\kappa_{n}$). Then there exists a cardinal preserving generic extension $V_{1}$ of $V$ satisfying $GCH$ and having the same reals as $V$ does, so that adding $\kappa$-many random reals over $V_{1}$ produces $\kappa^{+}$-many random reals over $V$.

• (b)

Suppose $V$ is a model of $GCH$. Then there is a generic extension $V_{1}$ of $V$ satisfying $GCH$ so that the only cardinal of $V$ which is collapsed in $V_{1}$ is $\aleph_{1}$ and such that adding $\aleph_{\omega}$-many random reals to $V_{1}$ produces $\aleph_{\omega+1}$-many of them over $V$.

• (c)

Suppose $V$ satisfies $GCH$. Then there is a generic extension $V_{1}$ of $V$ satisfying $GCH$ and having the same reals as $V$ does, so that the only cardinals of $V$ which are collapsed in $V_{1}$ are $\aleph_{2}$ and $\aleph_{3}$ and such that adding $\aleph_{\omega}$-many random reals to $V_{1}$ produces $\aleph_{\omega+1}$-many of them over $V$.

• (d)

Suppose that $\kappa$ is a strong cardinal, $\lambda\geq\kappa$ is regular and $GCH$ holds. Then there exists a cardinal preserving generic extension $V_{1}$ of $V$ having the same reals as $V$ does, so that adding $\kappa$-many random reals over $V_{1}$ produces $\lambda$-many of them over $V$.

• (e)

Suppose that there is a strong cardinal and $GCH$ holds. Let $\alpha<\omega_{1}$. Then there is a model $V_{1}\supset V$ having the same reals as $V$ and satisfying $GCH$ below $\aleph^{V_{1}}_{\omega}$ such that adding $\aleph^{V_{1}}_{\omega}$-many random reals to $V_{1}$ produces $\aleph^{V_{1}}_{\alpha+1}$-many of them over $V$.

We can also use ideas of the proof of Theorem 3.1 to get the following theorem, which is an analogue of [1, Theorem 3.1] for random reals.

Theorem 3.8**.**

Suppose that $V$ satisfies $GCH$. Then there is a cofinality preserving generic extension $V_{1}$ of $V$ satisfying $GCH$ so that adding a random real over $V_{1}$ produces $\aleph_{1}$-many random reals over $V$.

4. The second general fact about adding many random reals

In this section, we prove our second general result which is an analogue of Theorem 2.1 form [2]. Then we use the result to obtain similar results as in [2] for random reals.

Theorem 4.1**.**

Suppose $\kappa<\lambda$ are infinite (regular or singular) cardinals, and let $V_{1}$ be an extension of $V.$ Suppose that in $V_{1}:$

• $(a)$

$\kappa<\lambda$* are still infinite cardinals.*

• $(b)$

there exists an increasing sequence $\langle\kappa_{n}:n<\omega\rangle$ of regular cardinals, cofinal in $\kappa.$ In particular $cf(\kappa)=\omega.$

• $(c)$

there is an increasing (mod finite) sequence $\langle f_{\alpha}:\alpha<\lambda\rangle$ of functions in the product $\prod_{n<\omega}(\kappa_{n+1}\setminus\kappa_{n}).$

• $(d)$

there is a partition $\langle S_{\sigma}:\sigma<\kappa\rangle$ of $\lambda$ into sets of size $\lambda$ such that for every countable set $I\in V$ and every $\sigma<\kappa$ we have $|I\cap S_{\sigma}|<\aleph_{0}.$

Then adding $\kappa$-many random reals over $V_{1}$ produces $\lambda$-many random reals over $V.$

Proof.

Force to add $\kappa$-many random reals over $V_{1}$. Let us write them as $\langle r_{i,\sigma}:i,\sigma<\kappa\rangle$. Also in $V,$ split $\kappa$ into $\kappa$-blocks $B_{\sigma},\sigma<\kappa,$ each of size $\kappa,$ and let $\langle f_{\alpha}:\alpha<\lambda\rangle\in V_{1}$ be an increasing (mod finite) sequence in $\prod_{n<\omega}(\kappa_{n+1}\setminus\kappa_{n})$. Let $\alpha<\lambda$. We define a real $s_{\alpha}$ as follows. Pick $\sigma<\kappa$ such that $\alpha\in S_{\sigma}.$ Let $k_{\alpha}=min\{k<\omega:r_{\sigma,\sigma}(k)\}=1$ and set

$\forall n<\omega$, $s_{\alpha}(n)=r_{f_{\alpha}(n+k_{\alpha}),\sigma}(0)$.

