# Approximating diamond principles on products at an inaccessible cardinal

Omer Ben-Neria and Jing Zhang

###### Abstract.

We isolate _the approximating diamond principles_ , which are consequences of
the diamond principle at an inaccessible cardinal. We use these principles to
find new methods for negating the diamond principle at large cardinals. Most
notably, we demonstrate, using Gitik’s overlapping extenders forcing, a new
method to get the consistency of the failure of the diamond principle at a
large cardinal $\theta$ without changing cofinalities or adding fast clubs to
$\theta$. In addition, we show that the approximating diamond principles
necessarily hold at a weakly compact cardinal. This result, combined with the
fact that in all known models where the diamond principle fails the
approximating diamond principles also fail at an inaccessible cardinal,
exhibits essential combinatorial obstacles to make the diamond principle fail
at a weakly compact cardinal.

††2010 _Mathematics Subject Classification_. Primary: 03E02, 03E35, 03E55.
The first author was partially supported by the Israel Science Foundation
(Grant 1832/19).
The second author was supported by the European Research Council (grant
agreement ERC-2018-StG 802756).
The authors would like to thank Spencer Unger and Miha Habic for valuable
discussions and comments on this work. E-mails<EMAIL_ADDRESS><EMAIL_ADDRESS>
##  1\. Introduction

Fix a regular uncountable cardinal $\theta$ and a stationary set
$S\subset\theta$. A sequence $\langle x_{\alpha}\subset\alpha:\alpha\in
S\rangle$ is a _$\diamondsuit(S)$ -sequence_ if for any $X\subset\theta$, the
set $\\{\alpha\in S:X\cap\alpha=x_{\alpha}\\}$ is stationary. This principle
is due to Jensen [Jen72] and was discovered during the course of his fine
structural analysis of the constructible universe. Note that
$\diamondsuit(\theta)$ implies $\theta^{<\theta}=\theta$.

The history of the relation between the diamond and compactness principles,
goes back to the work of Kunen and Jensen [JK69], who showed that
$\diamondsuit(\theta)$ must hold at every subtle cardinal. In fact, they prove
that the stronger property $\diamondsuit(Reg^{\theta})$ holds at such
cardinals, where $Reg^{\theta}$ is the stationary collection of regular
cardinals below $\theta$.

The consistency of $\neg\diamondsuit(Reg^{\theta})$ along with $\theta$ being
weakly compact, $\Pi^{m}_{n}$-indescribable, strongly unfolddable was
established by Woodin, Hauser [Hau92] and Džamonja-Hamkins [DH06]
respectively. Each of these consistency results is established from its
minimal corresponding large cardinal assumption.

In contrast, $\neg\diamondsuit(\theta)$ at Mahlo $\theta$, is known to have a
significantly stronger consistency strength. Jensen [Jen91] has shown that
$\neg\diamondsuit(\theta)$ at a Mahlo cardinal $\theta$ implies the existence
of $0^{\\#}$. Zeman [Zem00] improved the lower bound to the existence of an
inner model with a cardinal $\theta$, in which for every $\gamma<\theta$, the
set $\\{\alpha<\theta\mid o(\alpha)\geq\gamma\\}$ is stationary in $\theta$.

There are also tight connections between the diamond principles and cardinal
arithmetic. Shelah [She10], building on the previous work by Gregory [Gre76]
and after a series of partial results, showed that for any uncountable
cardinal $\lambda$, $2^{\lambda}=\lambda^{+}$ iff $\diamondsuit(\lambda^{+})$.
On the other hand, it was previously known from work of Jensen [DJ74] that CH
does not imply $\diamondsuit(\omega_{1})$. See [Rin11] for a comprehensive
survey regarding the diamond principle at successor cardinals. The situation
at inaccessible cardinals is different. In [She03], Shelah showed that it is
consistent with GCH that $\theta$ is inaccessible and $\neg\diamondsuit(S)$
for some stationary $S$ such that $\theta-S$ is fat, from the minimal
assumption. Works of Gitik (see for example [Git86]) demonstrate the
consistency of GCH along with the saturation of
$\mathrm{NS}_{\theta}\restriction S$, hence $\neg\diamondsuit(S)$, where $S$
is also co-fat. However, in all of these scenarios, the diamond principle only
fails partially. In other words, $\diamondsuit(\theta)$ holds in all of these
models mentioned above.

Two of the main problems in this area are:

  1. (1)

Is it consistent that $\diamondsuit(\theta)$ fails at a weakly compact
$\theta$?

  2. (2)

Is it consistent with GCH that $\diamondsuit(\theta)$ fails?

The only previously known method to violate the diamond principle fully at an
inaccessible cardinal is due to Woodin [Cum], based on the analysis of Radin
forcing extensions. This approach has been further developed in [BN19] and
[BNZ] to obtain the failure of the full diamond at a stationary reflecting
cardinal and even at a cardinal carrying no amenable C-sequences.

Nevertheless, this method is not applicable for solving the two problems
above, due to the following. In the following, when we say “Radin forcing at
$\theta$”, we mean “Radin forcing defined from a measure sequence on
$\theta$”.

  1. (1)

It was shown in [BNZ] that in any Radin forcing extension at $\theta$, if
$\theta$ is weakly compact, then $\diamondsuit(\theta)$ holds.

  2. (2)

It was shown in [Cum13] that in the Radin forcing extension at $\theta$ over a
model of GCH, if $\theta$ is regular, then $\diamondsuit(\theta)$ holds.

These results motivate the need for new methods to make the diamond principle
fail at an inaccessible cardinal.

In order to state the main results of this paper, we need some definitions.

######  Definition 1.1.

Let $\delta$ be a limit ordinal and $f,g:\delta\to\operatorname{Ord}$ be
functions. We write $f<^{*}g$ if $\\{\alpha<\delta:f(\alpha)<g(\alpha)\\}$
contains a closed unbounded set in $\delta$ if $cf(\delta)>\omega$ and
contains co-bounded set if $cf(\delta)=\omega$.

Let $\theta$ be an inaccessible cardinal,
$\vec{\lambda}=\langle\lambda_{i}<\theta:i<\theta\rangle$ be an increasing
sequence of cardinals cofinal in $\theta$ and $S\subset\theta$ be a stationary
set. We define what it means to be an _approximating diamond sequence on
$\Pi_{i<\theta}\lambda_{i}$_ with support $S$.

######  Definition 1.2.

We say $\langle g_{\gamma}\in\Pi_{i<\gamma}\lambda_{i}:\gamma\in S\rangle$ is
a $\diamondsuit_{\vec{\lambda}}(S)$-sequence if for any
$f\in\Pi_{i<\theta}\lambda_{i}$, the collection $\\{\gamma\in
S:g_{\gamma}\not<^{*}f\restriction\gamma\\}$ is stationary.

The principle can be considered as the inaccessible analogue of the
_parametrized diamond principle_ corresponding to the bounding number
$\mathfrak{b}$ studied by Moore, Hrušák and Džamonja [MHD04] at the level of
$\omega_{1}$.

Easy observation reveals that for every sequence
$\vec{\lambda}=\langle\lambda_{i}:i<\theta\rangle$,
$\diamondsuit_{\vec{\lambda}}(S)$ follows from $\diamondsuit(S)$ (see Lemma
2.4), suggesting to force $\neg\diamondsuit_{\vec{\lambda}}(\theta)$ as a new
route for obtaining $\neg\diamondsuit(\theta)$ at a large cardinal $\theta$.
Indeed, we show in Theorem 3.1 that $\diamondsuit_{\vec{\beth}}(\theta)$ fails
in Woodin’s model [Cum], where $\vec{\beth}$ denote the sequence $\langle
2^{\alpha}:\alpha<\theta\rangle$. A similar argument shows that
$\diamondsuit_{\vec{\beth}}(\theta)$ fails in various Extender-based Radin
forcing of Merimovich [Mer03]. More surprising to the authors was that one can
force the failure of $\diamondsuit_{\vec{\beth}}(\theta)$ without changing
cofinalities or adding fast clubs. This is fundamentally different from
Woodin’s method. To obtain this result we utilize a version of the recent
forcing discovered by Gitik [Git19] using overlapping extenders.

######  Theorem 1.3.

Suppose $\theta$ is an inaccessible limit of $\theta^{++}$-strong cardinals,
then there exists a cofinality and cardinal preserving forcing extension
$V[G]$ of $V$ in which:

  1. (1)

$\diamondsuit_{\vec{\beth}}(\theta)$ fails,

  2. (2)

$V,V[G]$ agree on cardinals and cofinalities, and

  3. (3)

any closed unbounded subset of $\theta$ in $V[G]$ contains a closed unbounded
subset of $\theta$ from $V$.

The third condition in Theorem 1.3 holds suggests that $V[G]$ is a relatively
tame forcing extension of $V$ in some sense and will be useful in proving
certain compactness properties of $\theta$ in $V$ remain valid in $V[G]$. This
reasoning is shown to be true to an extent in Section 6. However, this
approach of forcing $\neg\diamondsuit(\theta)$ must not succeed at the level
of weak compactness, as evidenced by the following theorem.

######  Theorem 1.4.

If $\theta$ is weakly compact, then
$\diamondsuit_{\vec{\lambda}}(Reg^{\theta})$ holds for any increasing sequence
$\vec{\lambda}$ in $\theta$. Moreover, $\Diamond_{\vec{\lambda}}(S)$ holds for
every $S$ in the weak compact filter on $\theta$.

The last theorem prompts the investigation of a possible implication from
$\diamondsuit_{\vec{\lambda}}(S)$ to $\diamondsuit(\theta)$, for some sequence
$\vec{\lambda}$ and stationary set $S\subset\theta$. We do not know of such
implication in general but are able to prove an implication under GCH, in the
case $S\subset Sing^{\theta}_{>\omega}$, where $Sing^{\theta}_{>\omega}$ is
the set of singular ordinals of uncountable cofinalities.

######  Theorem 1.5.

$\diamondsuit_{\vec{\beth}}(Sing^{\theta}_{>\omega})$ and
$\diamondsuit(Sing^{\theta}_{>\omega})$ are equivalent under GCH.

The organization of the paper is:

  1. (1)

Section 2 contains the proofs of Theorem 1.4 and 1.5 as well as some
generalizations.

  2. (2)

Section 3 outlines a few results in the Radin forcing extension. In
particular, the consistency of $\neg\diamondsuit_{\vec{\lambda}}(\theta)$ and
the consistency of
$\diamondsuit_{\vec{\beth}}(Sing^{\theta}_{>\omega})+\neg\diamondsuit(\theta)$
are demonstrated.

  3. (3)

Section 4 gives a short description of Gitik’s forcing with overlapping
extenders of inaccessible length and review some of its key properties.

  4. (4)

Section 5 focuses on the proof of Theorem 1.3. It also includes a proof
showing that the assumption of the extenders being strong past $\theta$ is
needed.

  5. (5)

Section 6 illustrates the possibilities of preserving certain compactness
properties of $\theta$ from the ground model, extending some results from
[BNZ].

  6. (6)

Section 7 concludes with open questions.

##  2\. ZFC Results

Let $\theta$ be a strongly inaccessible cardinal.

######  Definition 2.1.

Let $J$ be an ideal on $\gamma$ and $f,g\in{}^{\gamma}\operatorname{Ord}$,
then $f<_{J}g$ abbreviates $\\{\alpha\in\gamma:f(\alpha)\geq g(\alpha)\\}\in
J$. Similary we can define $\leq_{J},=_{J}$.

Unless otherwise noted, all the ideals we consider are proper.

######  Notation 2.2.

Given an ideal $J$, let

  * •

$J^{+}$ denote the collection of positive sets, namely $J^{+}=\\{A:A\not\in
J\\}$ and

  * •

$J^{*}$ denote its dual filter, namely $J^{*}=\\{A:A^{c}\in J\\}$.

######  Definition 2.3.

Suppose we are given

  1. (1)

$\vec{J}=\langle J_{\gamma}\subset P(\gamma):\gamma\leq\theta\rangle$ where
each $J_{\gamma}$ is an ideal on $\gamma$,

  2. (2)

an increasing sequence $\vec{\lambda}=\langle\lambda_{i}:i<\theta\rangle$
cofinal in $\theta$, and

  3. (3)

$S\in J_{\theta}^{+}$.

Then a _$\diamondsuit_{\vec{J},\vec{\lambda}}(S)$ -sequence_ is $\langle
g_{\gamma}\in\Pi_{i<\gamma}\lambda_{i}:\gamma\in S\rangle$ satisfying: for any
$f\in\Pi_{i<\theta}\lambda_{i}$, $\\{\gamma\in
S:g_{\gamma}\not<_{J_{\gamma}}f\restriction\gamma\\}\in J_{\theta}^{+}$.

We say $\diamondsuit_{\vec{J},\vec{\lambda}}(S)$ holds if there exists a
$\diamondsuit_{\vec{J},\vec{\lambda}}(S)$-sequence. In this paper, unless
noted otherwise, we will assume the index set $S$ is a subset of the club set
$\\{\gamma<\theta:\forall i<\gamma,\lambda_{i}<\gamma\\}$.

######  Lemma 2.4.

$\diamondsuit(S)$ implies $\diamondsuit_{\vec{J},\vec{\lambda}}(S)$ for any
$\vec{J},\lambda,S\in\mathrm{NS}_{\theta}^{+}$ satisfying that
$J_{\theta}\subseteq\mathrm{NS}_{\theta}$.

###### Proof.

Let $\langle g^{\prime}_{\gamma}\in{}^{\gamma}\gamma:\gamma\in S\rangle$ be a
given $\diamondsuit(S)$-sequence. Define a
$\diamondsuit_{\vec{J},\vec{\lambda}}(S)$ by letting
$g_{\gamma}=g^{\prime}_{\gamma}$ if
$g^{\prime}_{\gamma}\in\Pi_{i<\gamma}\lambda_{i}$, and $g_{\gamma}\equiv 0$
otherwise. Given $f\in\Pi_{i<\theta}\lambda_{i}$, there exists a stationary
set $S^{\prime}\subset S$ such that for all $\gamma\in S^{\prime}$,
$f\restriction\gamma=g^{\prime}_{\gamma}=g_{\gamma}$, where the last equality
follows from the definition of $g_{\gamma}$. As
$J_{\theta}\subseteq\mathrm{NS}_{\theta}$, $S^{\prime}\in J_{\theta}^{+}$. ∎

######  Definition 2.5.

A transitive model $M\models\mathrm{ZFC}^{-}$ ($\mathrm{ZFC}-$ Power Set
Axiom) is a _$\theta$ -model_ if it contains $\theta$ and
${}^{<\theta}M\subset M$.

A typical $\theta$-model will be the transitive collapse of some $X\prec
H(\chi)$ where $\chi$ is a large enough regular cardinal, $\theta+1\subset X$
and ${}^{<\theta}X\subset X$.

######  Definition 2.6.

A sequence of ideals $\vec{J}=\langle J_{\gamma}:\gamma\leq\theta\rangle$ is

  1. (1)

_uniformly definable_ if there exist a formula $\varphi(x,y)$ and a parameter
$a\subset\theta$, such that for any large enough $\gamma$, $A\in J_{\gamma}$
iff $H(\gamma^{+})\models\varphi(A,a\cap\gamma)$.

  2. (2)

_$\theta$ -upward uniformly definable_ if it is uniformly definable as
witnessed by $\varphi(x,y),a$ and for any $\theta$-model $N$ containing $a$,
$(J_{\theta})^{N}\subset J_{\theta}$. More precisely, the latter asserts for
any $A\in N\cap P(\theta)$, if $N\models``H(\theta^{+})\models\varphi(A,a)"$,
then $H(\theta^{+})\models\varphi(A,a)$, namely, $A\in J_{\theta}$.

######  Remark 2.7.

Usually it suffices to work with $\langle J_{\gamma}:\gamma\in
A\cup\\{\theta\\}\rangle$ where $A\in J_{\theta}^{*}$. Hence, for the rest of
this article, whenever we define a sequence of ideals on a large subset of
$\theta$, we implicitly mean that any ideal can be filled in for missing
coordinates.

######  Example 2.8.

We give two concrete examples of $\theta$-upward uniformly definable sequence
of ideals:

  1. (1)

$\vec{J}_{bdd}=_{\mathrm{def}}\langle
P_{bdd}(\gamma):\gamma\in\lim\theta\rangle$, where $P_{bdd}(\gamma)$ is the
collection of all bounded subsets of $\gamma$. Let $\varphi(x)$ be “there
exists $y$ which is the largest cardinal and there exists $\eta<y$ such that
for all $\gamma^{\prime}>\gamma,\gamma^{\prime}\not\in x$”. Then we see that
$A\in J_{\gamma}$ iff $H(\gamma^{+})\models\varphi(A)$. It is easy to check
that this definition is upward absolute for $\theta$-models.

  2. (2)

$\vec{\mathrm{NS}}=\langle\mathrm{NS}_{\gamma}:\gamma\in\theta\cap\mathrm{cof}(>\omega)\rangle$,
where $\mathrm{NS}_{\gamma}$ is the nonstationary ideal on $\gamma$. Let
$\varphi(x)$ be “there exists $y$ which is the largest cardinal and there
exists a closed unbounded $c\subset y$ such that $c\cap x=\emptyset$”. Then we
see that $A\in J_{\gamma}$ iff $H(\gamma^{+})\models\varphi(A)$. The key point
to see that this definition is upward absolute for $\theta$-models is that for
any $\theta$-model $N$, if $C\in N$ such that $N\models C$ is club in
$\theta$, then $H(\theta^{+})\models C$ is club in $\theta$.

