# Axiom $\mathcal{A}$ and supercompactness

Alejandro Poveda  Harvard University, Department of Mathematics and Center of
Mathematical Sciences and Applications, Cambridge (MA), 02138, USA
<EMAIL_ADDRESS>https://scholar.harvard.edu/apoveda/homewww.alejandropovedaruzafa.com

###### Abstract.

We produce a model where every supercompact cardinal is $C^{(1)}$-supercompact
with inaccessible targets. This is a significant improvement of the main
identity-crises configuration obtained in [HMP22] and provides a definitive
answer to a question of Bagaria [Bag12, p.19]. This configuration is a
consequence of a new axiom we introduce –called $\mathcal{A}$– which is showed
to be compatible with Woodin’s $I_{0}$ cardinals. We also answer a question of
V. Gitman and G. Goldberg on the relationship between supercompactness and
cardinal-preserving extendibility. As an incidental result, we prove a theorem
suggesting that supercompactness is the strongest large-cardinal notion
preserved by Radin forcing.

###### Key words and phrases:

Supercompactness, $C^{(1)}$-supercompactness, Identity crises phenomena in the
large cardinal hierarchy, Prikry-type forcings.

###### 2020 Mathematics Subject Classification:

Primary 03E35; Secondary 03E04

The author acknowledges support from the Department of Mathematics at Harvard
University as well as from the Harvard Center of Mathematical Sciences and
Applications. This version is from June 6, 2024. Updates may be posted in the
authors’s webpage www.alejandropovedaruzafa.com

###### Contents

  1. 1 Introduction
  2. 2 Prelimminaries
    1. 2.1 Some large cardinal notions
    2. 2.2 A forcing adding a $\kappa$-regressive $\kappa$-Kurepa tree
    3. 2.3 The coding poset
    4. 2.4 Further coding
  3. 3 Axiom $\mathcal{A}$ and its consequences
  4. 4 The proof of the main theorem
    1. 4.1 Forcing $\mathcal{A}$
    2. 4.2 $\mathcal{A}$ is consistent with Vopěnka’s Principle
    3. 4.3 $\mathcal{A}$ and $I_{0}$ cardinals
    4. 4.4 Open questions
    5. 4.5 Axiom $\mathcal{A}$ and the Ultimate-$L$ conjecture
  5. 5 On a question of Gitman and Goldberg

##  1\. Introduction

In his pioneering and influential work [Mag76], Magidor discovered the so-
called _identity crises phenomenon_ for the classes of measurable, strong
compact and supercompact cardinals. An uncountable cardinal $\kappa$ is called
_measurable_ if there is a $\kappa$-complete ultrafilter over $\kappa$.
Similarly, $\kappa$ is called _strongly compact_ (_supercompact_) if for each
$\lambda\geq\kappa$ every $\kappa$-complete filter over
$\mathcal{P}_{\kappa}(\lambda):=\\{x\subseteq\lambda\mathrel{|}\allowbreak|x|<\kappa\\}$
extends to a $\kappa$-complete (normal) ultrafilter. It is clear that every
supercompact is strongly compact and that the latter are measurable. Standard
arguments also show that every supercompact is a limit of measurables. Way
more elusive is whether a supercompact cardinal (resp. a strongly compact) has
to be a limit of strongly compacts (resp. measurables). In his groundbreaking
work [Mag76], Magidor proved the consistency (relative to ZFC) of the next
configurations; namely, (1) the first strongly compact is the first
supercompact; (2) the first strongly compact is the first measurable. This is
known as the _identity crises phenomenon_.

Ever since the study of the identity crises phenomena have captivated the
attention of prominent set theorist, such as Apter, Cummings, Gitik, Magidor,
Shelah and Woodin, among others. Extending Magidor’s original theorem from
[Mag76], Kimchi and Magidor [KM80] proved the consistency of every strongly
compact cardinal being supercompact, except for certain limits of strong
compacts. Other major works in the area are due to Apter and Shelah [AS97a,
AS97b], Apter and Gitik [AG98], Apter and Cummings [AC00, AC01] and Woodin
[CW]. More recently, Hayut, Magidor and the author [HMP22] studied the
identity crises phenomenon at higher regions of the large cardinal hierarchy;
specifically, in the region comprised between the first supercompact and the
so-called _Vopěnka’s Principle_.

Recall that for $n<\omega$, $C^{(n)}$ denotes the club class of all ordinals
$\theta$ such that $V_{\theta}$ is a $\Sigma_{n}$-elementary substructure of
$V$ (in symbols, $V_{\theta}\prec_{\Sigma_{n}}\mathrm{V}$). A cardinal
$\kappa$ is _$C^{(n)}$ -supercompact_ if for all $\lambda>\kappa$ there is an
elementary embedding $j\colon V\rightarrow M$ with
$\operatorname{crit}(j)=\kappa$, $\lambda<j(\kappa)$, $M^{\lambda}\subseteq M$
and $j(\kappa)\in C^{(n)}$ [Bag12]. $C^{(0)}$-supercompact cardinals are
exactly the usual _supercompact_ cardinals.