The following lemma completes the proof.

Lemma 4.2**.**

$\langle s_{\alpha}:\alpha<\lambda\rangle$* is a sequence of $\lambda$-many random reals over $V$.*

Proof.

First note that $\langle r_{i,\sigma}:i,\sigma<\kappa\rangle$ is ${{\mathbb{R}}}(\kappa\times\kappa)$-generic over $V_{1}$. By Lemma 2.3, it suffices to show that for any countable set $I\subseteq\lambda$, $I\in V$, the sequence $\langle s_{\alpha}:\alpha\in I\rangle$ is ${{\mathbb{R}}}(I)$-generic over $V$. Thus it suffices to prove the following

For every $p\in{{\mathbb{R}}}(\kappa\times\kappa)$ and every open dense subset $D\in V$

$(*)$            of ${{\mathbb{R}}}(I)$,  there is $\bar{p}\leq p$ such  that \bar{p}\Vdash\ulcorner\langle\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{\sim}}{{s}}}_{\alpha}:\alpha\in I\rangle extends

some element of $D\urcorner.$

Let $p$ and $D$ be as above and for simplicity suppose that $p=1_{{\mathbb{R}}(\kappa\times\kappa)}=[2^{\kappa\times\kappa\times\omega}]$. Let $b=[T_{b}]\in D,$ where $T_{b}\subseteq 2^{I\times\omega}$. As $I$ is countable, we can find $\{\sigma_{j}:j<\bar{\omega}\leq\omega\}\subseteq\lambda$ such that

[TABLE]

and each $I\cap S_{\sigma_{j}}$ is non-empty. By $(d)$, each $I\cap S_{\sigma_{j}}$ is finite, say

[TABLE]

For every $j_{1},j_{2}<\bar{\omega},$ $l_{1}<k_{j_{1}},l_{2}<k_{j_{2}}$ and $n_{1},n_{2}<\omega$ set

[TABLE]

The following can be proved as in Claim 3.3.

Claim 4.3**.**

The set $\Delta=\{(j_{1},j_{2},l_{1},l_{2},n_{1},n_{2}):b\leq c(j_{1},j_{2},l_{1},l_{2},n_{1},n_{2})\}$ is finite. Also, $(j_{1},j_{2},l_{1},l_{2},n_{1},n_{2})\in\Delta$ implies $(j_{2},j_{1},l_{2},l_{1},n_{2},n_{1})\in\Delta.$

Let $\Lambda=\{j<\bar{\omega}:$ there exists $(j_{1},j_{2},l_{1},l_{2},n_{1},n_{2})\in\Delta$ with $j=j_{1}\}$. Then $\Lambda$ is finite. For each $j\in\Lambda,$ by $(c)$, we can find $n^{*}_{j}<\omega$ such that for all $n\geq n^{*}_{j}$ and $\alpha^{*}_{1}<\alpha^{*}_{2}$ in $I\cap S_{\sigma_{j}}$ we have $f_{\alpha^{*}_{1}}(n)<f_{\alpha^{*}_{2}}(n).$

Let

$S^{\prime}=[\{x\in 2^{\kappa\times\kappa\times\omega}:\forall j\in\Lambda(\forall n<n^{*}_{j},x(\sigma_{j},\sigma_{j},n)=0$ and $x(\sigma_{j},\sigma_{j},n^{*}_{j})=1)\}]$

Then $\mu_{\kappa\times\kappa}(S^{\prime})=2^{-|\Lambda|(n^{*}_{j}+1)}>0$, and so $p^{\prime}=[S^{\prime}]\in{\mathbb{R}}(\kappa\times\kappa).$ Also, for each $j\in\Lambda$ and $l<k_{j}$, $p^{\prime}\Vdash\ulcorner k_{\alpha_{jl}}=n^{*}_{j}\urcorner$. Let

[TABLE]