Note that $\diamondsuit_{\vec{\lambda}}(S)$ is exactly
$\diamondsuit_{\vec{J}^{*},\vec{\lambda}}(S)$ where
$J^{*}_{\gamma}=NS_{\gamma}$ when $cf(\gamma)>\omega$ and
$J^{*}_{\gamma}=P_{bdd}(\gamma)$ otherwise. Using $\vec{J}^{*}$, the following
Theorem implies Theorem 1.4.

######  Theorem 2.9.

If $\theta$ is weakly compact, then $\diamondsuit_{\vec{J},\vec{\lambda}}(S)$
holds when

  1. (1)

$\vec{\lambda}=\langle\lambda_{i}:i<\theta\rangle$ is increasing and cofinal
in $\theta$,

  2. (2)

$\vec{J}$ is $\theta$-upward uniformly definable,

  3. (3)

$J_{\theta}$ is $\sigma$-complete,

  4. (4)

$J_{\theta}\subset\mathrm{WC}_{\theta}$, where $\mathrm{WC}_{\theta}$ is the
weakly compact ideal on $\theta$,

  5. (5)

$S\in\mathrm{WC}_{\theta}^{+}$.

###### Proof.

Suppose for the sake of contradiction that the conclusion fails. More
explicitly, this means for any sequence $\langle
g_{\gamma}\in\Pi_{i<\gamma}\lambda_{i}:\gamma\in S\rangle$, there exist
$f\in\Pi_{i<\theta}\lambda_{i}$ and $A\in J^{*}_{\theta}$ such that for all
$\gamma\in A\cap S$, $g_{\gamma}<_{J_{\gamma}}f\restriction\gamma$. Note that
by the assumption that $J_{\theta}\subset\mathrm{WC}_{\theta}$,
$A\in\mathrm{WC}_{\theta}^{*}$. Let $\varphi(x,y)$ and $a\subset\theta$
witness the $\theta$-upward uniform definability of $\vec{J}$.

Recursively define $\langle
g_{\gamma}\in\Pi_{i<\gamma}\lambda_{i}:\gamma\in\lim(\theta+1)\rangle$ such
that for each $\gamma\in\lim\theta+1$, if there exists
$g\in\Pi_{i<\gamma}\lambda_{i}$ such that:

  1. (1)

for $J_{\gamma}^{*}$-many $\eta\in S\cap\gamma$,
$g\restriction\eta>_{J_{\eta}}g_{\eta}$ and

  2. (2)

there does not exist $f<_{J_{\gamma}}g$ satisfying the property above,

then $g_{\gamma}$ is any such $g$. Otherwise, $g_{\gamma}\equiv 0$ and we say
$g_{\gamma}$ is _defined vacuously_.

######  Claim 2.10.

If $J_{\gamma}$ is $\sigma$-complete and there exists $g$ satisfying (1), then
there exists $g$ satisfying (2).

###### Proof of the Claim.

Suppose for the sake of contradiction, no $g$ satisfies (2). Define
$g_{0}>_{J_{\gamma}}g_{1}>_{J_{\gamma}}g_{2}>_{J_{\gamma}}\cdots
g_{n}>_{J_{\gamma}}\cdots$ with each $g_{n}$ satisfying (1). Since
$J_{\gamma}$ is $\sigma$-complete, there exists $\gamma^{\prime}<\gamma$ such
that for any $m<n$, $g_{m}(\gamma^{\prime})>g_{n}(\gamma^{\prime})$.
Contradiction. ∎

Note that by Claim 2.10 and the initial assumption, $g_{\theta}$ is not
defined vacuously. Therefore, there exists $A\in J_{\theta}^{*}$, such that
for any $\tau\in A\cap S$, $g_{\tau}<_{J_{\tau}}g_{\theta}\restriction\tau$.

As $A\cap S\in\mathrm{WC}^{+}_{\theta}$, we can find

  * •

a transitive $\theta$-model $M$ containing $\vec{g}=\langle
g_{\gamma}:\gamma\in S\cup\\{\theta\\}\rangle$, $\vec{J}\restriction\theta$
and $a$,

  * •

a transitive $\theta$-model $N$ and an elementary embedding $j:M\to N$ with
critical point $\theta$,

  * •

$\theta\in j(A\cap S)$.

Let $f_{\theta}=j(\vec{g})(\theta)$. Observe that $f_{\theta}$ is not defined
vacuously since $j(g_{\theta})\restriction\theta=g_{\theta}\in N$ is a witness
to property (1) and Claim 2.10 implies there is a function in $N$ satisfying
both (1) and (2). The key point is: due to the fact that
$(J_{\theta})^{N}\subset J_{\theta}$, if we can find an infinite
$<_{(J_{\theta})^{N}}$-decreasing sequence, then it is an infinite
$<_{J_{\theta}}$-decreasing sequence, contradicting with Claim 2.10. By the
elementarity of $j$ and the fact that $\theta\in j(S\cap A)$, we have that
$N\models f_{\theta}<_{J_{\theta}}j(g_{\theta})\restriction\theta=g_{\theta}$.
Since $\vec{J}$ is $\theta$-upward uniformly definable, we have
$(J_{\theta})^{N}\subset J_{\theta}$. As a result, in $V$, we have that
$f_{\theta}<_{J_{\theta}}g_{\theta}$. This contradicts with the minimality of
$g_{\theta}$. ∎

######  Remark 2.11.

Under the same hypothesis as in Theorem 1.4, the previous proof gives the
following strengthening of $\diamondsuit_{\vec{J},\vec{\lambda}}(S)$, which
can be seen as a symmetric version of our approximating property: there exists
a sequence $\langle g_{\gamma}\in\Pi_{i<\gamma}\lambda_{i}:\gamma\in S\rangle$
such that for any $f\in\Pi_{i<\theta}\lambda_{i}$, the set $\\{\gamma\in
S:f\restriction\gamma\not<_{J_{\gamma}}g_{\gamma}\text{ and
}g_{\gamma}\not<_{J_{\gamma}}f\restriction\gamma\\}\in J^{+}_{\theta}$.

######  Remark 2.12.

Similar proof shows that: if $\theta$ is a $\Pi^{1}_{n+1}$-indescribable
cardinal, then for any increasing $\vec{\lambda}$ cofinal in $\theta$, there
exists $\langle g_{\gamma}\in\Pi_{i<\gamma}\lambda_{i}:\gamma<\theta\rangle$
such that for any $f\in\Pi_{i<\theta}\lambda_{i}$, the collection
$\\{\gamma:g_{\gamma}\not<_{\mathbf{\Pi^{1}_{n}}(\gamma)}f\restriction\gamma\\}$
is positive with respect to the ideal $\mathbf{\Pi^{1}_{n}}(\theta)$. Here
$\mathbf{\Pi^{1}_{n}}(\gamma)$ denotes the $\Pi^{1}_{n}$-indescribable ideal
on subsets of $\gamma$.

######  Remark 2.13.

Recall a theorem of Woodin (see [Hau92]) that it is consistent that $\theta$
is weakly compact and $\diamondsuit(\mathrm{Reg}^{\theta})$ fails. As a
result, $\diamondsuit(\mathrm{Reg}^{\theta})$ is not implied by
$\diamondsuit_{\vec{\mathrm{NS}},\vec{\lambda}}(\mathrm{Reg}^{\theta})$, in
contrast with the following proposition.

######  Proposition 2.14 (GCH).

Let $S$ be a stationary subset of
$S^{\theta}_{>\omega}=\theta\cap\\{\gamma:\omega<cf(\gamma)<\gamma\\}$. Then
$\diamondsuit_{\vec{\mathrm{NS}},\vec{\beth}}(S)$ implies $\diamondsuit(S)$.

###### Proof.

We may assume any $\gamma\in S$ is a singular cardinal. Fix a
$\diamondsuit_{\vec{J},\vec{\lambda}}(S)$-sequence $\vec{g}=\langle
g_{\gamma}:\gamma\in S\rangle$. For each $\gamma\in S$, let
$c_{\gamma}\subset\gamma$ be a club of order type $cf(\gamma)$. For each
$\gamma\in S$, let
$F_{bdd}(\gamma)=\\{h\in{}^{c_{\gamma}}\gamma:\mathrm{ran}(h)\subset\gamma^{\prime}<\gamma\\}$.
By the cardinal arithmetic assumption, $|F_{bdd}(\gamma)|=\gamma$ for each
strong limit singular cardinal $\gamma$. For each ordinal $\beta$, let
$f_{\beta}:|\beta|\to\beta$ be some bijection and let $\langle
x^{\beta}_{i}:i<\beta^{+}\rangle$ be an enumeration of $P(\beta)$.

Define $\mathcal{C}_{\gamma}=\\{\bigcup_{\beta\in
x}x^{\beta}_{f_{g_{\gamma}(\beta)+1}(h(\beta))}:x\subset c_{\gamma},h\in
F_{bdd}(\gamma),\operatorname{dom}(h)=x\\}$. Note that
$|\mathcal{C}_{\gamma}|=\gamma$. We check that
$\langle\mathcal{C}_{\gamma}:\gamma\in S\rangle$ is a
$\diamondsuit^{-}(S)$-sequence. Let $X\subset\theta$ be given. Let
$l_{X}\in\Pi_{i<\theta}i^{+}$ be such that for each $i$, $l_{X}(i)=j<i^{+}$
where $x^{i}_{j}=X\cap i$. By the fact that $\vec{g}$ is a
$\diamondsuit_{\vec{J},\vec{\lambda}}(S)$-sequence, $S^{\prime}=\\{\gamma\in
S:\exists A\in\mathrm{NS}^{+}_{\gamma},\forall\beta\in A,l_{X}(\beta)\leq
g_{\gamma}(\beta)\\}\in\mathrm{NS}_{\theta}^{+}$. It suffices to show that for
each $\gamma\in S^{\prime}$, $X\cap\gamma\in\mathcal{C}_{\gamma}$. Fix
$\gamma\in S^{\prime}$ as witnessed by $A\in\mathrm{NS}^{+}_{\gamma}$.
Consider the following regressive function: $\beta\in A\cap c_{\gamma}\mapsto
i_{\beta}<\beta$ such that $f_{g_{\gamma}(\beta)+1}(i_{\beta})=l_{X}(\beta)$.
By the Pressing Down lemma, there exists a stationary $x\subset c_{\gamma}\cap
A$ and some $\gamma^{\prime}<\gamma$ such that for all $\beta\in x$,
$i_{\beta}<\gamma^{\prime}$. Consider $h\in F_{bdd}(\gamma)$ defined on $x$
mapping $\beta\in x$ to $i_{\beta}$. Then $\bigcup_{\beta\in
x}x^{\beta}_{f_{g_{\gamma}(\beta)+1}(h(\beta))}=\bigcup_{\beta\in
x}x^{\beta}_{f_{g_{\gamma}(\beta)+1}(i_{\beta})}=\bigcup_{\beta\in
x}x^{\beta}_{l_{X}(\beta)}=X\cap\gamma\in\mathcal{C}_{\gamma}$. ∎

Theorem 1.5 is an immediate consequence of the above.

###### Proof.

(Theorem 1.5) $\diamondsuit(Sing^{\theta}_{>\omega})$ implies
$\diamondsuit_{\vec{\beth}}(Sing^{\theta}_{>\omega})$ by Lemma 2.4. Assuming
GCH, $\diamondsuit_{\vec{\beth}}(Sing^{\theta}_{>\omega})$ implies
$\diamondsuit(Sing^{\theta}_{>\omega})$ by Proposition 2.14. ∎

######  Remark 2.15.

Theorem 3.6 in the next section will show that the GCH assumption in
Proposition 2.14 cannot be completely waived. However, the cardinal arithmetic
in the model of Theorem 3.6 is quite specific, and it is possible that the
result of Proposition 2.14 can be obtained from mild cardinal arithmetic
assumptions such as $2^{\alpha}$ is regular for most $\alpha<\theta$. See
problems 7.3 and 7.4 in the open problems section.

In the rest of the section we discuss circumstances where weak compactness is
strong enough to give rise to a diamond sequence. These results will be used
in the next section to give a new and simple proof that diamond must hold in
Radin generic extensions in which the critical point is weakly compact, which
is the main result from [BNZ].

######  Definition 2.16.

We say $\mathcal{F}\subset P(\theta)$ is

  1. (1)

_closed under restrictions_ if for any $X\in\mathcal{F}$ and any
$\alpha<\theta$, $X\cap\alpha\in\mathcal{F}$;

  2. (2)

_closed under branching_ if for any $X\subset\theta$ such that for each
$\alpha<\theta$, $X\cap\alpha\in\mathcal{F}$, then $X\in\mathcal{F}$.

######  Definition 2.17.

Let $\mathcal{F}\subset P(\theta)$. We say $\mathcal{F}$ admits a _$\theta$
-upward uniformly definable well-order_ $<_{\theta}$ if

  1. (1)

there exist a formula $\varphi(x_{0},x_{1},y)$ and a parameter
$a\subset\theta$ such that for $A,B\in\mathcal{F}$, $A<_{\theta}B$ iff
$H(\theta^{+})\models\varphi(A,B,a)$;

  2. (2)

$<_{\theta}$ is a well-order;

  3. (3)

for any $\theta$-model $N$ containing $a$,
$(<_{\theta})^{N}\subset<_{\theta}$. More precisely, for any $A,B\in
N\cap\mathcal{F}$, if $N\models``H(\theta^{+})\models\varphi(A,B,a)"$, then
$H(\theta^{+})\models\varphi(A,B,a)"$.

######  Theorem 2.18.

Suppose $\theta$ is weakly compact. Let $\mathcal{F}\subset P(\theta)$ be a
collection closed under restrictions and branching admitting a $\theta$-upward
uniformly definable well-order $<_{\theta}$. Then $\diamondsuit(\theta)$ holds
for $\mathcal{F}$. Namely, there exists $\langle
S_{\gamma}:\gamma<\theta\rangle$ such that for any $A\in\mathcal{F}$, there
exists stationarily many $\gamma<\theta$ such that $A\cap\gamma=S_{\gamma}$.

###### Proof.

Let $\varphi(x_{0},x_{1},a)$ be the $\theta$-upward uniform definition for
$<_{\theta}$. Suppose for the sake of contradiction that there does not exist
a $\diamondsuit(\theta)$-sequence for sets in $\mathcal{F}$. Define a sequence
$\langle S_{\gamma}\subset\gamma:\gamma<\theta\rangle$ recursively as follows:
at stage $\alpha$, if there exists $X\in\mathcal{F}\cap P(\alpha)$ such that

  1. (1)

there exists a club $c\subset\alpha$ such that for all $i\in c$, $X\cap i\neq
S_{i}$ and

  2. (2)

there is no $Y\in\mathcal{F}\cap P(\alpha)$ with
$H(\alpha^{+})\models\varphi(Y,X,a\cap\alpha)$ satisfying the property above,

then define $S_{\alpha}$ as one such $X$. Otherwise, just let
$S_{\alpha}=\emptyset$. By the assumption, $\langle
S_{\gamma}:\gamma<\theta\rangle$ as defined is not a
$\diamondsuit(\theta)$-sequence for $\mathcal{F}$, as a result, there exists
$X\in\mathcal{F}$ and a club $C\subset\theta$ such that for all $\gamma\in C$,
$X\cap\gamma\neq S_{\gamma}$. Since $<_{\theta}$ is a well order, we may pick
such $X$ such that it is $<_{\theta}$-least.

By the fact that $\theta$ is weakly compact, we may find $\theta$-models $M,N$
with $M$ containing $X,\langle
S_{\gamma}:\gamma<\theta\rangle,C,\mathcal{F}^{\prime}=_{def}\mathcal{F}\cap
V_{\theta}$ and an elementary embedding $j:M\to N$ with critical point
$\theta$. Let $S_{\theta}=j(\vec{S})(\theta)$. Since $\theta\in j(C)$, we have
that $j(X)\cap\theta=X\neq S_{\theta}$. Note that $X\in
j(\mathcal{F}^{\prime})$ by the fact that every proper initial segment of $X$
is in $\mathcal{F}^{\prime}$ and the elementarity of $j$, we observe that $X$
satisfies (1) with respect to $\langle S_{\gamma}:\gamma<\theta\rangle$ as
witnessed by the club $C\subset\theta$. By the choice of $S_{\theta}$ and the
fact that $S_{\theta}\neq X$, we know that
$N\models``H(\theta^{+})\models\varphi(S_{\theta},X,a)$”. Since
$(<_{\theta})^{N}\subset<_{\theta}$ by the $\theta$-upward absoluteness
assumption and that $S_{\theta}\in\mathcal{F}$ since $\mathcal{F}$ is closed
under branching, we have that $S_{\theta}<_{\theta}X$ in $V$, contradicting
with the $<_{\theta}$-minimality of $X$. ∎

##  3\. Combinatorics in the Radin forcing extensions

We follow [BNZ, Section 2] for the definitions and notations regarding Radin
forcing. Let $\bar{U}$ be a measure sequence on $\theta$ derived from an
embedding $j:V\to M$. To be consistent with notations of this paper, we note
two notation differences from [BNZ].

  1. (1)

We use $\theta$ to denote the top cardinal that measure sequences are defined
on. Namely, the measure sequences are derived from the elementary embedding
$j:V\to M$ with critical point $\theta$. For a measure sequence $\bar{w}$, we
then use $\theta(\bar{w})$ to denote the critical point associated with
$\bar{w}$. In [BNZ], we used $\kappa$.