Answering a question of Bagaria from [Bag12, §5], in [HMP22] it was showed
that, for each $n<\omega$, the first supercompact cardinal can be the first
$C^{(n)}$-supercompact. A key component of the proof in [HMP22, §3] is that
$C^{(n)}$-supercompat cardinals are preserved by certain products of Prikry-
type forcings. Unlike with supercompacts or strongly compacts (see [Git10,
§6]), $C^{(n)}$-supercompacts are substantially much harder to preserve. We
refer to the last section of [HMP22] for a thorough discussion on this matter.

The preservation argument provided in [HMP22, §3] depends upon a fairly
peculiar feature of Prikry forcing; namely, that generic filters can be
constructed explicitly via iterated ultrapowers. In fact, no other method is
known to preserve $C^{(1)}$-supercompacts. The downside of this is that the
analysis of [HMP22, §3] is not exportable to asses other plausible
configurations for the class of $C^{(n)}$-supercompacts. For instance, the
strategy of [HMP22, §3] fails (dramatically) to make the first two
supercompact cardinals be $C^{(1)}$-supercompact. This is a consequence of a
well-known fact about Prikry forcing – namely, that this poset kills all the
strongly compacts below the cardinal where the Prikry sequence has been added.
As a result neither [HMP22] nor the technologies developed so far (e.g.
[AS97b, AC00, AC01]) were amenable to produce the equivalence between
supercompactness and $C^{(1)}$-supercompactness. In this paper we pursue a
completely different approach. Instead of preserving $C^{(1)}$-supercompact
cardinals, first, we isolate a new axiom –called _$\mathcal{A}$_ – and show
that it provides the sought one-to-one equivalence. Second, we prove that
$\mathcal{A}$ is compatible with very large cardinals – specifically, with a
proper class of supercompact cardinals.

######  Main Theorem.

Suppose the $\rm{GCH}$ holds and that suitable large cardinals exist. Then
there is a model of $\mathrm{ZFC}+\mathcal{A}$ with a stationary class of
supercompacts. Thus, in this model every supercompact cardinal is
$C^{(1)}$-supercompact.

In fact we show that axiom $\mathcal{A}$ is compatible with _Woodin’s axiom
$I_{0}$_ – this entails the existence of much stronger large cardinals than
those claimed above [Kan09, §24]. One thus infers that the equivalence between
supercompactness and $C^{(1)}$-supercompactness is not precluded by the
strongest known large cardinal axiom (compatible with the Axiom of Choice).

The study of axiom $\mathcal{A}$ as well as other proposed strengthenings (see
§4.4) has set-theoretic interests on its own. For instance, $\mathcal{A}$ is
intimately connected to the following natural inner-model-theoretic question:

######  Question.

Suppose $j\colon V\rightarrow M$ is an elementary embedding with
$\operatorname{crit}(j)=\kappa$, $M^{\kappa}\subseteq M$ and $j(\kappa)$ a
strong limit cardinal. Must it be that $\kappa$ is superstrong with target
$j(\kappa)$ (possibly witnessed by a different embedding)?

The above inquires whether the class of _superstrong cardinals_ coincides with
the _tall cardinals_ with strong limit targets. This in turn is inspired by
the known equivalence between _strong_ and _tall cardinals_ in iterable inner
models $L[E]$ proved by Fernandes and Schindler in [FS21]. The _Extender
Embedding Axiom_ (EEA) –due to Woodin (2018)– postulates an affirmative answer
to the above-mentioned question. Moreover, Woodin has conjectured that EEA
holds in his canonical inner model for supercompactness [Woo24].

######  Conjecture (Woodin).

$``\mathrm{ZFC}+V=\mathrm{Ultimate-}L$” proves $\mathrm{EEA}$.

In contrast, in this paper we prove two things; namely, (1) axiom
$\mathcal{A}$ is consistent with Woodin’s axiom $I_{0}$; (2) the theory
$``\mathcal{A}+$There is a supercompact cardinal” disproves EEA (modulo ZFC).
If Woodin’s prospects are true, our Main Theorem provides a fairly exotic
model for supercompactness. In §4.5 we briefly discuss the divergences between
$\mathcal{A}$ and $``V=\mathrm{Ultimate-}L$”.

In the second part of the manuscript we analyze the relationship between
supercompactness and cardinal-preserving extendibility. A cardinal $\kappa$ is
said to be _cardinal-preserving extendible_ if for all $\lambda\geq\kappa$
there is an elementary embedding $j\colon V_{\lambda}\rightarrow M$ with
$\operatorname{crit}(j)=\kappa$, $j(\kappa)>\lambda$ and
$\mathrm{Card}^{M}=\mathrm{Card}\cap M.$ This notion is due to V. Gitman and
J. Osinski. If $j\colon V_{\lambda}\rightarrow M$ is as above, clearly,
$j(\kappa)$ is a limit cardinal in $V$. Also, every extendible cardinal (see
[Kan09, §23]) is cardinal-preserving extendible.