By our choice of $n^{*}_{j}$ there are no collisions and the above definition is well-formed. Also, by the same arguments as before, $\mu_{\kappa\times\kappa}(\bar{S})=\mu_{\kappa\times\kappa}(S^{\prime})>0.$

Let $\bar{p}=[\bar{S}]$. Then $\bar{p}\in{\mathbb{R}}(\kappa\times\kappa)$ is well-defined and for all $\alpha=\alpha_{jl}\in I$, where $j\in\Lambda$ and $l<k_{j}$, and all $y\in S_{\bar{p}}$ we can find $x\in T_{b}$ such that for $m<n$,

\bar{p}\Vdash~{}\text{``}\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{\sim}}{{s}}}_{\alpha}(m)=\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{\sim}}{{s}}}_{\alpha_{jl}}(m)

=\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{\sim}}{{r}}}_{f_{\alpha_{jl}}(n^{*}_{j}+m),\sigma_{j}}(0)

$=y(f_{\alpha_{jl}}(n^{*}_{j}+m),\sigma_{j},0)$

$=x(\alpha_{jl},m)$

$=x(\alpha,m)\text{''.}$

This implies

\bar{p}\Vdash\ulcorner\langle\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{\sim}}{{s}}}_{\alpha_{jl}}:j\in\Lambda,l<k_{j}\rangle extends $b\urcorner.$

Now, as in the proof of Claim 3.6, we have the following:

Claim 4.4**.**

\bar{p}\Vdash\ulcorner\langle\smash{\underset{\raisebox{1.2pt}[0.0pt][0.0pt]{\sim}}{{s}}}_{\alpha}:\alpha\in I\rangle* extends $b\urcorner.$*

$(*)$ follows and we are done. ∎

The theorem follows. ∎

The following theorem follows from Theorem 4.1 and the arguments from [2].

Theorem 4.5**.**

• (a)

Suppose that $GCH$ holds in $V,$ $\kappa$ is a cardinal of countable cofinality and there are $\kappa$ many measurable cardinals below $\kappa$. Then there is a cardinal preserving forcing extension $V_{1}$ of $V$ not adding new reals and such that adding $\kappa$-many random reals random reals over $V_{1}$ produces $\kappa^{+}$-many random reals over $V$.

• (b)

Suppose that $V_{1}\supseteq V$ are such that:

1. (1)

$V_{1}$* and $V$ have the same cardinals and reals,* 2. (2)

$\kappa<\lambda$* are infinite cardinals of $V_{1}$,* 3. (3)

there is no partition $\langle S_{\sigma}:\sigma<\kappa\rangle$ of $\lambda$ in $V_{1}$ as in Theorem 3.1$(d).$**

Then adding $\kappa$-many random reals over $V_{1}$ cannot produce $\lambda$-many random reals over $V.$

• (c)

The following are equiconsistent:

1. (1)

There exists a pair $(V_{1},V_{2}),V_{1}\subseteq V_{2}$, of models of set theory with the same cardinals and reals and a cardinal $\kappa$ of cofinality $\omega$ (in $V_{2}$) such that adding $\kappa$-many random reals over $V_{2}$ adds more than $\kappa$-many random reals over $V_{1}.$ 2. (2)

There exists a cardinal $\delta$ which is a limit of $\delta$-many measurable cardinals.

• (d)

Suppose that $V_{1}\supseteq V$ are such that $V_{1}$ and $V$ have the same cardinals and reals and $\aleph_{\delta}$ is less than the first fixed point of the $\aleph$-function. Then adding $\aleph_{\delta}$-many random reals over $V_{1}$ cannot produce $\aleph_{\delta+1}$-many random reals over $V.$

• (e)

Suppose $GCH$ holds and there exists a cardinal $\kappa$ which is of cofinality $\omega$ and is a limit of $\kappa$-many measurable cardinals. Then there is pair $(V_{1},V_{2})$ of models of $ZFC,V_{1}\subseteq V_{2}$ such that:

1. (1)

$V_{1}$* and $V_{2}$ have the same cardinals and reals.* 2. (2)

$\kappa$* is the first fixed point of the $\aleph$-function in $V_{1}$ (and hence in $V_{2}$).* 3. (3)

Adding $\kappa$-many random reals over $V_{2}$ adds $\kappa^{+}$-many random reals over $V_{1}.$