  2. (2)

$p\geq q$ means $p$ is a stronger condition. In [BNZ], $p\leq q$ means $p$ is
stronger.

###  3.1. The failure of $\diamondsuit_{\vec{J},\vec{\lambda}}$

In this subsection, we provide an extension of Woodin’s theorem that in
suitable Radin extensions, $\diamondsuit(\theta)$ fails at an inaccessible
$\theta$ and show that in these models, actually the weaker
$\diamondsuit_{\vec{J},\vec{\lambda}}(\theta)$ fails for certain $\vec{J}$ and
$\vec{\lambda}$. Then we demonstrate that in any Radin extension, the
hypotheses of Theorem 2.18 holds.

For each $\alpha<\theta$, fix an enumeration $\vec{x}^{\alpha}=\langle
x^{\alpha}_{j}:j<2^{\alpha}\rangle$. Let
$\vec{x}^{\theta}=j(\langle\vec{x}^{\alpha}:\alpha<\theta\rangle)(\theta)$.
For any $X\subset\theta$ and $\alpha\leq\theta$, let $index_{\alpha}(X)$ be
the unique $j$ such that $X\cap\alpha=x^{\alpha}_{j}$. For each
$X\subset\theta$ and $X\in V$, let $f_{X}:\theta\to\theta$ mapping $\alpha$ to
$index_{\alpha}(X)$.

######  Theorem 3.1.

Suppose $lh(\bar{U})<2^{\theta}$ and $lh(\bar{U})\in\mathrm{cof}(>\theta)$,
then in $V^{R_{\bar{U}}}$,
$\diamondsuit_{\vec{\mathrm{NS}},\vec{\lambda}}(\theta)$ fails for some
$\vec{\lambda}$.

###### Proof.

Let $G\subset R_{\bar{U}}$ be generic over $V$. Let
$C_{G}=\\{C_{G}(i):i<\theta\\}$ is an increasing enumeration of the generic
club. Let $g:\theta\to\operatorname{Ord}$ be such that
$j(g)(\theta)=lh(\bar{U})^{+}\leq 2^{\theta}$. We may assume that
$g(\alpha)\leq 2^{\alpha}$ for all $\alpha<\theta$. Then $\vec{\lambda}$ will
be chosen such that $\lambda_{i}=g(C_{G}(i))$.

######  Claim 3.2.

For $\eta\leq lh(\bar{U})$, in $V^{R_{\bar{U}\restriction\eta}}$, for any
$\alpha_{0}<\alpha_{1}<lh(\bar{U})^{+}$,
$f_{x^{\theta}_{\alpha_{0}}}\restriction
C_{G_{\eta}}<^{*}f_{x^{\theta}_{\alpha_{1}}}\restriction C_{G_{\eta}}$.

###### Proof.

Note that
$\\{\bar{w}:index_{\theta(\bar{w})}(x^{\theta}_{\alpha_{0}})<index_{\theta(\bar{w})}(x^{\theta}_{\alpha_{1}})\\}\in\bigcap\bar{U}$,
since $index_{\theta}(j(x^{\theta}_{\alpha_{k}}))=\alpha_{k}$ for $k<2$. As a
result, the conclusion follows by genericity. ∎

######  Claim 3.3.

For $\eta\leq lh(\bar{U})$, in $V^{R_{\bar{U}\restriction\eta}}$, $\langle
f_{x^{\theta}_{k}}\restriction C_{G_{\eta}}:k<lh(\bar{U})^{+}\rangle$ is
cofinal in $(\Pi_{\alpha\in C_{G_{\eta}}}g(\alpha),<^{*})$.

###### Proof.

Given $p=p_{0}{}^{\frown}(\bar{U},A_{p})\in R_{\bar{U}\restriction\eta}$ and
$\dot{t}\in\Pi_{\alpha\in C_{G_{\eta}}}g(\alpha)$, we may directly extend $p$
if necessary and assume that for any $\bar{w}\in A_{p}$, there exists a
$R_{\bar{w}}$-name $h(\bar{w})$ such that
$p{}^{\frown}\bar{w}\Vdash\dot{t}(\theta(\bar{w}))=h({\bar{w}})$. For any
$\eta^{\prime}<\eta$, $j(h)(\bar{U}\restriction\eta^{\prime})$ is an
$R_{\bar{U}\restriction\eta^{\prime}}$-name for an ordinal in
$j(g)(\theta)=lh(\bar{U})^{+}$. By the fact that
$R_{\bar{U}\restriction\eta^{\prime}}$ is $\theta^{+}$-c.c for any
$\eta^{\prime}<\eta$, we can find some $k<lh(\bar{U})^{+}$ large enough such
that for any $\eta^{\prime}<\eta$,
$\Vdash_{R_{\bar{U}\restriction\eta^{\prime}}}j(h)(\bar{U}\restriction\eta^{\prime})<k=j(f_{x^{\theta}_{k}})(\theta)$.
As a result, $\\{\bar{w}\in
A_{p}:p{}^{\frown}\bar{w}\Vdash\dot{t}(\theta(\bar{w}))<f_{x^{\theta}_{k}}(\theta(\bar{w}))\\}\in\bigcap\bar{U}\restriction\eta$.
The conclusion follows from genericity. ∎

Suppose we are given $\langle h_{\gamma}\in\Pi_{\alpha\in
C_{G}\cap\gamma}g(\alpha):\gamma\in\lim C_{G}\rangle$, we need to find some
$f\in\Pi_{\alpha\in C_{G}}g(\alpha)$ such that on a tail $\gamma\in\lim
C_{G}$, $h_{\gamma}<^{*}f\restriction\gamma$.

We proceed with the density argument. Given $p=p_{0}{}^{\frown}(\bar{U},A)\in
R_{\bar{U}}$, by taking a direct extension if necessary, we may assume that
for any $\bar{w}\in A$,
$p{}^{\frown}\bar{w}\Vdash\dot{h}_{\theta(\bar{w})}=\dot{h}_{\bar{w}}$, where
$\dot{h}_{\bar{w}}$ is a $R_{\bar{w}}$-name. As a result, for any
$\eta<lh(\bar{U})$, there is an $R_{\bar{U}\restriction\eta}$-name
$\dot{h}_{\bar{U}\restriction\eta}$ such that
$j(p){}^{\frown}\bar{U}\restriction\eta$ forces that
$j(\dot{h})_{\theta}=\dot{h}_{\bar{U}\restriction\eta}$. By Claim 3.3 and the
$\theta^{+}$-c.c of $R_{\bar{U}\restriction\eta}$, there exists
$i_{\eta}<lh(\bar{U})^{+}$ such that
$\Vdash_{R_{\bar{U}\restriction\eta}}\dot{h}_{\bar{U}\restriction\eta}<^{*}f_{x^{\theta}_{i_{\eta}}}\restriction
C_{G_{\eta}}$. Let $i=\sup_{\eta<lh(\bar{U})}i_{\eta}+1$, by Claim 3.2 we know
that
$j(p){}^{\frown}\bar{U}\restriction\eta\Vdash_{j(\vec{R}_{\bar{U}})}j(\dot{h})_{\theta}<^{*}f_{x^{\theta}_{i}}\restriction
C_{G_{\eta}}$. As a result, $A^{\prime}=\\{\bar{w}\in
A:p{}^{\frown}\bar{w}\Vdash\dot{h}_{\theta(\bar{w})}<^{*}f_{x^{\theta}_{i}}\restriction
C_{G}\cap\theta(\bar{w})\\}\in\bigcap\bar{U}$. Note that
$j(f_{x^{\theta}_{i}})\restriction\theta=f_{x^{\theta}_{i}}$ since
$j(x^{\theta}_{i})\cap\theta=x^{\theta}_{i}$. As a result, the extension
$p_{0}{}^{\frown}(\bar{U},A^{\prime})$ of $p$ forces the desired conclusion. ∎

######  Proposition 3.4.

Suppose $lh(\bar{U})\in\mathrm{cof}(>\theta)$. Then in $V^{R_{\bar{U}}}$,
$\mathcal{F}=_{\mathrm{def}}(P(\theta))^{V}$

  1. (1)

is closed under restrictions and branching and

  2. (2)

admits a $\theta$-upward uniformly definable well-order $<_{\theta}$.

###### Proof.

Let $G\subset R_{\bar{U}}$, and work in $V[G]$. Let $\mathcal{F}=V\cap
P(\theta)$. Note that $\mathcal{F}$ is closed under restrictions and
branching. The latter is due to the theorem by Cummings and Woodin that
forcing with $R_{\bar{U}}$ does not add fresh subsets to $\theta$. For
$X,Y\in\mathcal{F}$, $X<_{\theta}Y$ iff for a tail of $\alpha\in C_{G}$,
$f_{X}(\alpha)<f_{Y}(\alpha)$. Let us verify that $<_{\theta}$ is indeed a
$\theta$-upward uniformly definable well-order.

  1. (1)

It can be readily checked that $<_{\theta}$ is definable in $H(\theta^{+})$
using parameters $C_{G}$ and $\vec{x}=\langle
x^{\alpha}_{i}:i<2^{\alpha},\alpha<\theta\rangle$.

  2. (2)

We check that $<_{\theta}$ is a well-order. It suffices to check that for any
$i_{0},i_{1}<2^{\theta}$, $x^{\theta}_{i_{0}}<_{\theta}x^{\theta}_{i_{1}}$ iff
$i_{0}<i_{1}$. This follows from Claim 3.2.

  3. (3)

Let $N$ be a $\theta$-model containing $C_{G}$ and $\vec{x}$. Then for any
$X,Y\in N\cap V$, $X<_{\theta}Y$ iff for a tail $\alpha\in C_{G}$,
$f_{X}(\alpha)<f_{Y}(\alpha)$, which is absolute between $N$ and $V[G]$.

∎

######  Remark 3.5.

By Proposition 3.4, Theorem 2.18 and a theorem of Magidor asserting that in
any Radin forcing extension where $\theta$ remains regular
$V$-$\diamondsuit(\theta)$ is equivalent to $\diamondsuit(\theta)$, we can
conclude that in any Radin forcing extension, if $\theta$ is weakly compact
then $\diamondsuit(\theta)$ holds. This was originally proved in [BNZ], where
the proof there is different and contains some other information. For example,
it contains correspondences between certain large cardinal properties in the
Radin model and measure sequences satisfying various properties in the ground
model.

###  3.2. Separating $\diamondsuit_{\vec{\mathrm{NS}},\vec{\beth}}$ from
$\diamondsuit$

In this subsection, we show that without the GCH assumption, the conclusion of
Proposition 2.14 may not hold.

######  Theorem 3.6.

Suppose there is a strong cardinal $\theta$ in $V$. Then in some forcing
extension of $V$, we have
$\diamondsuit_{\vec{\mathrm{NS}},\vec{\beth}}(Sing^{\theta}_{>\omega})$ holds
but $\diamondsuit(\theta)$ fails.

###### Proof.

By some preliminary forcing, we may suppose in $V$, there is a strong cardinal
$\theta$ with $2^{\theta}=\lambda>cf(\lambda)=\theta^{+}$. Let $\bar{U}$ be a
measure sequence of length $\theta^{+}$ derived from an embedding $j:V\to M$.
We show that in $V^{R_{\bar{U}}}$,
$\diamondsuit_{\vec{\mathrm{NS}},\vec{\beth}}(Sing^{\theta}_{>\omega})$ holds.
In fact, we show something stronger. Note that by a theorem of Woodin,
$\diamondsuit(\theta)$ fails in this model.

For each $\alpha<\theta$, let $\langle x^{\alpha}_{j}:j<2^{\alpha}\rangle$
enumerate the subsets of $\alpha$. Let $l_{\alpha}:cf(2^{\alpha})\to
2^{\alpha}$ be a cofinal map. By the hypothesis, we know a
$\bigcap\bar{U}$-measure one set of $\alpha$ satisfies
$cf(2^{\alpha})=\alpha^{+}$. For each measure sequence $\bar{w}$, define
$t_{\bar{w}}$ to be a function on $\mathcal{MS}\cap V_{\theta(\bar{w})}$ such
that $\bar{v}$ is mapped to the index of
$x^{\theta(\bar{w})}_{l_{\theta(\bar{w})}(lh(\bar{w}))}\cap\theta(\bar{v})$.
We argue that in $V^{R_{\bar{U}}}$, the sequence $\langle
t_{\bar{w}}\restriction\mathcal{MS}_{G}:\bar{w}\in
G,\omega<cf^{V[G]}(\theta(\bar{w}))<\theta(\bar{w})\rangle$ is a
$\diamondsuit_{\vec{\mathrm{NS}},\vec{\beth}}(Sing^{\theta}_{>\omega})$-sequence.

Given $p\in R_{\bar{U}}$ and $\dot{g}\in\Pi_{\alpha\in C_{G}}2^{\alpha}$, by
directly extending $p$ if necessary, we may assume that for any $\bar{w}\in
A_{p}$,

  * •

$cf(2^{\theta(\bar{w})})=\theta(\bar{w})^{+}$,

  * •

$p{}^{\frown}\bar{w}\Vdash\dot{g}(\theta(\bar{w}))=h(\bar{w})$ for some
$R_{\bar{w}}$-name $h(\bar{w})$ for an element in $2^{\theta(\bar{w})}$.

Since $R_{\bar{w}}$ satisfies $\theta(\bar{w})^{+}$-c.c, we may find a
function $m$ with domain $A_{p}$ such that for each $\bar{w}\in A_{p}$,
$m(\bar{w})\in\theta(\bar{w})^{+}$ such that
$p{}^{\frown}\bar{w}\Vdash\dot{g}(\theta(\bar{w}))<l_{\theta(\bar{w})}(m(\bar{w}))$.
It suffices to prove the following claim.

######  Claim 3.7.

The collection $B=\\{\bar{w}:\\{\bar{v}\in
V_{\theta(\bar{w})}:p{}^{\frown}\bar{w}{}^{\frown}\bar{v}\Vdash\dot{g}(\theta(\bar{v}))<t_{\bar{w}}(\bar{v})\\}\in\bigcap\bar{w}\\}$
is $\bar{U}$-positive.

###### Proof.

Let $f:\theta^{+}\to\theta^{+}$ be defined such that
$f(\xi)=j(m)(\bar{U}\restriction\xi)$. Let $C$ be the collection closure
points of $f$. Namely, $\tau\in C$ iff for any $\gamma<\tau$,
$f(\gamma)<\tau$. Fix $\tau\in C$ with $cf(\tau)\in(\omega,\theta)$. We want
to show that $B\in U(\tau)$. It suffices to show $\\{\bar{v}\in
V_{\theta}:j(p){}^{\frown}\bar{U}\restriction\tau{}^{\frown}\bar{v}\Vdash
j(\dot{g})(\theta(\bar{v}))<j(t)_{\bar{U}\restriction\tau}(\bar{v})\\}\in\bigcap\bar{U}\restriction\tau$.
By the definition, it suffices to show that for any $\xi<\tau$,

$j(j(p)){}^{\frown}j(\bar{U}\restriction\tau){}^{\frown}\bar{U}\restriction\xi\Vdash
j(j(\dot{g}))(\theta)<j(j(t)_{\bar{U}\restriction\tau})(\bar{U}\restriction\xi).$

Fix $\xi<\tau$. By elementarity,

  * •

$j(j(p)){}^{\frown}\bar{U}\restriction\xi\Vdash
j(j(\dot{g}))(\theta)<j(j(l))_{\theta}(j(j(m))(\bar{U}\restriction\xi))=j(l)_{\theta}(j(m)(\bar{U}\restriction\xi))=j(l)_{\theta}(f(\xi))$,
and

  * •

$j(p){}^{\frown}\bar{U}\restriction\tau\Vdash\forall\alpha<\theta,j(t)_{\bar{U}\restriction\tau}(\alpha)=index_{\alpha}(x^{\theta}_{i}\cap\alpha)$
where $i=j(l)_{\theta}(\tau)$.

As a result,
$j(j(p)){}^{\frown}j(\bar{U}\restriction\tau){}^{\frown}\bar{U}\restriction\xi$
forces that

$j(j(\dot{g}))(\theta)<j(l)_{\theta}(f(\xi))<j(l)_{\theta}(\tau)=index_{\theta}(j(x^{\theta}_{i})\cap\theta)=j(j(t)_{\bar{U}\restriction\tau})(\bar{U}\restriction\xi),$

as desired (we use the fact that $\tau$ is a closure point for $f$ in the
second inequality). ∎

∎

##  4\. Overlapping Extenders Forcing of Inaccessible length

We give a brief account of Gitik’s overlapping extenders forcing from [Git19]
with several minor changes in terminology and notations, which we adopt for
ease of presentation of later sections. We follow the Jerusalem forcing
convention of [Git19], by which a condition $p$ is stronger (more informative)
than $q$ is denoted by $p\geq q$.

For all relevant notations involving measures, extenders, and Prikry-type
forcings, we refer the reader to [Git10].

The forcing $\mathbb{P}_{\vec{E}}$ is based on a sequence of overlapping
extenders $\vec{E}=\langle E_{\alpha}\mid\alpha<\theta\rangle$ in $V$ that
satisfy several simple conditions. We list these conditions below and refer to
a sequence $\vec{E}$ which satisfies them as a _forcing suitable_ sequence.