In private communication [GG23] V. Gitman and G. Goldberg posed us the
following question:

######  Question.

Is every supercompact a cardinal-preserving extendible?

We answer the above in the negative. Specifically the following is proved:

######  Theorem.

If there is a supercompact cardinal then there is a model where the first
supercompact is not cardinal-preserving extendible.

The proof of the above suggests that the strongest large-cardinal notion
preserved by Radin forcing is supercompactness (Theorem 5.4). Specifically, we
show that in the typical Radin extension where a cardinal remains supercompact
there are no tall embeddings with a limit cardinal as a target.

In §2 we set notations and provide the reader with some forcing preliminaries.
In §3 we introduce axiom $\mathcal{A}$ and show that it yields the equivalence
between supercompacts and $C^{(1)}$-supercompacts (with inaccessible target).
We also show that $\mathcal{A}$ disproves the equivalence betwen superstrong
and tall cardinals with strong limit targets postulated by $\mathrm{EEA}$. The
main result of the section is Theorem 4.1 which yields the consistency of
$\mathcal{A}$ with _Vopěnka’s Principle_. In §4.3 we show that $\mathcal{A}$
is in fact compatible with $I_{0}$ cardinals. In §4.4 we propose a few
questions in regards to strengthenings of $\mathcal{A}$. Finally, in §5 we
answer the question proposed by Gitman and Goldberg.

##  2\. Prelimminaries

In this section we garner some definitions and preliminary results that will
be used in the proof of our main theorem. We begin with §2.1 by presenting a
few relevant large cardinal notions. §§2.2 and 2.3 revolve around two posets
instrumental for the proof of our main theorem.

###  2.1. Some large cardinal notions

Here we collect some relevant large-cardinal notions. For further information
on this issue we refer our readers to Kanamori’s excellent text on large
cardinals [Kan09].

######  Definition 2.1.

We say that $\kappa$ is _tall with target $\lambda$_ if there is an elementary
embedding $j\colon V\rightarrow M$ with $\operatorname{crit}(j)=\kappa$,
$M^{\kappa}\subseteq M$ and $j(\kappa)=\lambda$. A cardinal $\kappa$ is said
to be _tall_ if $\kappa$ is tall with target $\lambda$ for proper-class many
$\lambda>\kappa$.

The above notion admits a natural extender-like characterization; namely,
$\kappa$ is tall with target $\lambda$ if and only if there is a
$(\kappa,\lambda)$-extender $E$ such that $(M_{E})^{\kappa}\subseteq M_{E}$
and $j_{E}(\kappa)=\lambda$. For details, see [Kan09, §26].

The notion of $C^{(n)}$-tallness is defined analogously by requiring
$j(\kappa)$ be a member of the class
$C^{(n)}:=\\{\theta\in\operatorname{Ord}\mathrel{|}\allowbreak
V_{\theta}\prec_{\Sigma_{n}}V\\}$. Note that $C^{(0)}$ and $C^{(1)}$ are
respectively the class of all ordinals and all strong limit cardinals.
Oftentimes members in $C^{(n)}$ will be referred as _$\Sigma_{n}$ -correct
cardinals_.

In this paper we shall be interested in the following version of tallness:

######  Definition 2.2.

Fix $n<\omega$. A cardinal $\kappa$ is _enhanced $C^{(n)}$-tall_ if for each
$\lambda>\kappa$ there is $\theta\in C^{(n)}$ with
$\operatorname{cf}(\theta)>\lambda$ such that $\kappa$ is tall with target
$\theta$.

Another important large-cardinal notion is superstrongness:

######  Definition 2.3.

A cardinal $\kappa$ is called _superstrong with target $\lambda$_ if there is
an elementary embedding $j\colon V\rightarrow M$ with
$\operatorname{crit}(j)=\kappa$, $j(\kappa)=\lambda$ and $V_{\lambda}\subseteq
M$. A cardinal $\kappa$ is _superstrong_ if it is superstrong with target
$\lambda$ for some $\lambda$.

As before, the superstrongness of a cardinal $\kappa$ can be characterized via
the existence of a $(\kappa,\lambda)$-extender $E$ such that
$j_{E}(\kappa)=\theta$ and $V_{\theta}\subseteq M_{E}$.

In general we do not require the target model of a superstrong embedding be
closed under $\kappa$-sequences in $V$. Whenever this happens we will say that
$\kappa$ is an _enhanced superstrong_ cardinal. A key fact about superstrong
cardinals with inaccessible targets is the following:

######  Fact 2.4.

Suppose that $\kappa$ is superstrong with inaccessible target $\lambda$. Then,
$\\{\tau<\lambda\mathrel{|}\allowbreak\text{$\kappa$ is superstrong with
target $\tau$}\\}$ contains a club in $\lambda.$ Moreover,
$\\{\tau<\lambda\mathrel{|}\allowbreak\text{$\kappa$ is an enhanced
superstrong with target $\tau$}\\}$ is a ${\geq}\kappa^{+}$-club.111A set
$C\subseteq\lambda$ is called a _${\geq}\kappa^{+}$ -club_ if it is unbounded
in $\lambda$ and for every increasing sequence
$\langle\alpha_{\xi}\mathrel{|}\allowbreak\xi<\theta\rangle\subseteq C$ with
$\theta\in[\kappa^{+},\lambda)$ regular, $\sup_{\xi<\theta}\alpha_{\xi}\in C.$

We refrain to prove the previous fact. For details we invite our readers to
consult [Tsa12, Proposition 2.6 and Corollary 2.10].