######  Definition 4.1.

(Forcing suitable extender sequence)
Let $\vec{E}=\langle E_{\alpha}:\alpha<\theta\rangle$ be a sequence of
extenders whose critical points are
$\kappa_{\alpha}=\operatorname{cp}(E_{\alpha})$. The ultrapower embedding of
$E_{\alpha}$ is denoted by $j_{E_{\alpha}}:V\to M_{E_{\alpha}}$.

We say that $\vec{E}$ is _forcing suitable_ if it has the following
properties:

  1. (1)

The sequence of critical points
$\langle\kappa_{\alpha}\mid\alpha<\theta\rangle$ is strictly increasing and
discrete. Namely, for every $\beta<\theta$,
$\kappa_{\beta}>\vec{\kappa}_{\beta}:=\cup_{\alpha<\beta}\kappa_{\alpha}$.

  2. (2)

There exists some $\lambda\geq\cup_{\alpha<\theta}\kappa_{\alpha}$ such that
for each $\alpha<\theta$, $E_{\alpha}$ is a
$(\kappa_{\alpha},\lambda)$-extender consisting of a directed system of
$\kappa_{\alpha}$ complete measures
$E_{\alpha}=\\{E_{\alpha}(a),\pi_{a,b}:a\subseteq b\in[\lambda]^{<\omega}\\}$
whose direct limit embedding $j_{E_{\alpha}}:V\to M_{E_{\alpha}}$ satisfies
$H_{\lambda}\subseteq M_{E_{\alpha}}$.

  3. (3)

$E_{\alpha}\triangleleft E_{\beta}$ for all $\alpha<\beta$. Moreover, there
exists a function $e^{\beta}_{\alpha}:\kappa_{\beta}\to V_{\beta}$ such that
$E_{\alpha}=j_{E_{\beta}}(e^{\beta}_{\alpha})(\kappa_{\beta})$.

Fix a sequence of functions $\vec{e}=\langle
e^{\beta}_{\alpha}\mid\alpha<\beta<\theta\rangle$ as above. It follows that
for each $\beta<\theta$, the function
$\lambda^{\beta}:\kappa_{\beta}\to\kappa_{\beta}$ which maps
$\rho<\kappa_{\beta}$ to the length of the extender $e^{\beta}_{0}(\rho)$
(i.e., $e^{\beta}_{0}(\rho)$ is a
$(\kappa_{0},\lambda^{\beta}(\rho))$-extender) represents
$\lambda=j_{E_{\beta}}(\lambda^{\beta})(\kappa_{\beta})$ in $M_{E_{\beta}}$.

We denote $\lambda$ by $\lambda(\vec{E})$, and for $\beta<\theta$, denote the
restricted sequences $\langle E_{\gamma}\mid\gamma\geq\beta\rangle$ by
$\vec{E}\setminus\beta$, and $\langle E_{\alpha}\mid\alpha<\beta\rangle$ by
$\vec{E}\restriction\beta$.

We start with an auxiliary poset $\mathbb{P}^{*}_{\vec{E}}$.

######  Definition 4.2.

  1. (1)

The poset $\mathbb{P}^{*}_{E}$ associated to a $(\kappa,\lambda)$-extender $E$
is the Cohen-type poset, consisting of partial functions $f:\lambda\to\kappa$
of size $|f|\leq\kappa$, and further satisfy that $f(\kappa)<\kappa$ is an
inaccessible cardinal. The order of $\mathbb{P}^{*}_{E}$ is containment.

We will occasionally write $\operatorname{Add}(\kappa^{+},\lambda)$ to denote
this forcing.

  2. (2)

For a sequence of extenders $\vec{E}=\langle
E_{\alpha}\mid\alpha<\theta\rangle$ the poset
$\mathbb{P}^{*}_{\vec{E}}=\prod_{\alpha<\theta}\mathbb{P}^{*}_{E_{\alpha}}$,
with the usual full-support product ordering.

It will also be useful to consider the tail-ordering
$\leq_{\operatorname{bd}}$ on $\mathbb{P}^{*}_{\vec{E}}$ by which a sequence
$\vec{f}=\langle f_{\alpha}\mid\alpha<\theta\rangle$ of
$f_{\alpha}\in\mathbb{P}^{*}_{E_{\alpha}}$ is
$\leq_{\operatorname{bd}}$-stronger than a sequence $\vec{g}=\langle
g_{\alpha}\mid\alpha<\theta\rangle$ if
$f_{\alpha}\geq_{\mathbb{P}^{*}_{E_{\alpha}}}g_{\alpha}$ for all but boundedly
many $\alpha<\theta$.

  3. (3)

Let $\vec{f}=\langle f_{\alpha}\mid\alpha<\theta\rangle$ be a sequence of
functions. For each $\beta<\theta$ for which
$\kappa_{\beta}\in\operatorname{dom}(f_{\beta})$, define

$\rho^{\vec{f}}_{\beta}=f_{\beta}(\kappa_{\beta}),$

and

$\lambda^{\vec{f}}_{\beta}=\lambda^{\beta}(\rho^{\vec{f}}_{\beta}).$

  4. (4)

Let $\vec{f}=\langle f_{\alpha}\mid\alpha<\theta\rangle$ be a sequence of
functions and $s\in[\theta]^{<\omega}$ such that
$\kappa_{\beta}\in\operatorname{dom}(f_{\beta})$ for each $\beta\in s$.

Define the sequence of extender $\vec{E}^{\vec{f},s}=\langle
E^{\vec{f},s}_{\alpha}\mid\alpha<\theta\rangle$ and sequence of lengths
$\vec{\lambda}^{\vec{f},s}$ as follows:

     * •

$E^{\vec{f},s}_{\alpha}=E_{\alpha}$ and
$\lambda^{\vec{f},s}_{\alpha}=\lambda(\vec{E})$ for every $\alpha\geq\max(s)$,

     * •

for $\alpha<\max(s)$, if $\beta=\min(s\setminus(\alpha+1))$ then
$E^{\vec{f},s}_{\alpha}=e^{\beta}_{\alpha}(\rho^{\vec{f}}_{\beta})$ and
$\lambda^{\vec{f},s}=\lambda_{\beta}^{\vec{f}}$.

  5. (5)

We say that a sequence of functions $\vec{f}=\langle
f_{\alpha}\mid\alpha<\theta\rangle$ is $(\vec{E},s)$-compatible if it
satisfies the following requirements:

     * •

$\kappa_{\beta}\in\operatorname{dom}(f_{\beta})$ for every $\beta\in s$,

     * •

for every $\alpha<\theta$, $E^{\vec{f},s}_{\alpha}$ is a
$(\kappa_{\alpha},\lambda^{\vec{f},s}_{\alpha})$-extender, and
$f_{\alpha}\in\mathbb{P}^{*}_{E^{\vec{f},s}_{\alpha}}$.

To move forward and define the domain of the poset $\mathbb{P}_{\vec{E}}$, one
needs to introduce the measures $E_{\alpha}(f_{\alpha})$ for
$f_{\alpha}\in\mathbb{P}^{*}_{E_{\alpha}}$.

######  Definition 4.3.

Let $E$ be a $(\kappa,\lambda)$-extender, with an associate ultrapower
embedding $j_{E}:V\to M_{E}$.

  1. (1)

For a subset $d\in[\lambda]^{\leq\kappa}$ with $\kappa\in d$, define:

     * •

$mc_{E}(d)=\\{(j_{E}(\tau),\tau)\mid\tau\in d\\}$, and

     * •

$E(d)=\\{X\subseteq[\lambda\times\kappa]^{<\kappa}\mid mc_{E}(d)\in
j_{E}(X)\\}$.

     * •

The set of “typical” $E(d)$ objects, denoted $OB_{E}(d)$ is the set of all
partial functions $\nu:d\to\kappa$ with the following properties:

       1. (a)

$\kappa\in\operatorname{dom}(\nu)$,

       2. (b)

$|\nu|=\nu(\kappa)$,

       3. (c)

$\nu$ is order-preserving.

  2. (2)

For a function $f\in\mathbb{P}^{*}_{E}$ with $\kappa\in\operatorname{dom}(f)$,
define $E(f)=E(\operatorname{dom}(f))$ and
$OB_{E}(f)=OB_{E}(\operatorname{dom}(f))$.

######  Definition 4.4.

(The domain of $\mathbb{P}_{\vec{E}}$)

As a set,
$\mathbb{P}_{\vec{E}}=\biguplus_{s\in[\theta]^{<\omega}}\mathbb{P}^{s}_{\vec{E}}$
is union of disjoint parts $\mathbb{P}^{s}_{\vec{E}}$,
$s\in[\theta]^{<\omega}$. For each $s\in[\theta]^{<\omega}$,
$\mathbb{P}^{s}_{\vec{E}}$ consists of sequences $p=\langle
p_{\alpha}\mid\alpha<\theta\rangle$, for which there exists a
$(\vec{E},s)$-compatible sequence of functions $\vec{f}=\langle
f_{\alpha}\mid\alpha<\theta\rangle$, and the following properties hold:

  1. (1)

For each $\alpha\in s$, $p_{\alpha}=\langle f_{\alpha}\rangle$,

  2. (2)

for each $\alpha\in\theta\setminus s$, $p_{\alpha}=\langle
f_{\alpha},A_{\alpha}\rangle$ where $A_{\alpha}\in
E^{\vec{f},s}_{\alpha}(f_{\alpha})$ is contained in
$OB_{E^{\vec{f},s}_{\alpha}}(f_{\alpha})$, and

  3. (3)

for any $\alpha<\alpha^{\prime}\in\theta$ and any $\nu\in
A_{\alpha^{\prime}}$,
$\operatorname{dom}(f_{\alpha})\subset\operatorname{dom}(\nu)$. In particular,
$\langle\operatorname{dom}(f_{\alpha}):\alpha<\theta\rangle$ is increasing.

For a condition $p\in\mathbb{P}_{\vec{E}}$ as above, we define

  * •

The support of $p$, denoted $s(p)$, is the unique $s\in[\theta]^{<\omega}$
such that $p\in\mathbb{P}^{s}_{\vec{E}}$.

  * •

$\vec{f}=\vec{f}^{p}$ and denote each $f_{\alpha}$ by $f^{p}_{\alpha}$, and
$A_{\alpha}$ by $A^{p}_{\alpha}$.

  * •

$\vec{E}^{p}=\vec{E}^{\vec{f},s}$ and $E^{p}_{\alpha}=E^{\vec{f},s}_{\alpha}$
for each $\alpha<\theta$.

  * •

$\vec{\lambda}^{p}=\vec{\lambda}^{\vec{f},s}$ and
$\lambda^{p}_{\alpha}=\lambda^{\vec{f},s}_{\alpha}$ for every $\alpha<\theta$.

######  Definition 4.5.

(The projection operator $\pi_{\nu}^{*}$ on functions and sets)

  1. (1)

Let $\nu\in OB_{E^{\prime}}(f^{\prime})$ for some extender $E^{\prime}$ which
is a $(\kappa^{\prime},\lambda)$-extender and
$f^{\prime}\in\mathbb{P}^{*}_{E^{\prime}}$ with
$\kappa^{\prime}\in\operatorname{dom}(\nu)$.

We define the projection operator $\pi^{*}_{\nu}$ on suitable functions $g$,
sets $A$, and extenders $E\triangleleft E^{\prime}$ with a designated
representing function $e$.

    1. (a)

For a function $g:\lambda\to\kappa$ for some $\kappa$, so that
$\operatorname{dom}(g)\subseteq\operatorname{dom}(\nu)$, we define
$\pi^{*}_{\nu}g$ to be the function $g^{\prime}$ given by
$\operatorname{dom}(g^{\prime})=\\{\nu(\tau)\mid\tau\in\operatorname{dom}(g)\\}$
and $g^{\prime}(\nu(\tau))=g(\tau)$ for every $\tau\in\operatorname{dom}(g)$.

    2. (b)

For a set $A$ consisting of functions $\mu\in OB_{E}(g)$ for some extender $E$
with critical point $\kappa<\nu(\kappa^{\prime})$, and a function $g$ with
$\operatorname{dom}(g)\subseteq\operatorname{dom}(\nu)$, (which implies that
$\operatorname{dom}(\mu)\subseteq\operatorname{dom}(\nu)$ for all $\mu\in A$)
we define

$\pi^{*}_{\nu}A=\\{\pi^{*}_{\nu}\mu\mid\mu\in A\\}.$

    3. (c)

Suppose that $E\triangleleft E^{\prime}$, with
$E=j_{E^{\prime}}(e)(\kappa^{\prime})$. Define
$\pi^{*}_{\nu}(E)=e(\nu(\kappa^{\prime}))$.

  2. (2)

Suppose that $\vec{E}=\langle E_{\alpha}\mid\alpha<\beta\rangle$ is a sequence
of length $\beta<\kappa^{\prime}$ with critical points
$\kappa_{\alpha}<\kappa^{\prime}$, so that $\vec{E}\in M_{E^{\prime}}$ is
represented by a prescribed sequence of functions $\vec{e}=\langle
e_{\alpha}\mid\alpha<\beta\rangle$. For every $\nu\in OB_{E^{\prime}}(g)$ for
some $g$ such that $\beta<\nu(\kappa^{\prime})$, define

$\pi^{*}_{\nu}(\vec{E})=\langle\pi^{*}_{\nu}(E_{\alpha})\mid\alpha<\beta\rangle=\langle
e_{\alpha}(\nu(\kappa^{\prime}))\mid\alpha<\beta\rangle$

and

$\pi^{*}_{\nu}(\mathbb{P}_{\vec{E}})=\mathbb{P}_{\pi^{*}_{\nu}(\vec{E})}.$

  3. (3)

Let $\vec{E}=\langle E_{\alpha}\mid\alpha<\beta\rangle$ be a sequence of
extenders as above, with a prescribed representing sequence of functions
$\vec{e}=\langle e_{\alpha}\mid\alpha<\beta\rangle$, and $p=\langle
p_{\alpha}\mid\alpha<\beta\rangle\in\mathbb{P}_{\vec{E}}$ so that
$\operatorname{dom}(f^{p}_{\alpha})\subseteq\operatorname{dom}(\nu)$ for every
$\alpha<\beta$.

Define $\pi^{*}_{\nu}(p)$ to be the sequence
$\langle\pi^{*}_{\nu}(p_{\alpha})\mid\alpha<\beta\rangle$, such that for every
$\alpha<\beta$,
$\pi^{*}_{\nu}(p_{\alpha})=\begin{cases}\langle\pi^{*}_{\nu}(f^{p}_{\alpha}),\pi^{*}_{\nu}(A^{p}_{\alpha})\rangle&\alpha\not\in
s(p)\\\ \langle\pi^{*}_{\nu}(f^{p}_{\alpha})\rangle&\alpha\in
s(p)\par\end{cases}$, with the above defined projections of functions and
measure-one sets of objects.

######  Definition 4.6 (One-point extensions and End extensions).

Given a condition $p\in\mathbb{P}_{\vec{E}}$ and an object $\nu\in
A^{p}_{\alpha}$ for some $\alpha\in\theta\setminus s$ define the one-point
extension $p^{+\nu}$ of $p$ by $\nu$ to be given by

  1. (1)

$p^{+\nu}\restriction\alpha=\pi^{*}_{\nu}(p\restriction\alpha)$,

  2. (2)

$p^{+\nu}_{\alpha}=(f^{p}_{\alpha})_{\nu}$, where for any
$i\in\operatorname{dom}(f^{p}_{\alpha})=\operatorname{dom}((f^{p}_{\alpha})_{\nu})$,
$(f^{p}_{\alpha})_{\nu}(i)=\begin{cases}\nu(i)&i\in\operatorname{dom}(\nu)\\\
f^{p}_{\alpha}(i)&i\not\in\operatorname{dom}(\nu)\end{cases}$.

  3. (3)

For $\beta>\alpha$, $p^{+\nu}_{\beta}=\langle
f^{p}_{\beta},A^{\prime}_{\beta}\rangle$, where $A^{\prime}_{\beta}=\\{\mu\in
A^{p}_{\beta}\mid\mu(\kappa_{\beta})>\lambda(\nu(\kappa_{\alpha}))\\}$

We say that a condition $p^{\prime}$ is an end extension of a condition $p$ if
it is a condition resulting from a sequence of one-point extensions by objects
$\nu_{0},\nu_{1},\dots,\nu_{k}$ for some finite $k$. We denote $p^{\prime}$ by
$p^{+\vec{\nu}}$ where $\vec{\nu}=\langle\nu_{0},\dots,\nu_{k}\rangle$.

######  Definition 4.7 (Direct extensions in $\mathbb{P}_{E}$ and the ordering
of $\mathbb{P}_{E}$).

Let $p,p^{*}\in\mathbb{P}_{\vec{E}}$. We say that $p^{*}$ is a direct
extension of $p$ (denoted $p^{*}\geq p$) if $s(p)=s(p^{*})$ and for every
$\alpha\in\theta\setminus s(p)$, $f^{p}_{\alpha}\subseteq f^{p^{*}}_{\alpha}$
and

$A^{p^{*}}_{\alpha}\subseteq\\{\nu\in
OB_{E^{p}_{\alpha}}(f^{p^{*}}_{\alpha}):\nu\restriction\operatorname{dom}(f^{p}_{\alpha})\in
A^{p}_{\alpha}\\}$

For conditions $p,q\in\mathbb{P}_{\vec{E}}$, $q$ extends $p$ (namely,
$q\geq_{\mathbb{P}_{\vec{E}}}p$) if it is obtained from $p$ by finitely many
applications of direct extensions and end extensions.

######  Remark 4.8.

Assuming GCH, $\mathbb{P}_{\vec{E}}$ has the following basic properties:

  1. (1)

$\mathbb{P}_{\vec{E}}$ has size $|\mathbb{P}_{\vec{E}}|=\lambda$ and satisfies
the $\theta^{++}$-c.c.

  2. (2)

for every $p\in\mathbb{P}_{\vec{E}}$ and $\alpha\not\in s(p)$ and $\nu\in
A^{p}_{\alpha}$, $\mathbb{P}_{E}/p^{+\nu}$ factors to a product
$\mathbb{P}_{\pi_{\nu}^{*}(\vec{E}\restriction\alpha)}/(p\restriction\alpha)\times\mathbb{P}_{\vec{E}\backslash\alpha}/(p\downharpoonright\alpha)$,
whose first component satisfies $\bar{\kappa}_{\alpha}^{++}$-c.c, and the
direct extension of the second component is $\kappa_{\alpha}^{+}$-closed.

###  4.1. Prikry-type properties and capturing dense open sets

######  Definition 4.9.

Let $p\in\mathbb{P}_{\vec{E}}$ be a condition. An $(p,n)$-fat-tree is a tree
$T$ of height $n$, with the following properties:

  1. (1)

$stem(T)=\langle\rangle$ is the empty sequence,

  2. (2)

for every sequence $\vec{\nu}=\langle\nu_{i}\mid i<k\rangle\in T$,
$p^{+\vec{\nu}}$ is an end-extension of $p$

  3. (3)

for every sequence $\vec{\nu}=\langle\nu_{i}\mid i<k\rangle\in T$, of non-
maximal length $k<n$, there exists some $\alpha_{\vec{\nu}}<\theta$ so that
$\operatorname{succ}_{T}(\vec{\nu})\in
E^{p}_{\alpha_{\vec{\nu}}}(f^{p}_{\alpha_{\vec{\nu}}})$,

  4. (4)

$\alpha_{\vec{\nu}}<\alpha_{\vec{\nu}{}^{\frown}\langle\mu\rangle}$ for every
$\vec{\nu}\in T$ and $\mu\in\operatorname{succ}_{T}(\vec{\nu})$.

We say that a $T$ is fully compatible with $p$ if
$\operatorname{succ}_{T}(\vec{\nu})=A^{p}_{\alpha_{\vec{\nu}}}$ for every non-
maximal sequence $\vec{\nu}\in T$.

The following is a simple consequence of the order definition of
$\mathbb{P}_{\vec{E}}$.

######  Lemma 4.10.

  1. (1)

If $T$ is a $(p,n)$-fat-tree which is fully compatible with $p$, then the set
of end-extensions $\\{p^{+\vec{\nu}}\mid\vec{\nu}\in T,|\vec{\nu}|=n\\}$ is
predense in $\mathbb{P}_{\vec{E}}/p$.

  2. (2)

For every condition $p\in\mathbb{P}_{\vec{E}}$ and a $(p,n)$-fat-tree $T$,
there exists a direct extension $p^{*}\geq^{*}p$ and a $(p^{*},n)$-fat subtree
$T^{*}$ of $T$ so that $T^{*}$ is fully compatible with $p^{*}$.

The following statement is known as the strong Prikry Lemma. We omit the proof
as it is similar to the argument in Gitik’s singular form from [Git19].

######  Proposition 4.11.

(Strong Prikry Property)
Let $D\subseteq\mathbb{P}_{\vec{E}}$ be a dense open set. For every condition
$p\in\mathbb{P}_{\vec{E}}$ there is a direct extension $p^{*}\geq^{*}p$ and a
$(p^{*},k)$-fat-tree $T$ which is fully compatible with $p^{*}$ such that for
every $\vec{\nu}\in T$ of maximal length $|\vec{\nu}|=k$,
$(p^{*})^{+\vec{\nu}}\in D$.

From the strong Prikry Lemma, it is easy to deduce the Prikry property.

######  Corollary 4.12.

$\mathbb{P}_{\vec{E}}$ satisfies the Prikry Property, namely, for any
condition $p\in\mathbb{P}_{\vec{E}}$ and any statement $\varphi$ in the
forcing language, there exists an direct extension $q$ of $p$ such that
$q\Vdash\varphi$ or $q\Vdash\neg\varphi$.

Combining Remark 4.8 and the Prikry property 4.12 shows that $\mathbb{P}_{E}$
does not collapse cardinals up to $\theta$ or above $\theta^{+}$

######  Corollary 4.13.

$\mathbb{P}_{\vec{E}}$ preserves all cardinals $\tau\leq\theta$ or
$\tau>\theta^{+}$.

Our next goal is to verify $\mathbb{P}_{\vec{E}}$ does not collapse
$\theta^{+}$ or change the cofinality of cardinals. For this we expand the
description of the strong Prikry property for meeting a single dense open set
$D$, to one which simultaneously deals with $\theta$-many dense open sets.

######  Definition 4.14.

Let $\vec{D}=\langle
D_{\beta}\subseteq\mathbb{P}_{\vec{E}}:\beta<\theta\rangle$ be a sequence of
dense open sets, $p\in\mathbb{P}_{\vec{E}}$, We say that $\vec{D}$ is strongly
captured by $p$ if for every $\beta>\max(s(p))$ and $\nu\in A^{p}_{\beta}$
there exists a dense open subset
$D_{\nu,<\beta}\subseteq\pi^{*}_{\nu}(\mathbb{P}_{\vec{E}^{p}\restriction\beta})$
such that for every $w\in D_{\nu,<\beta}$, there is a
$(p^{+\nu}_{\geq\beta},n)$-fat tree $T^{\nu,w}$, for some $n<\omega$, such
that for every maximal sequence $\vec{\mu}\in T^{\nu,w}$,

$w{}^{\frown}p_{\geq\beta}^{+\langle\nu\rangle{}^{\frown}\vec{\mu}}\in
D_{\beta}.$

######  Lemma 4.15.

For every sequence of dense open sets $\vec{D}=\langle
D_{\beta}\subseteq\mathbb{P}_{\vec{E}}:\beta<\theta\rangle$ and
$p\in\mathbb{P}_{\vec{E}}$ there exists a direct extension $p^{*}\geq^{*}p$
which strongly captures $\vec{D}$.

Before turning to the proof of the Lemma, we introduce a useful terminology of
totally $N$-generic conditions.

######  Definition 4.16.

Let $(\mathbb{P}^{*},\leq^{*})$ be a poset which belongs to $H_{\chi}$ for
some regular cardinal $\chi$, and
$N\prec(H_{\chi},\in,\mathbb{P}^{*},\leq^{*})$. We say that a condition
$f\in\mathbb{P}$ is totally $N$-generic if it bounds an $N$-generic filter for
$\mathbb{P}^{*}$. Namely, the set $\\{f^{\prime}\in
N\cap\mathbb{P}^{*}:f^{\prime}\leq f\\}$ is generic over $N$.

###### Proof.

To simplify the presentation, we assume that $s(p)=\emptyset$.

We recursively define a $\leq^{*}$-increasing $\langle
q(\beta):\beta\leq\theta\rangle$ such that

  1. (1)

$q_{0}=p$,

  2. (2)

$q(\beta)\restriction\alpha=q(\alpha)\restriction\alpha$ for any
$\alpha<\beta<\theta$.

At limit stages $\delta\leq\theta$, granted the existence of such a sequence,
we can define

$q(\delta)_{<\delta}=\bigvee_{\alpha<\delta}q(\alpha)_{<\alpha}$

and

$q(\delta)_{\geq\delta}=\bigvee_{\alpha<\delta}q(\alpha)_{\geq\delta}$

where $\bigvee$ is the natural limit condition obtained by taking coordinate-
wise unions of the Cohen functions and intersecting the appropriate pullbacks
of the measure one sets. Note that $q(\delta)$ is easily seen to be a
condition which satisfies the requirements above.

At stage $\beta+1$, given $q(\beta)$, we take an elementary submodel $N\prec
H(\chi)$ containing relevant objects including
$\mathbb{P}_{\vec{E}},D=D_{\beta}$, and $q(\beta)$ with $|N|=\kappa_{\beta}$
and ${}^{<\kappa_{\beta}}N\subset N$. By the $\kappa_{\beta}^{+}$-closure of
the direct extension ordering of
$\mathbb{P}^{*}_{E^{p}(\beta)}\times\mathbb{P}^{*}_{\vec{E}\setminus\beta}$,
we can find a totally $N$-generic condition
$f^{*}_{\beta}{}^{\frown}r^{*}_{>\beta}$ with
$\operatorname{dom}(f^{*}_{\beta})=N\cap\lambda(\vec{E}^{p})$. Let

$A_{\beta}^{*}=\\{\nu\in
OB_{E_{\beta}}(f^{*}_{\beta})\mid\nu\restriction\operatorname{dom}(f^{q(\beta)}_{\beta})\in
A^{q(\beta)}_{\beta}\\},$

and define

$q(\beta+1)=q(\beta)\restriction\beta{}^{\frown}\langle
f^{*}_{\beta},A^{*}_{\beta}\rangle{}^{\frown}r^{*}_{>\beta}.$

For each $\nu\in A^{*}_{\beta}$, the forcing above $q(\beta+1)^{+\nu}$ breaks
into a product
$\pi^{*}_{\nu}(\mathbb{P}_{\vec{E}^{p}\restriction\beta})\times\mathbb{P}_{\vec{E}^{p}\setminus\beta}$.
For each $w\in\pi^{*}_{\nu}(\mathbb{P}_{\vec{E}^{p}\restriction\beta})$, let
$D_{w,\geq\beta}\subseteq\mathbb{P}_{\vec{E}^{p}\setminus\beta}$ be the set of
all condition $q\geq q(\beta)^{+\nu}_{\geq\beta}$ such that either

  * •

$q$ forces that there is no
$q^{\prime}\in\dot{G}(\mathbb{P}_{\vec{E}^{p}\setminus\beta})$ for which
$w{}^{\frown}q^{\prime}\in D$, or

  * •

$w{}^{\frown}q\in D$.

$D_{w,\geq\beta}$ is clearly dense open in
$\mathbb{P}_{\vec{E}^{p}\setminus\beta}$ and belongs to $N$.

By the strong Prikry property, there exists a dense open set
$D^{*}_{w,\geq\beta}\subseteq\mathbb{P}^{*}_{E_{\beta}^{p}}\times\mathbb{P}_{\vec{E}^{p}\setminus(\beta+1)}$
in the direct extension ordering, such that for every $q^{\nu}_{\geq\beta}\in
D^{*}_{w,\geq\beta}$ there is a $(q^{\nu}_{\geq\beta},n_{\nu,w})$-fat-tree
$T^{\nu,w}$, such that for every maximal sequence $\vec{\mu}\in T^{\nu,w}$,
$(q^{\nu}_{\geq\beta})^{+\vec{\mu}}\in D_{w,\geq\beta}$.

Since $\nu\in N$, $q(\beta)^{+\nu},D_{w,\geq\beta}$, and $D^{*}_{w,\geq\beta}$
belong to $N$ as well. It follows from our $N$-genericity assumption of
$q(\beta+1)$, that $q(\beta+1)^{+\nu}_{\geq\beta}\in D^{*}_{w,\geq\beta}$.

Let

$D_{\nu,<\beta}=\\{w\in\pi^{*}_{\nu}(\mathbb{P}_{\vec{E}^{p}\restriction\beta})\mid\exists
q^{\prime}\geq q(\beta+1)^{+\nu}_{\geq\beta}.\ w{}^{\frown}q^{\prime}\in
D.\\}$

$D_{\nu,<\beta}$ is clearly dense open in
$\pi^{*}_{\nu}(\mathbb{P}_{\vec{E}^{p}\restriction\beta})$. Fix $w\in
D_{\nu,<\beta}$. We verify that for every maximal sequence in the associated
$(q(\beta+1)^{+\nu}_{\geq\beta},n_{\nu,w})$-fat-tree $T^{\nu,w}$, $\vec{\mu}$,
the condition
$w{}^{\frown}q(\beta+1)^{+\langle\nu\rangle{}^{\frown}\vec{\mu}}_{\geq\beta}$
belongs to $D$.

By the definition of $D_{w,\geq\beta}$, it suffices to verify that the
condition $q(\beta+1)^{+\langle\nu\rangle{}^{\frown}\vec{\mu}}_{\geq\beta}$
forces the statement $\sigma_{w}$ in forcing language of
$\mathbb{P}_{\vec{E}^{p}\setminus\beta}$,

$\sigma_{w}:=\exists q^{\prime}\in\dot{G}.\ w{}^{\frown}q^{\prime}\in D.$

As a matter of fact, we show that the weaker condition
$q(\beta+1)^{+\langle\nu\rangle}$ forces $\sigma_{w}$. To see this, recall
that by the Prikry Lemma applied to
$\mathbb{P}^{*}_{E_{\beta}^{p}}\times\mathbb{P}_{\vec{E}^{p}\setminus(\beta+1)}$,
the set of direct extensions of
$q(\beta)^{+\nu\restriction\operatorname{dom}(f^{q}_{\beta})}_{\geq\beta}$
deciding $\sigma_{w}$ is dense open above
$q(\beta)^{+\nu\restriction\operatorname{dom}(f^{q}_{\beta})}_{\geq\beta}$ in
the direct extension ordering, and belongs to $N$. It follows that
$q(\beta+1)^{+\nu}_{\geq\beta}$ decides $\sigma_{w}$, and since $w\in
D_{\nu,<\beta}$ it must be that

$q(\beta+1)^{+\nu}_{\geq\beta}\Vdash\sigma_{w}.$

∎

######  Proposition 4.17.

For any $p\in\mathbb{P}_{\vec{E}}$ and for any name
$\dot{f}:\theta\to\operatorname{Ord}$, there exists a direct extension
$q\geq^{*}p$ and a function $g:\theta\to[\operatorname{Ord}]^{<\theta}$ such
that $q\Vdash\dot{f}(\alpha)\in g(\alpha)$ for all $\alpha<\theta$.

###### Proof.

We may without loss of generality assume $s(p)=\emptyset$. Since otherwise,
letting $\gamma=\max s(p)$, then we can decompose $\mathbb{P}_{\vec{E}}/p$
into
$\mathbb{P}_{0}\times\mathbb{P}_{1}=\mathbb{P}_{\vec{E}^{p}\restriction\gamma+1}/p\restriction\gamma+1\times\mathbb{P}_{\vec{E}\backslash\gamma+1}/p_{>\gamma}$.
Since $\mathbb{P}_{0}$ has $\kappa_{\gamma}^{+}$-c.c, we can work in
$\mathbb{P}_{1}$ above $p_{>\gamma}$ where the support is empty to obtain the
desired conclusion.

Let $D_{\beta}=\\{r\in\mathbb{P}_{\vec{E}}:\exists\delta\
r\Vdash\dot{f}(\beta)=\delta\\}$ for $\beta<\theta$. It is easy to check that
$D_{\beta}$ is a dense open set. Apply Lemma 4.15 to $p$ and $\vec{D}=\langle
D_{\beta}:\beta<\theta\rangle$ to get

  * •

$q\geq^{*}p$ and

  * •

for any $\nu\in A^{q^{*}}_{\beta}$, a dense open subset $D_{\nu,<\beta}$ of
$\pi_{\nu}^{*}(\mathbb{P}_{\vec{E}\restriction\beta})$ such that for any $w\in
D_{\nu,<\beta}$, there exists a $(q^{+\nu}_{\geq\beta},n^{\nu,w})$-fat tree
$T^{\nu,w}$ such that for any maximal sequence $\vec{\mu}\in T^{\nu,w}$,
$(w{}^{\frown}q)^{+\langle\nu\rangle{}^{\frown}\vec{\mu}}\in D_{\beta}$, as
witnessed by $\delta^{\nu,w,\vec{\mu}}\in\operatorname{Ord}$.

Define $g(\beta)=\bigcup\\{\delta^{\nu,w,\vec{\mu}}:\nu\in A^{q}_{\beta},w\in
D_{\nu,<\beta},\vec{\mu}$ is a maximal sequence from $T^{\nu,w}\\}$. It is
easy to see that $q$ and $g$ are as sought. ∎

######  Corollary 4.18.

  1. (1)

The cofinalities and cardinalities of $\theta$ and $\theta^{+}$ are preserved,

  2. (2)

$\mathbb{P}_{\vec{E}}$ preserves all cofinalities and cardinalities, and

  3. (3)

For every $\mathbb{P}_{\vec{E}}$-name of a closed unbounded subsets $\dot{D}$
of $\theta$ and a condition $p$, there is a direct extensions $q$ of $p$ and a
closed unbounded set $C\subseteq\theta$ such that
$q\Vdash\check{C}\subseteq\dot{D}$.

###### Proof.

  1. (1)

Let $p\in\mathbb{P}_{\vec{E}}$ and $\dot{f}:\rho\to\theta$, $\rho<\theta$ be
given. By Proposition 4.17, there is $q\geq p$ and
$g:\rho\to[\theta]^{<\theta}$ such that $q\Vdash\dot{f}(\alpha)\in g(\alpha)$
for any $\alpha<\rho$. Then $q\Vdash
ran(\dot{f})\subset\sup\bigcup\\{g(\alpha):\alpha<\rho\\}<\theta$ since
$\theta$ is regular. Similar argument shows that the regularity of
$\theta^{+}$ is preserved.