Other large cardinals considered in the manuscript are _supercompacts_ ,
_cardinal-preserving extendibles_ , _$C^{(n)}$ -extendibles_, _Vopěnka
cardinals_ and _$I_{0}$ cardinals_. The first two were presented in §1 so we
are left with the others.

######  Definition 2.5 (Bagaria [Bag12]).

A cardinal $\delta$ is called _$\lambda$ -$C^{(n)}$-extendible_ for
$\lambda>\delta$ if there is $\theta\in\mathrm{Ord}$ and an elementary
embedding $j\colon V_{\lambda}\rightarrow V_{\theta}$ such that
$\operatorname{crit}(j)=\delta$, $j(\delta)>\lambda$ and $j(\delta)\in
C^{(n)}$. A cardinal $\delta$ is called _$C^{(n)}$ -extendible_ if it is
$\lambda$-$C^{(n)}$-extendible for a proper class of $\lambda$.

A cardinal $\delta$ extendible in the classical sense (see [Kan09]) if and
only if it is $C^{(1)}$-extendible. The next easy observation shows that mild
versions of extendibility yield rich superstrong embeddings:

######  Proposition 2.6.

Suppose that $j\colon V_{\delta+1}\rightarrow V_{j(\delta)+1}$ witnesses
$(\delta+1)$-extendibility of $\delta$. Then there is an embedding
$\iota\colon V\rightarrow M$ witnessing that $\delta$ is superstrong with
target $j(\delta)$ and $M$ is correct about stationaries of $j(\delta)$.

###### Proof.

Let $\mathcal{I}:=V_{j(\delta)}\cup\mathrm{Cub}_{j(\delta)}$ where
$\mathrm{Cub}_{j(\delta)}$ denotes the club filter at $j(\delta)$.

Let us consider the extender $\mathcal{E}:=\langle E_{a}\mathrel{|}\allowbreak
a\in[\mathcal{I}]^{<\omega}\rangle$ where

$X\in E_{a}\;\Longleftrightarrow\;X\subseteq[V_{\delta+1}]^{|a|}\,\wedge\,a\in
j(X).$

As usual, there is a natural projection map $\pi_{ab}$ between $E_{a}$ and
$E_{b}$ whenever $a\subseteq b$ (see [Kan09, §23]) so we may form the
corresponding extender ultrapower $\iota:=j_{\mathcal{E}}\colon V\rightarrow
M_{\mathcal{E}}$. It is routine to check that $\iota(\delta)=j(\delta)$ and
$V_{\iota(\delta)}\cup\mathrm{Cub}_{\iota(\delta)}\subseteq M_{\mathcal{E}}.$
In particular, $M_{\mathcal{E}}$ is stationary correct at $j(\delta)$. ∎

######  Definition 2.7.

Superstrong cardinals $\delta$ as those in Proposition 2.6 will be referred as
_stationary-correct superstrong (with target $j(\delta)$)_.

We will also consider a strengthening of $\mathop{C^{(n)}}$-extendibility:

######  Definition 2.8.

Fix proper classes $A$ and $\mathbb{P}$. A cardinal $\delta$ is
_$C^{(n)}_{\mathbb{P}}$ -$A$-extendible_ if for all $\lambda>\delta$ there are
$\theta\in\mathrm{Ord}$ and an elementary embedding

$j\colon\langle V_{\lambda},\in,A\cap\lambda\rangle\rightarrow\langle
V_{\theta},\in,A\cap\theta\rangle$

such that $\operatorname{crit}(j)=\delta$, $j(\delta)>\lambda$ and
$j(\delta)\in C^{(n)}_{\mathbb{P}}$ where $C^{(n)}_{\mathbb{P}}$ is the club
class

$\\{\mu\in\operatorname{Ord}\mathrel{|}\allowbreak\langle
V_{\mu},\in,\mathbb{P}\cap V_{\mu}\rangle\prec_{\Sigma_{n}}\langle
V,\in,\mathbb{P}\rangle\\}.$

######  Fact 2.9 ([BP, Lemma 3.16]).

If $\mathbb{P}$ is an adequate class forcing iteration, $j(\delta)\in
C^{(n)}_{\mathbb{P}}$ and the trivial condition of $\mathbb{P}$ forces
$``V_{j(\delta)}[\dot{G}_{j(\delta)}]=V[\dot{G}]_{j(\delta)}$” then
$\text{$\mathop{1\hskip-3.0pt{\rm
l}}\Vdash_{\mathbb{P}}``j(\delta)\in\dot{C}^{(n)}$''}.$