  2. (2)

This is an immediate consequence of previous item (1) and the factoring
property of $\mathbb{P}_{\vec{E}}$ from Remark 4.8.

  3. (3)

Given $p\in\mathbb{P}_{\vec{E}}$ and $\dot{D}$ a $\mathbb{P}_{\vec{E}}$-name
for a club in $\theta$, applying Proposition 4.17, there exist $q\geq p$ and
$g:\theta\to[\theta]^{<\theta}$ such that $q\Vdash\min\dot{D}-\alpha\in
g(\alpha)$ for all $\alpha<\theta$. Let $h(\alpha)=\sup g(\alpha)<\theta$.
Then $q\Vdash cl_{h}\subset\dot{D}$ where
$cl_{h}=\\{\eta<\theta:\forall\gamma<\eta\ h(\gamma)<\eta\\}$ is a club in
$\theta$ in $V$.

∎

##  5\. The failure of
$\diamondsuit_{\vec{\mathrm{NS}},\vec{\lambda}}(\theta)$ in
$V^{\mathbb{P}_{\vec{E}}}$

######  Lemma 5.1.

Given a condition $p\in\mathbb{P}_{\vec{E}}$ and a $\mathbb{P}_{\vec{E}}$-name
for a sequence $\langle\dot{X}_{\alpha}\subset\alpha:\alpha<\theta\rangle$,
there exists a direct extension $q$ of $p$ and a sequence $\langle
X^{\alpha}_{\nu}:\nu\in X^{q}_{\alpha},\alpha<\theta\rangle$ such that for
each $\alpha<\theta$, $\alpha\not\in s(p)$,

  1. (1)

for each $\nu\in A^{q}_{\alpha}$, $X^{\alpha}_{\nu}$ is a
$\pi^{*}_{\nu}(\mathbb{P}_{\vec{E}})$-name,

  2. (2)

for each $\nu\in A^{q}_{\alpha}$,
$q^{+\nu}\Vdash\dot{X}^{\alpha}_{\nu}=\dot{X}_{\alpha}$.

###### Proof.

To simplify the presentation, we may assume that $s(p)=\emptyset$. This is
without loss of generality, since we can proceed by induction on the size of
$s(p)$. More precisely, suppose $\alpha=\max s(p)$, then
$\mathbb{P}_{\vec{E}}/p$ could be broken into
$(\mathbb{P}_{\vec{E}^{p}\restriction\alpha}/p\restriction\alpha)\times(\mathbb{P}_{\vec{E}\setminus\alpha}/p\downharpoonright\alpha)$.
Directly extending $p\downharpoonright\alpha$ if necessary, we may assume that
$\langle\dot{X}_{\beta}:\beta<\alpha\rangle$ is a
$\mathbb{P}_{\vec{E}^{p}\restriction\alpha}/p\restriction\alpha$-name. Then
the conclusion follows from the induction hypothesis applied to
$\mathbb{P}_{\vec{E}^{p}\restriction\alpha}/p\restriction\alpha$ and the
simplified case applied to
$\mathbb{P}_{\vec{E}\setminus\alpha}/p\downharpoonright\alpha$.

We recursively define a $\leq^{*}$-increasing $\langle
q_{\alpha}:\alpha<\theta\rangle$ such that

  1. (1)

$q_{0}=p$,

  2. (2)

$q_{\beta}\restriction\alpha=q_{\alpha}\restriction\alpha$ for any
$\alpha<\beta<\theta$,

  3. (3)

for each $\nu\in A^{q_{\alpha+1}}_{\alpha}$, there exists a
$\pi^{*}_{\nu}(\mathbb{P}_{\vec{E}\restriction\alpha})$-name
$\dot{X}^{\alpha}_{\nu}$ such that $q_{\alpha+1}^{+\nu}$ forces that
$\dot{X}_{\alpha}=\dot{X}^{\alpha}_{\nu}$.

Granted the existence of such a sequence, we can define
$q=\bigcup_{\alpha<\theta}q_{\alpha}\restriction\alpha$, which is easily seen
to be as sought. To construct the sequence, at limit stage $\alpha<\theta$, we
just let $q_{\alpha}=\bigvee_{\beta<\alpha}q_{\beta}$. At stage $\alpha+1$,
let $N\prec H(\chi)$ be an elementary submodel containing relevant objects
including $\mathbb{P}_{\vec{E}}$, $q_{\alpha}$ and $\dot{X}_{\alpha}$, such
that $|N|=\kappa_{\alpha}$ and ${}^{<\kappa_{\alpha}}N\subset N$. By the
$\kappa_{\alpha}^{+}$-closure of
$(\mathbb{P}^{*}_{E(\alpha)}/f^{q_{\alpha}}_{\alpha}\times\mathbb{P}^{*}_{\vec{E}\backslash\alpha}/\vec{f}^{q_{\alpha}}\downharpoonright\alpha)$,
we can find an $N$ totally generic condition $f^{*}_{\alpha}{}^{\frown}\langle
r^{*}_{\beta}:\beta>\alpha\rangle$ with
$\operatorname{dom}(f^{*}_{\alpha})=N\cap\lambda$. Let $A_{\alpha}^{*}$ be
$\\{\nu\in
E(f^{*}_{\alpha}):\nu\restriction\operatorname{dom}(f^{q_{\alpha}}_{\alpha})\in
A^{q_{\alpha}}_{\alpha}\\}$. Let us show that
$q_{\alpha+1}=q_{\alpha}\restriction\alpha{}^{\frown}(f^{*}_{\alpha},A^{*}_{\alpha}){}^{\frown}\langle
r^{*}_{\beta}:\beta>\alpha\rangle$ is as desired. Let $\nu\in A^{*}_{\alpha}$
and $q^{*}=\pi^{*}_{\nu}(q_{\alpha}\restriction\alpha)$. Notice that since
$|\nu|<\kappa_{\alpha}$, $\nu\subset N$, ${}^{<\kappa_{\alpha}}N\subset N$ and
$q^{*}\in V_{\theta}$, we have that $\nu,q^{*}\in N$. In $N$, consider the set

$D=\\{(f,\vec{r})\in(\mathbb{P}^{*}_{E_{\alpha}}/f^{q_{\alpha}}_{\alpha})\times(\mathbb{P}^{*}_{\vec{E}\backslash\alpha}/\vec{f}^{q_{\alpha}}\downharpoonright\alpha):$
$\text{there exists a
}\pi^{*}_{\nu}(\mathbb{P}_{\vec{E}\restriction\alpha})\text{-name }\dot{B},\
q^{*}{}^{\frown}f_{\nu}{}^{\frown}\vec{r}\Vdash\dot{B}=\dot{X}_{\alpha}\\}.$

Note that $D$ is dense in
$\mathbb{P}^{*}_{E(\alpha)}/f^{q_{\alpha}}_{\alpha}\times\mathbb{P}^{*}_{\vec{E}\backslash\alpha}/\vec{f}^{q_{\alpha}}\downharpoonright\alpha$
and

  * •

the number of (nice)
$\pi^{*}_{\nu}(\mathbb{P}_{\vec{E}\restriction\alpha})$-names for subsets of
$\alpha$ is $<\kappa_{\alpha}$ and

  * •

$(\mathbb{P}^{*}_{E(\alpha)}/f^{q_{\alpha}}_{\alpha})\times(\mathbb{P}_{\vec{E}\backslash\alpha}/q_{\alpha}\downharpoonright\alpha)$
satisfies the Prikry Property by Corollary 4.12 and is
$\kappa^{+}_{\alpha}$-closed.

As a result, $f^{*}_{\alpha}{}^{\frown}\langle
r^{*}_{\beta}:\beta>\alpha\rangle$ meets $D$, which in turns implies that
$q^{*}{}^{\frown}(f^{*}_{\alpha})_{\nu}{}^{\frown}\langle
r^{*}_{\beta}:\beta>\alpha\rangle$ forces that
$\dot{X}_{\alpha}=\dot{X}^{\alpha}_{\nu}$ for some
$\pi^{*}_{\nu}(\mathbb{P}_{\vec{E}\restriction\alpha})$-name
$\dot{X}^{\alpha}_{\nu}$. ∎

######  Definition 5.2.

Let $G\subset\mathbb{P}_{\vec{E}}$ be generic and in $V[G]$, define:

  1. (1)

for $i<\theta$, $\lambda_{i}=\lambda^{i}(f^{q}_{i}(\kappa_{i}))$ for some
(any) $q\in G$ with $i\in s(q)$;

  2. (2)

for each $\tau<\lambda$, $t_{\tau}$ is a function on $\theta$ such that for
any $\gamma<\theta$, $t_{\tau}(\gamma)$ is defined and is equal to $\tau_{k}$
iff there exists some condition $p\in G$ satisfying:

     * •

$\gamma\in s(p)$,

     * •

if we enumerate decreasingly
$\\{\gamma_{0},\cdots,\gamma_{k-1}\\}=s(p)-\gamma$ (in particular
$\gamma=\gamma_{k-1})$, then
$\tau=\tau_{0}\in\operatorname{dom}(f^{p}_{\gamma_{0}})$,
$\tau_{1}=f^{p}_{\gamma_{0}}(\tau_{0})\in\operatorname{dom}(f^{p}_{\gamma_{1}})$,
$\tau_{2}=f^{p}_{\gamma_{1}}(\tau_{1})\in\operatorname{dom}(f^{p}_{\gamma_{2}})$,
$\cdots,\tau_{k}=f^{p}_{\gamma_{k-1}}(\tau_{k-1})$.

######  Proposition 5.3.

Assume that $\lambda\geq\theta^{+}$. For any $p\in\mathbb{P}_{\vec{E}}$ and a
$\mathbb{P}_{\vec{E}}$-name for a function in
$\dot{g}\in\Pi_{i<\theta}\dot{\lambda}_{i}$, it is true that
$p\Vdash\exists\tau<\lambda,\dot{g}<^{*}\dot{t}_{\tau}$.

###### Proof.

Given $p\in\mathbb{P}_{\vec{E}}$, and a name for a function
$\dot{g}\in\Pi_{i<\theta}\dot{\lambda}_{i}$, we find a direct extension $q$ of
$p$ and some $\tau<\lambda$ such that $q\Vdash\dot{g}<^{*}\dot{t}_{\tau}$. We
may without loss of generality assume that $s(p)=\emptyset$. The reason is
that if $\alpha=\max s(p)$, $\mathbb{P}_{\vec{E}}/p$ can be naturally
decomposed at $\alpha$ to
$\mathbb{P}_{\vec{E}^{p}\restriction\alpha}/p\restriction\alpha\times\mathbb{P}_{\vec{E}\backslash\alpha}/p\downharpoonright\alpha$
with the lower part of the decomposition being small. Then we can work with
the upper part of the decomposition.

Applying Lemma 5.1 and directly extending $p$ is necessarily, we may assume
that for any $\alpha<\theta$ and $\nu\in A^{p}_{\alpha}$,
$p^{+\nu}\Vdash\dot{g}(\alpha)=\dot{g}^{\alpha}_{\nu}$ where
$\dot{g}^{\alpha}_{\nu}$ is a
$\pi^{*}_{\nu}(\mathbb{P}_{\vec{E}\restriction\alpha})$-name for an ordinal
below $\lambda^{\alpha}(\nu(\kappa_{\alpha}))$. For each $\nu\in
A^{p}_{\alpha}$, since $\pi^{*}_{\nu}(\mathbb{P}_{\vec{E}\restriction\alpha})$
satisfies $\vec{\kappa}_{\alpha}^{++}$-c.c and
$\lambda^{\alpha}(\nu(\kappa_{\alpha}))$ is a regular cardinal
$>\vec{\kappa}_{\alpha}^{++}$, there exists an ordinal
$\gamma_{\nu}^{\alpha}<\lambda^{\alpha}(\nu(\kappa_{\alpha}))$ such that
$\Vdash_{\pi^{*}_{\nu}(\mathbb{P}_{\vec{E}\restriction\alpha})}\dot{g}^{\alpha}_{\nu}<\gamma_{\nu}^{\alpha}$.
For each $\alpha<\theta$, let
$\gamma_{\alpha}=j_{\alpha}(\nu\mapsto\gamma_{\nu}^{\alpha})(mc_{E_{\alpha}}(f^{p}_{\alpha}))<\lambda$.
Let $\tau\in\lambda-(\sup_{\alpha<\theta}\gamma_{\alpha}+1)$. Now we directly
extend $p$ to $q$ such that for each $\alpha<\theta$,

  * •

$\operatorname{dom}(f^{q}_{\alpha})$ contains $\tau$, and

  * •

for any $\nu\in A^{q}_{\alpha}$, it holds that
$\tau\in\operatorname{dom}(\nu)$ and $\nu(\tau)>\gamma_{\nu}^{\alpha}$.

It is now easy to see that $q$ is as desired, namely,
$q\Vdash\dot{g}<^{*}\dot{t}_{\tau}$. ∎

######  Proposition 5.4.

In $V[G]$, $\diamondsuit_{\vec{\mathrm{NS}},\vec{\lambda}}(\theta)$ fails,
where $\vec{\lambda}=\langle\lambda_{i}:i<\theta\rangle$.

###### Proof.

Given $p\in\mathbb{P}_{\vec{E}}$ and a $\mathbb{P}_{\vec{E}}$-name
$\langle\dot{g}_{\gamma}\in\Pi_{i<\gamma}\dot{\lambda}_{i}:\gamma\in\lim\theta\rangle$,
we need to find $q\geq p$ and $\tau<\lambda$ such that $q\Vdash``\text{for a
tail
}\gamma\in\lim\theta,\dot{g}_{\gamma}<^{*}\dot{t}_{\tau}\restriction\gamma$”.
As in the proof of Proposition 5.3, we may assume $s(p)=\emptyset$. By Lemma
5.1, directly extending $p$ if necessary, we may assume that for any
$\gamma\in\lim\theta$ and $\nu\in A^{p}_{\gamma}$,
$p^{+\nu}\Vdash\dot{g}_{\gamma}=\dot{g}^{\gamma}_{\nu}$ for some
$\pi^{*}_{\nu}(\mathbb{P}_{\vec{E}\restriction\gamma})$-name
$\dot{g}^{\gamma}_{\nu}$.

Fix $\nu\in A^{p}_{\gamma}$. By Proposition 5.3 applied to
$\pi^{*}_{\nu}(\mathbb{P}_{\vec{E}\restriction\gamma})$ and
$\pi^{*}_{\nu}(p\restriction\gamma)$, as well as by the fact that
$\pi^{*}_{\nu}(\mathbb{P}_{\vec{E}\restriction\gamma})$ satisfies
$\vec{\kappa}_{\gamma}^{++}$-c.c and
$\lambda^{\gamma}(\nu(\kappa_{\gamma}))>\vec{\kappa}^{+}_{\gamma}$ is regular,
there exists some $\tau_{\nu}^{\gamma}<\lambda^{\gamma}(\nu(\kappa_{\gamma}))$
such that
$\pi^{*}_{\nu}(p\restriction\gamma)\Vdash_{\pi^{*}_{\nu}(\mathbb{P}_{\vec{E}\restriction\gamma})}\dot{g}^{\gamma}_{\nu}<^{*}(\dot{t}_{\tau_{\nu}})^{\nu}$.
We need to clarify one notation here: $(\dot{t}_{\tau_{\nu}})^{\nu}$ refers to
the $t$-functions (see Definition 5.2) defined locally in
$V^{\pi^{*}_{\nu}(\mathbb{P}_{\vec{E}\restriction\gamma})}$.

For each $\gamma\in\lim\theta$, let
$\tau_{\gamma}=j_{\gamma}(\nu\mapsto\tau_{\nu})(mc_{E_{\gamma}}(f^{p}_{\gamma}))<\lambda$.
Let $\tau\in\lambda-(\sup_{\gamma\in\lim\theta}\tau_{\gamma}+1)$. Directly
extend $p$ to $q$ such that for any $\gamma\in\lim\theta$, $\nu\in
A^{q}_{\gamma}$,

  * •

$\tau\in\operatorname{dom}(\nu)\subset\operatorname{dom}(f^{q}_{\gamma})$,

  * •

$\nu(\tau)>\tau_{\nu}$.

Note that by Definition 5.2 applied to $\mathbb{P}_{\vec{E}}$ as well as
$\pi^{*}_{\nu}(\mathbb{P}_{\vec{E}_{<\gamma}})$ and the choice of $q$, we have
that: for any $\gamma\in\lim\theta$ and any $\nu\in A^{q}_{\gamma}$,
$q^{+\nu}\Vdash(\dot{t}_{\nu(\tau)})^{\nu}=^{*}\dot{t}_{\tau}\restriction\gamma$.
Since $\nu(\tau)>\tau_{\nu}$, we know that
$q^{+\nu}\Vdash\dot{g}_{\gamma}=\dot{g}^{\gamma}_{\nu}<^{*}(\dot{t}_{\tau_{\nu}})^{\nu}<^{*}(\dot{t}_{\nu(\tau)})^{\nu}=^{*}\dot{t}_{\tau}\restriction\gamma$
for any $\gamma\in\lim\theta$ and $\nu\in A^{q}_{\gamma}$. Therefore, $q$ and
$\tau$ are as desired.∎

With Proposition 5.4, we can complete the proof of Theorem 1.3.

###### Proof.

(Theorem 1.3)
Assume without loss of generality that $V\models\mathrm{GCH}$. Let
$\langle\kappa_{\alpha}:\alpha<\theta\rangle$ be an increasing enumeration of
a cofinal discrete set of $\theta^{++}$-strong cardinals
$\kappa_{\alpha}<\theta$. Having each $\kappa_{\alpha}$ being
$\theta^{++}$-strong, with a standard preliminary $\theta$-c.c iteration of
Cohen sets that does not changes cofinalities,111Forcing with an Easton
support iteration of Cohen forcings, adding functions $f_{\nu}:\nu\to V_{\nu}$
for all $\nu<\theta$, a standard argument shows that for each $\beta<\theta$
the extender embedding $j_{E_{\beta}}:V\to M_{E_{\beta}}$ and each $x\in
V_{j_{E_{\beta}}(\kappa_{\beta})}^{M_{E_{\beta}}}$, the embedding extends to
an embedding $j^{*}$ which further satisfies
$j^{*}(f_{\kappa_{\beta}})(\kappa_{\beta})=x$. it is straightforward to
recursively define a forcing suitable sequence of extenders $\vec{E}=\langle
E_{\alpha}:\alpha<\theta\rangle$ such that each $E_{\alpha}$ is a
$(\kappa_{\alpha},\theta^{+})$-extender. Let $G\subseteq\mathbb{P}_{\vec{E}}$
be a generic filter and $\vec{\lambda}=\langle\lambda_{i}:i<\theta\rangle$ be
the sequence from Definition 5.2 of the different reflections of
$\lambda=\theta^{+}$ via points associated with $E_{i}$, $i<\theta$. Then in
$V[G]$,

  * •

By Corollary 4.18, $V[G]$ and $V$ agree on cardinals and cofinalities, and
every club $C\subseteq\theta$ in $V[G]$ contains a club $C^{\prime}$ in $V$.

  * •

for every limit ordinal $\delta\leq\theta$, $\lambda_{\delta}$ is a regular
cardinal, $\vec{\kappa}_{\delta}^{++}<\lambda_{\delta}<\kappa_{\delta}$. We
have $\lambda_{\delta}\geq 2^{\vec{\kappa}_{\delta}}$ is witnessed by the
sequence $\vec{t}^{\delta}\subseteq\prod_{i<\delta}\lambda_{i}$ generically
introduced by the $\vec{E}\restriction\delta$ reflected part of the forcing.
On the other hand, $2^{\vec{\kappa}_{\delta}}\leq\lambda_{\delta}$ since the
$E_{\delta}$-refelcted forcing satisfies $\vec{\kappa}_{\delta}^{++}$-c.c and
has size $\lambda_{\delta}$. Hence
$\lambda_{\delta}=2^{\vec{\kappa}_{\delta}}$ also equals to
$2^{\delta}=\beth(\delta)$ for club many $\delta<\theta$.

$\Diamond_{\vec{\mathrm{NS}},\vec{\lambda}}(\theta)$ fails by Proposition 5.4.
Therefore $\Diamond_{\vec{\beth}}(\theta)$ fails as well.

∎

In the last part of this section we show that the assumption
$\lambda\geq\theta^{+}$ is crucial for the argument. For convenience, for each
$\alpha<\theta$, $d\in[\lambda]^{\leq\kappa_{\alpha}}$ and $A\in
E_{\alpha}(d)$, we may assume that for any $\nu,\nu^{\prime}\in A$, if
$\nu(\kappa_{\alpha})=\nu^{\prime}(\kappa_{\alpha})$, then
$\operatorname{dom}(\nu)=\operatorname{dom}(\nu^{\prime})$. This is without
loss of generality, since for any $A^{\prime}\in E(d)$, we can always find
$A\subset A^{\prime}$ in $E(d)$ satisfying the requirement above. This is to
ensure that if the value at $\kappa_{\alpha}$ is determined, then there is a
unique $\nu$ responsible for this.

######  Proposition 5.5.

Let $\mathbb{P}_{\vec{E}}$ be defined such that $\lambda=\theta$. If
$\diamondsuit(\theta)$ holds in the ground model, then it continues to hold in
$V^{\mathbb{P}_{\vec{E}}}$.

###### Proof.

Assume the ground model $\diamondsuit(\theta)$ gives rise to a sequence
$\langle(p_{\alpha},f_{\alpha}):\alpha<\theta\rangle$ such that

  * •

$p_{\alpha}$ is a condition in $\mathbb{P}_{\vec{E}\restriction\alpha}$
satisfying that for each $\beta<\alpha$,
$\operatorname{dom}(f^{p_{\alpha}}_{\beta})\subset\alpha$,

  * •

$f_{\alpha}$ is a function with domain
$\bigcup_{\beta<\alpha}A^{p_{\alpha}}_{\beta}$ such that for any $\nu\in
A^{p_{\alpha}}_{\beta}$ where $\beta<\alpha$, $f_{\alpha}(\nu)$ is a nice
$\pi^{*}_{\nu}(\mathbb{P}_{\vec{E}\restriction\beta})$-name for a function
from $\beta\to 2$.

Let $G\subset\mathbb{P}_{\vec{E}}$ be generic over $V$. In $V[G]$, let us
define a sequence: at $\alpha$, we need to define a function $g_{\alpha}$ from
$\alpha\to 2$. For $\beta\in\alpha-s(p_{\alpha})$ and $\nu\in
A^{p_{\alpha}}_{\beta}$, we say $\nu$ is _good_ if
$\pi^{*}_{\nu}(p_{\alpha}\restriction\beta)\in
G_{\beta}=\\{p\restriction\beta:p\in G,\beta\in s(p)\\}$ and
$p_{\alpha}^{+\nu}$ is an initial segment of a condition in $G$. Then we
define $g_{\alpha}$ to be $\bigcup_{\beta,\nu\in
A^{p_{\alpha}}_{\beta},\nu\text{ is good}}(f_{\alpha}(\nu))^{G_{\beta}}$. We
show, using a density argument, that in $V[G]$, $\langle
g_{\alpha}:\alpha<\theta\rangle$ is a $\diamondsuit(\theta)$-sequence.

Suppose we are given $p\in\mathbb{P}_{\vec{E}}$, a $\mathbb{P}_{\vec{E}}$-name
$\dot{f}:\theta\to 2$ and a $\mathbb{P}_{\vec{E}}$-name $\dot{D}$ for a club
in $\theta$. By Lemma 5.1, we may assume that $p^{+\nu}$ forces
$\dot{f}\restriction\beta=\dot{f}^{\beta}_{\nu}$ for every $\nu\in
A^{p}_{\beta}$, where $\dot{f}^{\beta}_{\nu}$ is a nice
$\pi_{\nu}^{*}(\mathbb{P}_{\vec{E}\restriction\beta})$-name. Let $h$ be the
function whose domain is $\bigcup_{\beta<\theta}A^{p}_{\beta}$ and for
$\beta<\theta$ and $\nu\in A^{p}_{\beta}$, $h(\nu)=\dot{f}^{\beta}_{\nu}$. By
Corollary 4.18, we may assume there is a club $D\in V$ such that $p\Vdash
D\subset\dot{D}$.

By the property of the $\diamondsuit(\theta)$ in the ground model, we may
assume that $E=\\{\beta\in
D:(p_{\alpha},f_{\alpha})=(p\restriction\alpha,h\restriction V_{\beta})\\}$ is
a stationary subset of $\theta$. Fix $\alpha\in E$. Let
$G\subset\mathbb{P}_{\vec{E}}$ be generic with $p\in G$, then it is clear that
for any $\beta\in\alpha$, there is a unique $\nu=\nu_{\beta}\in A^{p}_{\beta}$
such that $p^{+\nu}$ is in $G$. As a result, by our choice,
$g_{\alpha}=\bigcup_{\beta<\alpha}(f^{\beta}_{\nu_{\beta}})^{G_{\beta}}=\bigcup_{\beta<\alpha}(\dot{f})^{G}\restriction\beta=(\dot{f})^{G}\restriction\alpha$.
∎

##  6\. Compactness properties in $V^{\mathbb{P}_{\vec{E}}}$ when the length
of $\vec{E}$ is measurable

Suppose $\theta$ is a measurable cardinal carrying a normal measure $U$. We
note that usually a weaker large cardinal hypothesis on $\theta$ suffices to
run the arguments in this section. We will not optimize in this respect for
simplicity.

######  Definition 6.1.

Given a sequence $\vec{r}=\langle r_{\alpha}:\alpha<\theta\rangle$ where
$r_{\alpha}:\lambda\to\kappa$ is a partial function of size $\leq\kappa$ where
$\kappa<\theta<\lambda$. We say $\vec{r}$ forms a _$\Delta$ -system with root
$f$_ if $\\{\operatorname{dom}(r_{\alpha}):\alpha<\theta\\}$ forms a
$\Delta$-system with root $d^{*}$ and $f=r_{\alpha}\restriction d^{*}$ for any
$\alpha<\theta$. We say $\vec{r}$ forms a _$\Delta$ -system_ if there exists
some $f$ such that $\vec{r}$ forms a $\Delta$-system with root $f$.

######  Lemma 6.2.

Suppose we are given a sequence $\vec{r}=\langle r_{\alpha}\in
Add(\kappa^{+},\lambda):\alpha<\theta\rangle$ where $\kappa<\theta<\lambda$.
Then there exists $B\in U$ such that $\\{r_{\alpha}:\alpha\in B\\}$ forms a
$\Delta$-system.

###### Proof.

Let $m_{\alpha}=\operatorname{dom}(r_{\alpha})$. It suffices to show that
there is $B\in U$ such that $\\{m_{\alpha}:\alpha\in B\\}$ forms a
$\Delta$-system. Since
$\bigcup_{\alpha<\theta}m_{\alpha}\in[\lambda]^{\leq\theta}$, by injecting
this set into $\theta$, we may assume without loss of generality that
$m_{\alpha}\in[\theta]^{\leq\kappa}$. Consider the map $\alpha\mapsto
m_{\alpha}\cap\alpha$. There exist $B\in U$ and $m\subset\theta$ such that for
any $\alpha\in B$, $m\cap\alpha=m_{\alpha}\cap\alpha$. Let $C\subset\theta$ be
the club consisting of ordinals $\alpha$ such that for any $\beta<\alpha$,
$\sup m_{\beta}<\alpha$. It can be checked easily that
$\\{m_{\alpha}:\alpha\in C\cap B\\}$ forms a $\Delta$-system. ∎

######  Lemma 6.3.

Given a sequence $\langle f_{\alpha}:\alpha<\theta\rangle$ such that for each
$\alpha$, $f_{\alpha}\in\Pi_{i<\alpha}Add(\kappa_{i}^{+},\lambda)$, there
exists $B\in U$ such that for any $\gamma\in\theta$,
$\\{f_{\alpha}(\gamma):\alpha\in B-(\gamma+1)\\}$ forms a $\Delta$-system.

###### Proof.

For each $\gamma<\theta$, apply Lemma 6.2 to find $B_{\gamma}\in U$ such that
$\\{f_{\alpha}(\gamma):\alpha\in B_{\gamma}\\}$ forms a $\Delta$-system. Let
$B=\Delta_{\gamma<\theta}B_{\gamma}$, which satisfies the conclusion as
desired. ∎

Let $G\subset\mathbb{P}_{\vec{E}}$ be generic. In $V[G]$, let $\bar{U}$ be the
filter generated by $U$. Namely, $\bar{U}=\\{B\subset\theta:\exists A\in
U,A\subset B\\}$.

######  Lemma 6.4.

$\bar{U}$ is $\theta$-complete and normal.

###### Proof.

Let $\langle A_{\alpha}\in\bar{U}:\alpha<\theta\rangle$ be a given sequence.
We may without loss of generality assume each $A_{\alpha}\in U$ by the
definition of $\bar{U}$. By Proposition 4.17, we know there exists
$\\{\mathcal{A}_{\alpha}\in[U]^{<\theta}:\alpha<\theta\\}\in V$ such that for
each $\alpha<\theta$, $A_{\alpha}\in\mathcal{A}_{\alpha}$. By
$\theta$-completeness of $U$ in $V$, we can let
$B_{\alpha}=\bigcap\mathcal{A}_{\alpha}\in U$ and we know that
$B_{\alpha}\subset A_{\alpha}$. By the normality of $U$ in $V$,
$\Delta_{\alpha<\theta}B_{\alpha}\in U$. It is clear that
$\Delta_{\alpha<\theta}B_{\alpha}\subset\Delta_{\alpha<\theta}A_{\alpha}$,
implying that $\Delta_{\alpha<\theta}A_{\alpha}\in\bar{U}$. Hence $\bar{U}$ is
normal in $V[G]$. ∎

######  Lemma 6.5.

Suppose we are given

  1. (1)

$p\Vdash\dot{S}\in\bar{U}^{+}$ and

  2. (2)

for each $\beta\in\lim\theta$ and $\nu\in A^{p}_{\beta}$, a dense open set
$D_{\nu,<\beta}\subset\pi^{*}_{\nu}(\mathbb{P}_{\vec{E}\restriction\beta})$.

Then there exists a direct extension $q\geq^{*}p$, $B\in U$ and $\langle
p_{\beta}\in\mathbb{P}_{\vec{E}\restriction\beta}:\beta\in B\rangle$ such that
for each $\beta\in B$,

  1. (1)

$p_{\beta}\geq_{\mathbb{P}_{\vec{E}\restriction\beta}}q\restriction\beta$,

  2. (2)

$p_{\beta}{}^{\frown}q\downharpoonright\beta\in\mathbb{P}_{\vec{E}}$ forces
that $\beta\in\dot{S}$ and

  3. (3)

for each $\nu\in A^{q}_{\beta}$, $\pi^{*}_{\nu}(p_{\beta})\in D_{\nu,<\beta}$.

###### Proof.

For the sake of simplicity, we assume $s(p)=\emptyset$ and $\Vdash\dot{S}$
consists of limit ordinals. We construct recursively $\langle
q_{\beta}\in\mathbb{P}_{\vec{E}}:\beta<\theta\rangle$, $\langle
p_{\beta}\in\mathbb{P}_{\vec{E}\restriction\beta}:\beta\in\lim\theta\rangle$
satisfying that

  1. (1)

for any $\beta_{0}<\beta_{1}$,
$q_{\beta_{1}}\restriction\beta_{0}=q_{\beta_{0}}\restriction\beta_{0}$ and
$q_{\beta_{1}}\geq^{*}q_{\beta_{0}}$,

  2. (2)

for any $\beta$,
$p_{\beta}\geq_{\mathbb{P}_{\vec{E}\restriction\beta}}q_{\beta}\restriction\beta$
and
$p_{\beta}{}^{\frown}q_{\beta}\downharpoonright\beta\in\mathbb{P}_{\vec{E}}$,

  3. (3)

if $q_{\beta}\not\Vdash\beta\not\in\dot{S}$, then
$p_{\beta}{}^{\frown}q_{\beta}\downharpoonright\beta\Vdash\beta\in\dot{S}$,

  4. (4)

for any $\nu\in A^{q_{\beta}}_{\beta}$, $\pi^{*}_{\nu}(p_{\beta})\in
D_{\nu,<\beta}$.

Applying Lemma 5.1 and directly extending $p$ if necessarily, we may assume
that for any $\nu\in A^{q}_{\beta}$, $p^{+\nu}$ forces that the Boolean value
deciding $``\beta\in\dot{S}"$ is a
$\pi^{*}_{\nu}(\mathbb{P}_{\vec{E}\restriction\beta})$-name. Suppose we are at
stage $\beta$ of the construction, having already defined $\langle
q_{\alpha}:\alpha<\beta\rangle$ and $\langle p_{\alpha}:\alpha<\beta\rangle$
satisfying the requirements above. Let us now define $q_{\beta}$. If $\beta$
is a successor, then $q_{\beta}$ is $q_{\beta-1}$. If $\beta$ is a limit, then
$q^{*}$ is defined to be the condition such that
$q^{*}\restriction\alpha=q_{\alpha}\restriction\alpha$ for all $\alpha<\beta$
and for $\gamma\geq\beta$, $q^{*}(\gamma)$ is the least upper bound of
$\langle q_{\alpha}(\gamma):\alpha<\beta\rangle$. For each $\nu\in
A^{q^{*}}_{\beta}$, if there exists a condition extending
$\pi^{*}_{\nu}(q^{*}\restriction\beta)$ in $D_{\nu,<\beta}$ that forces in
$\pi^{*}_{\nu}(\mathbb{P}_{\vec{E}\restriction\beta})$ that $\beta\in\dot{S}$,
then we define $p_{\beta,\nu}$ to be this condition and call $\nu$
_successful_. Otherwise, $p_{\beta,\nu}$ is any extension of
$\pi^{*}_{\nu}(q^{*}\restriction\beta)$ in $D_{\nu,<\beta}$ and $\nu$ is
_unsuccessful_. Next we directly extend $q^{*}$ to $q_{\beta}$ such that

  * •

$q_{\beta}\restriction\beta=q^{*}\restriction\beta$,

  * •

either for all $\nu\in A^{q_{\beta}}_{\beta}$, $\nu$ is successful or for all
$\nu\in A^{q_{\beta}}_{\beta}$, $\nu$ is unsuccessful,

  * •

there is $p_{\beta}\in\mathbb{P}_{\vec{E}}$ such that
$p_{\beta}{}^{\frown}q_{\beta}\downharpoonright\beta\in\mathbb{P}_{\vec{E}}$
and for each $\nu\in A^{q_{\beta}}_{\beta}$,
$\pi^{*}_{\nu}(p_{\beta})=p_{\beta,\nu}$.