In [Pov24] the author introduced the notion of _almost-
$C^{(n)}$-extendibility_ aiming to asses whether every $C^{(n)}$-extendible is
a limit of $C^{(n)}$-supercompact cardinals. This new large-cardinal concept
is defined as follows:

######  Definition 2.10.

A cardinal $\delta$ is called _almost- $C^{(n)}$-extendible_ if for all
$\lambda>\delta$, $\delta$ is $\lambda$-supercompact and there is $\mu\in
C^{(n)}$ with $\operatorname{cf}(\mu)>\lambda$ such that $\delta$ is
superstrong with target $\mu.$

In [Pov24, §2] it is showed that every almost-$C^{(n)}$-extendible cardinal is
$\Sigma_{n+2}$-correct. Its important to observe that, for a given
almost-$C^{(n)}$-extendible cardinal, its supercompactness and superstrongness
are witnessed by (possibly) different elementary embeddings. In the cases
where both properties are witnessed by the same embedding, Tsaprounis [Tsa18]
has showed that this is equivalent to $C^{(n)}$-extendibility. Exploiting this
technical nuance the following were proved in [Pov24, §2]:

  1. (1)

Every almost-$C^{(n)}$-extendible is $C^{(n)}$-supercompact;

  2. (2)

Every almost-$C^{(1)}$-extendible is a limit of supercompacts;

  3. (3)

Every $C^{(n)}$-extendible is a limit of almost-$C^{(n)}$-extendibles.

In particular, every $C^{(n)}$-extendible cardinal is a limit of
$C^{(n)}$-supercompacts.

Next we discuss Vopěnka and almost huge cardinals. A set $X\subseteq
V_{\kappa}$ is called _Vopěnka in $\kappa$_ if for every natural sequence of
structures222A natural sequence of structures is a sequence
$\langle\mathcal{M}_{\alpha}\mathrel{|}\allowbreak\alpha<\kappa\rangle$ where
each $\mathcal{M}_{\alpha}$ takes the form $\langle
V_{f(\alpha)},\in,\\{\alpha\\},R_{\alpha}\rangle$ with $\alpha<f(\alpha)\leq
f(\beta)<\kappa$ and $R_{\alpha}\subseteq V_{f(\alpha)}$ [Kan09, p.336].
$\langle\mathcal{M}_{\alpha}\mathrel{|}\allowbreak\alpha<\kappa\rangle$ there
are ordinals $\alpha<\beta<\kappa$ and an elementary embedding
$j\colon\mathcal{M}_{\alpha}\rightarrow\mathcal{M}_{\beta}$ with
$\operatorname{crit}(j)\in X$. An inaccessible cardinal $\kappa$ is called
_Vopěnka_ if $\kappa$ is Vopěnka in $\kappa$. The _Vopěnka filter_ on $\kappa$
is $\mathcal{F}:=\\{X\subseteq\kappa\mathrel{|}\allowbreak\text{$\kappa-X$ is
not Vop\v{e}nka in $\kappa$}\\}.$ It is not hard to show that $\kappa$ is
Vopěnka if and only if $\kappa$ is inaccessible and $\mathcal{F}$ is a proper
filter. Whenever we refer to Vopěnka’s filter we will always be assuming that
it is proper. A classical result in set theory says that $\mathcal{F}$ is a
normal filter [Kan09, Proposition 24.14] and in particular it contains the
club filter on $\kappa$. In special circumstances $\mathcal{F}$ can be
extended to a normal (uniform) measure on $\kappa$. The next lemma pinpoints
one of those:

######  Lemma 2.11.

If there is an elementary embedding $j\colon V\rightarrow M$ such that
$\operatorname{crit}(j)=\kappa$ and $M^{<j(\kappa)}\subseteq M$ then
$\mathcal{F}\subseteq\mathcal{U}$ where
$\mathcal{U}:=\\{X\subseteq\kappa\mathrel{|}\allowbreak\kappa\in j(X)\\}.$

###### Proof.

By Powell’s theorem [Kan09, Theorem 24.18] there is $X\in\mathcal{U}$ such
that if
$\langle\mathcal{M}_{\alpha}\mathrel{|}\allowbreak\alpha<\kappa\rangle$ is a
natural sequence of structures and $\alpha<\beta$ are members of $X$ then
there is $\iota\colon\mathcal{M}_{\alpha}\rightarrow\mathcal{M}_{\beta}$ with
$\operatorname{crit}(\iota)=\alpha.$ By elementarity, for each $M$-natural
sequence
$\langle\mathcal{M}_{\alpha}\mathrel{|}\allowbreak\alpha<j(\kappa)\rangle$ and
an ordinal $\beta\in j(X)$ with $\kappa<\beta$ there is an $M$-elementary
embedding $\iota\colon\mathcal{M}_{\kappa}\rightarrow\mathcal{M}_{\beta}$ with
$\operatorname{crit}(\iota)=\kappa$.

Let $F\in\mathcal{F}$. Then $j(F)$ belongs to the $M$-Vopěnka filter
$j(\mathcal{F})$. That means that there is
$\langle\mathcal{N}_{\alpha}\mathrel{|}\allowbreak\alpha<j(\kappa)\rangle$ an
$M$-natural sequence such that if
$\iota\colon\mathcal{N}_{\alpha}\rightarrow\mathcal{N}_{\beta}$ is an
elementary embedding then $\operatorname{crit}(\iota)\in j(F)$. Applying the
previous observation we conclude that there is an elementary embedding
$\iota\colon\mathcal{N}_{\kappa}\rightarrow\mathcal{N}_{\beta}$ whose critical
point is $\kappa\in j(F)$. Therefore, $F\in\mathcal{U}$ as needed. ∎

The assumption employed in the previous lemma is referred in the large-
cardinal literature as _almost hugeness._ To grant the above hypothesis as
well as some extra properties we will demand a bit more about our $\kappa$:

######  Definition 2.12.

A cardinal $\kappa$ is called _huge_ if there is an elementary embedding
$j\colon V\rightarrow M$ such that $\operatorname{crit}(j)=\kappa$ and
$M^{j(\kappa)}\subseteq M$.

One of the strongest known large cardinals is _Woodin’s axiom $I_{0}$_:

######  Definition 2.13 (Woodin).

$I_{0}(\lambda)$ denotes the assertion “There is an elementary embedding
$j\colon L(V_{\lambda+1})\rightarrow L(V_{\lambda+1})$ with
$\operatorname{crit}(j)<\lambda.$”

To make explicit the critical point of an $I_{0}(\lambda)$ embedding

$j\colon L(V_{\lambda+1})\rightarrow L(V_{\lambda+1})$

we will write $I_{0}(\delta,\lambda)$ being $\delta:=\operatorname{crit}(j)$.
If $I_{0}(\delta,\lambda)$ holds as witnessed by $j$ we shall denote by
$\langle\delta_{n}\mathrel{|}\allowbreak n<\omega\rangle$ the _critical
sequence of $j$_; namely, the sequence defined by $\delta_{0}:=\delta$ and
$\delta_{n+1}:=j(\delta_{n})$ for each $n<\omega$.

######  Definition 2.14 (Woodin).

For a class $X$ one says that _$I_{0}(\delta,\lambda,X)$ holds_ if there is an
elementary embedding $j\colon L(V_{\lambda+1})\rightarrow L(V_{\lambda+1})$
with $\operatorname{crit}(j)=\delta$ below $\lambda$ and such that
$j(X\cap\lambda)=X\cap\lambda$.

###  2.2. A forcing adding a $\kappa$-regressive $\kappa$-Kurepa tree

Recall that a _tree_ is a partially ordered set $\mathbb{T}=(T,<)$ such that
for each $t\in T$ the set of predecessors
$\mathrm{pred}_{\mathbb{T}}(t):=\\{s\in T\mathrel{|}\allowbreak s<t\\}$ is
well-ordered by $<$. In a harmless abuse of notation one identifies the tree
$\mathbb{T}$ with its underlying universe $T$. For an ordinal $\alpha$, the
_$\alpha$ th-level level of $T$_ is defined to be

$T_{\alpha}:=\\{t\in
T\mathrel{|}\allowbreak\operatorname{otp}(\mathrm{pred}_{\mathbb{T}}(t))=\alpha\\}.$

The collection of first $\alpha$-many levels $T_{<\alpha}$ is $\\{t\in
T\mathrel{|}\allowbreak\operatorname{otp}(\mathrm{pred}_{\mathbb{T}}(t))<\alpha\\}$.

The _height of $T$_ (denoted $\mathrm{ht}(T)$) is the first ordinal $\alpha$
such that $T_{\alpha}=\varnothing.$ Finally, a _branch for $T$_ is a
$\subseteq$-maximal $<$-chain $b\subseteq T$.

For the scope of this section let us fix a Mahlo cardinal $\kappa$.333An
inaccessible cardinal that is limit of inaccessibles suffices. We shall
consider $\kappa$-trees on $\kappa$; namely, trees $T\subseteq\kappa$ such
that $\mathrm{ht}(T)=\kappa$ and $|T_{\alpha}|<\kappa$ for all
$\alpha<\mathrm{ht}(T)$. The following definition –modulo a slight tweak– is
borrowed from König and Yoshinobu’s paper [KY12]:

######  Definition 2.15.

A $\kappa$-tree $T$ is said to be a _$\kappa$ -regressive $\kappa$-Kurepa
tree_ if it has $\kappa^{+}$-many branches and for each inaccessible cardinal
$\delta<\kappa$ there is a regressive map $f\colon T_{\delta}\rightarrow
T_{<\delta}$; namely, a map satisfying the following clauses:

  1. (1)

$f(x)<_{T}x$ for all $x\in T_{\delta}$;

  2. (2)

if $x,y\in T_{\delta}$ are $<_{T}$-incompatible then $x\wedge y<_{T}f(x),f(y)$
where

$x\wedge y:=\sup_{<_{T}}\\{z\in T\mathrel{|}\allowbreak
z<_{T}x\,\wedge\,z<_{T}y\\}.$