If for all $\nu\in A^{q_{\beta}}_{\beta}$, $\nu$ is successful, then we say
that $\beta$ is _successful_. Otherwise, $\beta$ is _unsuccessful_. This
finishes the definition of $q_{\beta}$ and $p_{\beta}$. We need to verify the
requirements are met. It is clear from the definition that we only need to
check (3). If $q_{\beta}\not\Vdash\beta\not\in\dot{S}$, then $\beta$ is
successful. Since if $\beta$ is unsuccessful, then by the definition of
$q_{\beta}$, we have for all $\nu\in A^{q_{\beta}}_{\beta}$,
$q_{\beta}^{+\nu}\Vdash\beta\not\in\dot{S}$. Hence
$q_{\beta}\Vdash\beta\not\in\dot{S}$, which is impossible. As a result,
$\beta$ is successful, then for any $\nu\in A^{q_{\beta}}_{\beta}$,
$(p_{\beta}{}^{\frown}q_{\beta}\downharpoonright\beta)^{+\nu}$ forces that
$\beta\in\dot{S}$. As a result,
$p_{\beta}{}^{\frown}q_{\beta}\downharpoonright\beta\Vdash\beta\in\dot{S}$.

Finally, let $q=\lim_{\beta}q_{\beta}$. More precisely,
$q(\beta)=q_{\beta}(\beta)$ for each $\beta<\theta$. Then it is clear that for
each limit $\beta$, $q\geq q_{\beta}$,
$q\restriction\beta=q_{\beta}\restriction\beta$ and hence
$p_{\beta}\geq_{\mathbb{P}_{\vec{E}\restriction\beta}}q\restriction\beta$.

It is only left to check that $B=\\{\beta<\theta:\beta\text{ is
successful}\\}\in U$. Suppose not, then $B^{c}\in U$. We show that $q\Vdash
B^{c}\cap\dot{S}=\emptyset$, which contradicts with the assumption that
$q\Vdash\dot{S}\in\bar{U}^{+}$. To see this, note that for each $\beta\in
F^{c}\cap\lim\theta$, as $\beta$ is unsuccessful, we have
$q_{\beta}\Vdash\beta\not\in\dot{S}$. Since $q\geq q_{\beta}$,
$q\Vdash\beta\not\in\dot{S}$. ∎

The next proposition demonstrates the ineffability of $\theta$ relative to
sequences of sets individually lying in the ground model (not necessarily the
whole sequence) retains in the extension.

######  Proposition 6.6.

In $V[G]$, the following property holds: for any $S\in\bar{U}^{+}$, any
$\langle X_{\alpha}\in V\cap P(\alpha):\alpha\in S\rangle$, there exists
$X\subset\theta$ and $X\in V$ such that $\\{\alpha\in
S:X\cap\alpha=X_{\alpha}\\}\in\bar{U}^{+}$.

###### Proof.

Given $p\in\mathbb{P}_{\vec{E}}$ and a name
$\langle\dot{X}_{\alpha}:\alpha<\theta\rangle$ as in the hypothesis, we will
find an extension of $p$ forcing that “there is some $X\subset\theta$ in $V$
such that $\\{\alpha:X\cap\alpha=\dot{X}_{\alpha}\\}$ is in $\bar{U}^{+}$”.
For simplicity, we assume that $s(p)=\emptyset$. By Lemma 5.1 and Lemma 6.5,
we may assume that there are $q\geq^{*}p$, $B\in U$ and $\langle
p_{\beta}\in\mathbb{P}_{\vec{E}\restriction\beta}:\beta\in B\rangle$, such
that

  1. (1)

$p_{\beta}\geq_{\mathbb{P}_{\vec{E}\restriction\beta}}q\restriction\beta$,

  2. (2)

$p_{\beta}{}^{\frown}q\downharpoonright\beta$ forces that $\beta\in\dot{S}$,

  3. (3)

for each $\beta\in B$ and $\nu\in A^{q}_{\beta}$, $p^{+\nu}$ forces that
$\dot{X}_{\beta}=X_{\nu}^{\beta}$ for some $X_{\nu}^{\beta}\in V\cap
P(\beta)$.

For each $\beta<\theta$, there exists $X_{\beta}\subset\beta$ such that the
collection of $\nu\in A^{q}_{\beta}$ with $X_{\beta}=X_{\nu}^{\beta}$ is in
$E_{\beta}(f^{q}_{\beta})$, since $E_{\beta}(f^{q}_{\beta})$ is
$\kappa_{\beta}$-complete and $2^{\beta}<\kappa_{\beta}$. Extending $q$ if
necessary, we may assume $q$ already satisfies the property.

Shrink $B$ to $B^{\prime}\in U$ and find $X\subset\theta$,
$s\in[\theta]^{<\omega}$ and $t$ such that

  * •

for any $\beta\in B^{\prime}$, $X\cap\beta=X_{\beta}$,

  * •

for any $\beta\in B^{\prime}$, $s(p_{\beta})=s$ and $p_{\beta}\restriction\max
s=t$,

  * •

for any $\gamma<\theta$, $\\{f^{p_{\beta}}_{\gamma}:\beta\in
B^{\prime}-(\gamma+1)\\}$ forms a $\Delta$-system with root $f_{\gamma}$
(apply Lemma 6.3).

We can further extend $q$ to $q^{\prime}\in\mathbb{P}_{\vec{E}}$ such that

  1. (1)

$s(q^{\prime})=s$,

  2. (2)

$q^{\prime}\restriction\max s=t$,

  3. (3)

for any $\gamma\in\theta-\max s$, $f^{q^{\prime}}_{\gamma}=f_{\gamma}$.

######  Claim 6.7.

No direct extension of $q^{\prime}$ can force that
$\\{\beta\in\dot{S}:p_{\beta}{}^{\frown}q\downharpoonright\beta\in\dot{G}\\}$
is disjoint from some $E\in U$.

###### Proof of the Claim.

Suppose for the sake of contradiction that $q^{*}$ is a direct extension of
$q^{\prime}$ forcing that there is some $E^{*}\in U$ disjoint from
$\\{\beta:p_{\beta}{}^{\frown}q\downharpoonright\beta\in\dot{G}\\}$. Apply
Proposition 4.17, we can find $q^{\prime\prime}\geq^{*}q^{*}$ and
$\mathcal{E}\in[U]^{<\theta}$ such that $q^{\prime\prime}$ forces that
$E^{*}\in\mathcal{E}$. Since $U$ is $\theta$-complete, we know that
$E=\cap\mathcal{E}\in U$. In other words, we have found a direct extension
$q^{\prime\prime}$ of $q^{\prime}$ such that $q^{\prime\prime}\Vdash E$ is
disjoint from
$\\{\beta:p_{\beta}{}^{\frown}q\downharpoonright\beta\in\dot{G}\\}$.

For each $\gamma\in\theta-\max(s)$, there exists $g(\gamma)$ such that for all
$\alpha>g(\gamma)$ and $\alpha\in B^{\prime}$, it is the case that
$f^{q^{\prime\prime}}_{\gamma}$ and $f^{p_{\alpha}}_{\gamma}$ are compatible
as functions. Let $C\subset\theta$ be the set of closure points of $g$. Let
$\beta\in E\cap B^{\prime}\cap C$. Then $q^{\prime\prime}$ and
$p_{\beta}{}^{\frown}q\downharpoonright\beta$ are compatible since:

  * •

$q^{\prime\prime}\downharpoonright\beta\geq^{*}q\downharpoonright\beta$,

  * •

$q^{\prime\prime}\restriction\max(s)\geq^{*}q\restriction\max(s)=t=p_{\beta}\restriction\max(s)$,

  * •

for $\gamma\in(\max(s),\beta)$, $f^{p\beta}_{\gamma}$ is compatible with
$f^{q^{\prime\prime}}_{\gamma}$ since $\beta\in C\cap B^{\prime}$.

Any common extension of $q^{\prime\prime}$ and
$p_{\beta}{}^{\frown}q\downharpoonright\beta$ forces that
$\beta\in\\{\alpha\in\dot{S}:p_{\alpha}{}^{\frown}q\downharpoonright\alpha\in\dot{G}\\}\cap
E\neq\emptyset$, which is a contradiction. ∎

By Claim 6.7 and Corollary 4.12, we can take a direct extension $q^{*}$ of $q$
forcing that
$\\{\beta:p_{\beta}{}^{\frown}q\downharpoonright\beta\in\dot{G}\\}=\\{\beta:X\cap\alpha=\dot{X}_{\alpha}\\}\in\bar{U}^{+}$.∎

######  Corollary 6.8.

$\mathbb{P}_{\vec{E}}$ does not add a fresh subset of $\theta$.

######  Remark 6.9.

$\mathbb{P}_{\vec{E}}$ does add a fresh subset of $\theta^{+}$. In fact, it
adds $\lambda$ many $\theta^{+}$-Cohen sets.

######  Corollary 6.10.

In $V[G]$, for any $S\in\bar{U}^{+}$ and any C-sequence $\langle
C_{\alpha}:\alpha\in S\rangle$, there exists a club $D\subset\theta$ such that
$\\{\alpha\in S:D\cap\alpha\subset C_{\alpha}\\}\in\bar{U}^{+}$.

###### Proof.

This follows immediately from Corollary 4.18 (3), Lemma 5.1 and Proposition
6.6. ∎

######  Remark 6.11.

The conclusion of Corollary 6.10 implies that “there does not exist an
amenable C-sequence”. The concept of an amenable C-sequence appears in [BR19],
strengthening the principle $\otimes_{\overrightarrow{C}}$ from [She94, p.
134]. That “there is no amenable C-sequence on $\kappa$” is a non-trivial
compactness principle. See the discussion in [BNZ, Section 5] for more
details. Corollary 6.10 can be used to argue that with suitable $\vec{E}$, in
$V^{\mathbb{P}_{\vec{E}}}$, there is no amenable C-sequence on $\theta$ while
$\diamondsuit(\theta)$ fails. Such a model was first constructed in [BNZ],
using a different method based on an analysis of suitable Radin extensions.

Finally, we show that in $V[G]$, even though $2^{\theta}=\lambda$ and
$\lambda$ can be large, the degree of saturation of $\bar{U}$ remains small.
Assume $V\models\mathrm{GCH}$.

######  Proposition 6.12.

$\bar{U}$ is $\theta^{++}$-saturated.

###### Proof.

Suppose we are given $\langle\dot{S}_{i}\in\bar{U}^{+}:i<\theta^{++}\rangle$
and a condition $p\in\mathbb{P}_{\vec{E}}$. We need to find some $q\geq p$ and
$i\neq j<\theta^{++}$ such that
$q\Vdash\dot{S}_{i}\cap\dot{S}_{j}\in\bar{U}^{+}$. For simplicity, assume that
$s(p)=\emptyset$.

For each $i<\theta^{++}$, by Lemma 6.5, we find a direct extension
$q_{i}\geq^{*}p$, $B_{i}\in U$ and $\langle
p^{i}_{\beta}\in\mathbb{P}_{\vec{E}\restriction\beta}:\beta\in B_{i}\rangle$
such that for each $\beta\in B_{i}$,

  1. (1)

$p^{i}_{\beta}\geq_{\mathbb{P}_{\vec{E}\restriction\beta}}q_{i}\restriction\beta$,

  2. (2)

$p^{i}_{\beta}{}^{\frown}q_{i}\downharpoonright\beta$ forces that
$\beta\in\dot{S}_{i}$.

By shrinking each $B_{i}$ if necessary, we may assume

  * •

there exists $s_{i}\in[\theta]^{<\omega}$ and $t_{i}$ such that for all
$\beta\in B_{i}$, $s(p^{i}_{\beta})=s_{i}$ and $p^{i}_{\beta}\restriction\max
s_{i}=t_{i}$,

  * •

for any $\gamma\in\theta-\max(s_{i})$, $\\{f^{p_{\eta}^{i}}_{\gamma}:\eta\in
B_{i}-(\gamma+1)\\}$ forms a $\Delta$-system with root $f^{i}_{\gamma}$ (apply
Lemma 6.3).

We may extend $q_{i}$ if necessary so that

  1. (1)

$s(q_{i})=s_{i}$,

  2. (2)

$q_{i}\restriction\max s_{i}=t_{i}$, and

  3. (3)

for each $\gamma\in\theta-\max(s_{i})$, $f^{q_{i}}_{\gamma}=f^{i}_{\gamma}$.

Since $2^{\theta}=\theta^{+}$ in $V$, we can find
$A\in[\theta^{++}]^{\theta^{++}}$, $s\in[\theta]^{<\omega}$ and $t$ such that

  1. (1)

for any $i\in A$, $s_{i}=s$ and $t_{i}=t$,

  2. (2)

for any $i,j\in A$, $q_{i}$ is compatible with $q_{j}$,

  3. (3)

for any $i,j\in A$ and any $\beta\in B_{i}\cap B_{j}$, $p^{i}_{\beta}$ is
compatible with $p^{j}_{\beta}$ in $\mathbb{P}_{\vec{E}\restriction\beta}$.

Let $i<j\in A$. The following claim finishes the proof of the lemma. Let
$q^{*}$ be the least upper bound of $p_{i}$ and $p_{j}$.

######  Claim 6.13.

There exists a direct extension $q$ of $q^{*}$ forcing that
$\dot{S}_{i}\cap\dot{S}_{j}\in\bar{U}^{+}$.

###### Proof of the Claim.

Suppose not, then by Proposition 4.17, we can find a direct extension
$q^{\prime}$ of $q^{*}$ and $\mathcal{B}\in[U]^{<\theta}$ such that
$q^{\prime}\Vdash\exists B^{\prime}\in\mathcal{B}$,
$\dot{S}_{i}\cap\dot{S}_{j}\cap B=\emptyset$. Let $B=\bigcap\mathcal{B}\in U$,
then we know that $q^{\prime}\Vdash\dot{S}_{i}\cap\dot{S}_{j}\cap
B=\emptyset$. For each $\gamma\in\theta-\max s$, there exists some
$\eta_{\gamma}<\theta$, such that for all $\eta>\eta_{\gamma}$ in $B_{i}\cap
B_{j}$, $f^{q^{\prime}}_{\gamma}$, $f^{p^{i}_{\eta}}_{\gamma}$ and
$f^{p^{j}_{\eta}}_{\gamma}$ are mutually compatible. To see this, note that
$\operatorname{dom}(f^{q^{\prime}}_{\gamma})\supset\operatorname{dom}(f^{q_{i}}_{\gamma})=\operatorname{dom}(f^{i}_{\gamma})$,
hence, for sufficiently large $\eta$,
$\operatorname{dom}(f^{p^{i}_{\eta}}_{\gamma})\cap\operatorname{dom}(f^{q^{\prime}}_{\gamma})\subset\operatorname{dom}(f^{i}_{\gamma})$.
The same reasoning works for $j$. That $f^{p^{i}_{\eta}}_{\gamma}$ is
compatible with $f^{p^{j}_{\eta}}_{\gamma}$ is guaranteed by the fact that
$p^{i}_{\eta}$ is compatible with $p^{j}_{\eta}$. Let $C$ be the club
consisting of the closure points of the function $\gamma\mapsto\eta_{\gamma}$.
Take $\eta\in C\cap B\cap B_{i}\cap B_{j}$. As a result, there exists a direct
extension $r$ of $q^{\prime}$ extending both
$p^{i}_{\eta}{}^{\frown}q_{i}\downharpoonright\eta$ and
$p^{j}_{\eta}{}^{\frown}q_{j}\downharpoonright\eta$. Therefore,
$r\Vdash\eta\in\dot{S}_{i}\cap\dot{S}_{j}\cap B$, contradicting with the
assumption on $q^{\prime}$ and the choice of $B$. ∎

∎

##  7\. Open questions

######  Question 7.1.

Does $\diamondsuit_{\vec{\lambda}}(Reg^{\theta})$ imply
$\diamondsuit_{\vec{\lambda}}(Sing^{\theta}_{>\omega})$ outright (see Theorem
1.5 for relevant notations) or under additional assumptions such as GCH or
$\theta$ is weakly compact?

######  Question 7.2.

Does $\diamondsuit(Reg^{\theta})$ imply $\diamondsuit(Sing^{\theta})$? Does
$\theta$ being measurable imply $\diamondsuit(Sing^{\theta})$?

######  Question 7.3.

Is it consistent that
$\neg\diamondsuit_{\vec{\mathrm{NS}},\vec{\beth}}(\theta)+\Diamond(\theta)$ in
a model where $2^{\alpha}$ is regular for stationarily many $\alpha<\theta$
(See Theorem 3.6)?

######  Question 7.4.

Does $\forall\vec{\lambda}\ \diamondsuit_{\vec{\lambda}}(\theta)$ imply
$\diamondsuit(\theta)$?222Note that
$\diamondsuit_{\vec{\mathrm{NS}},\langle\alpha^{++}:\alpha<\theta\rangle}(\theta)$
fails in the model described in Theorem 3.6.

######  Question 7.5.

Is it consistent that a strongly inaccessible $\theta$ carries a
$\theta^{+}$-saturated ideal and $\neg\diamondsuit(\theta)$? How about if
$\theta$ carries a $\theta$-saturated ideal?

This is related to Proposition 6.12. Ketonen [Ket74] showed that if $\theta$
carries a $\gamma$-saturated ideal for some $\gamma<\theta$, then
$\diamondsuit(\theta)$ holds.

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