######  Remark 2.16.

The only (yet crucial) difference with the notion in [KY12] is that here we
require the regressive maps to exist only at inaccessible levels of $T$. This
will be important to ensure that the natural forcing introducing a
$\kappa$-regressive $\kappa$-Kurepa tree is _almost $\kappa$-directed closed_;
to wit, for each $\gamma<\kappa$, the poset contains a dense
$|\gamma|$-directed-closed subforcing. As shown in [KY12], the former notion
yields a poset which is not almost $\omega_{2}$-directed-closed for it
contradicts $\mathrm{MM}$. On the other hand, notice that we do not assume
anything upon the cardinality of the level sets of $T$. This contrast with the
usual demand on $\kappa$-Kurepa trees that
$|T_{\alpha}|\leq|\alpha|+\aleph_{0}$ for all $\alpha<\kappa.$

We wish to define a poset $\mathbb{K}_{\kappa}$ that kills the measurability
of $\kappa$ and it is both $\kappa$-closed and almost $\kappa$-directed-
closed. The usual forcing to get a $\kappa$-Kurepa tree (see [Cum10, §6]) has
the first two of these properties but fails to have the third one. It turns
out that this is tightly connected with the requirement that the $\alpha$th-
level set of the tree is not too big relative to $|\alpha|$. We do not require
this in purpose, aiming to have some wiggle room for our poset to fulfil the
third requirement. The forcing we shall use is a variation of the one
considered by König and Yoshinobu in [KY12, §3]. Namely,

######  Definition 2.17.

Let $\mathbb{K}_{\kappa}$ denote the poset consisting of $p=(t^{p},h^{p})$
where:

  1. (1)

$t^{p}$ is a normal tree on $\kappa$, the levels of $t^{p}$ have cardinality
${<}\kappa$ and its height $\mathrm{ht}(t^{p})$ is $\alpha^{p}+1$ for some
$\alpha^{p}<\kappa$;

  2. (2)

for each inaccessible $\delta\leq\alpha^{p}$ there is a regressive map
$f\colon t^{p}_{\delta}\rightarrow t^{p}_{<\delta}$;

  3. (3)

$h^{p}\colon t^{p}_{\alpha^{p}}\rightarrow\kappa^{+}$ is a one-to-one
function.

Given conditions $(t,h),(s,g)$ in $\mathbb{K}_{\kappa}$ we write
$(t,h)\leq(s,g)$ if $s=t\mathbin{\upharpoonright}\mathrm{ht}(s)$,
$\operatorname{ran}(g)\subseteq\operatorname{ran}(h)$ and
$g^{-1}(\xi)<_{t}h^{-1}(\xi)$ for each $\xi\in\operatorname{ran}(g)$.

######  Proposition 2.18 (Main properties of $\mathbb{K}_{\kappa}$).

  1. (1)

$\mathbb{K}_{\kappa}$ is $\kappa^{+}$-cc and $\kappa$-closed;

  2. (2)

$\mathbb{K}_{\kappa}$ adds a $\kappa$-regressive $\kappa$-Kurepa tree.

  3. (3)

$\mathbb{K}_{\kappa}$ is almost $\kappa$-directed-closed;

  4. (4)

$\mathop{1\hskip-3.0pt{\rm l}}\Vdash_{\mathbb{K}_{\kappa}}``\kappa$ is not
measurable”.

###### Proof.

(1) Since $2^{<\kappa}=\kappa$ standard arguments show that
$\mathbb{K}_{\kappa}$ is $\kappa^{+}$-cc. We skip the verification of
$\kappa$-closure as it is similar to the argument for (3).

(2) Let $G\subseteq\mathbb{K}_{\kappa}$ a $V$-generic filter and define
$T:=\bigcup\\{t^{p}\mathrel{|}\allowbreak p\in G\\}$. It is routine to check
that $T$ is a $\kappa$-regressive $\kappa$-Kurepa tree as witnessed by the
collection of branches
$\\{b_{\alpha}\mathrel{|}\allowbreak\alpha<\kappa^{+}\\}$ defined as

$b_{\alpha}:=\sup_{<_{T}}\\{h^{-1}(\alpha)\mathrel{|}\allowbreak\exists p\in
G\,(h^{p}=h\,\wedge\,\alpha\in\operatorname{ran}(h^{p}))\\}.$

(3) Fix $\gamma<\kappa$ and define
$\\{(t,h)\in\mathbb{K}_{\kappa}\mathrel{|}\allowbreak\text{The height of the
tree $t$ is ${>}\gamma$}\\}$. Clearly, this set is dense in
$\mathbb{K}_{\kappa}$. We claim that it is also $|\gamma|^{+}$-directed-
closed.

Let $\\{(t_{\xi},h_{\xi})\mathrel{|}\allowbreak\xi<|\gamma|\\}$ be any
directed set of conditions in the set. For each $\xi<|\gamma|$ let
$\alpha_{\xi}+1$ be the height of $t_{\xi}$ and set
$\alpha^{*}:=\sup_{\xi<\gamma}\alpha_{\xi}$. Define

$\textstyle
t^{*}:=\bigcup_{\xi<|\gamma|}t_{\xi}\,\wedge\,<_{t^{*}}:=\bigcup_{\xi<|\gamma|}<_{t_{\xi}}.$

Let us add to $(t^{*},\leq_{t^{*}})$ one more level so that the resulting tree
has successor height. Let $\mathcal{B}$ denote the collection of all branches
$b\subseteq t^{*}$ (i.e., $\subseteq$-maximal $<_{t^{*}}$-chains) and
$\Phi\colon\mathcal{B}\rightarrow\kappa\setminus t^{*}$ be an injective map
$b\mapsto\alpha_{b}$ (note that this exists as $\kappa$ is inaccessible). We
declare $\alpha_{b}$ to be bigger than $\eta$ for all $\eta\in b$ and declare
$\alpha_{b}$ and $\alpha_{b^{\prime}}$ to be incompatible provided $b\neq
b^{\prime}.$

The above choice yields a new tree $t$ whose first $\alpha^{*}$-many levels
are given by $t^{*}$ and the top one is given by the new ordinals
$\\{\alpha_{b}\mathrel{|}\allowbreak b\in\mathcal{B}\\}$. In symbols,

$t_{{<}\alpha_{*}}=t^{*}\;\text{and}\;t_{\alpha_{*}}=\\{\alpha_{b}\mathrel{|}\allowbreak
b\in\mathcal{B}\\}.$

The proposed one-to-one map $h\colon t_{\alpha^{*}}\rightarrow\kappa^{+}$ is
defined as follows. For each ordinal
$\nu\in\bigcup_{\xi<|\gamma|}\operatorname{ran}(h_{\xi})$ let us define a
$<_{t^{*}}$-chain through $t^{*}$ as

$c_{\nu}:=\sup_{<_{t^{*}}}\\{h_{\xi}^{-1}(\nu)\mathrel{|}\allowbreak\nu\in\operatorname{ran}(h_{\xi})\\}.$

It is conceivable that $c_{\nu}$ is not $\subseteq$-maximal but we can
$\subseteq$-extend it to some $b_{\nu}\in\mathcal{B}$. In that case
$\alpha_{b_{\nu}}$ is $<_{t}$-bigger than all members of $c_{\nu}$. Let

$\textstyle\Psi\colon\\{\alpha_{b}\mathrel{|}\allowbreak
b\in\mathcal{B}\\}\setminus\\{\alpha_{b_{\nu}}\mathrel{|}\allowbreak\nu\in\bigcup_{\xi<|\gamma|}\operatorname{ran}(h_{\xi})\\}\rightarrow\kappa^{+}\setminus(\sup\bigcup_{\xi<|\gamma|}\operatorname{ran}(h_{\xi}))+1$

be any injection. We define $h\colon t_{\alpha^{*}}\rightarrow\kappa^{+}$ as
follows:

$h(\alpha_{b}):=\begin{cases}\nu,&\text{if $\alpha_{b}=\alpha_{b_{\nu}}$ for
some $\nu\in\bigcup_{\xi<|\gamma|}\operatorname{ran}(h_{\xi})$;}\\\
\Psi(\alpha_{b}),&\text{otherwise.}\end{cases}$

We claim that $(t,h)$ is a condition in $\mathbb{K}_{\kappa}$. The only caveat
is Clause (2) in Definition 2.15: If $\delta\leq\alpha^{*}$ is an inaccessible
cardinal, since $\operatorname{cf}(\alpha^{*})=\gamma<\alpha^{*}$, it must be
the case that $\delta<\alpha^{*}$. Hence there is some $\xi$ such that
$\delta\leq\alpha_{\xi}$. Since $t_{\delta}=(t_{\xi})_{\delta}$ and
$t_{<\delta}=(t_{\xi})_{<\delta}$ we simply use the fact
$(t_{\xi},h_{\xi})\in\mathbb{K}_{\kappa}$ to get the sought regressive
function. Thus we conclude that $(t,h)\in\mathbb{K}_{\kappa}$. The
construction of $(t,h)$ guarantees that $(t,h)\leq(t_{\xi},h_{\xi})$ for all
$\xi<|\gamma|.$

(4) Suppose otherwise and let $G\subseteq\mathbb{K}_{\kappa}$ a $V$-generic
filter such that $\kappa$ is measurable in $V[G]$. Let $j\colon
V[G]\rightarrow M$ be an elementary embedding with
$\operatorname{crit}(j)=\kappa$ and $T$ be the $\kappa$-regressive
$\kappa$-Kurepa tree introduced by $G$. By elementarity, $M$ models that
$j(T)$ is a $j(\kappa)$-regressive $j(\kappa)$-Kurepa tree and that $\kappa$
is inaccessible. Let $f\colon j(T)_{\kappa}\rightarrow T$ be a regressive map
in $M$. For a branch $b$ of $T$, the $\kappa$th-member of $j(b)$ – denoted
$j(b)_{\kappa}$– belongs to $j(T)_{\kappa}$ and witnesses
$f(j(b)_{\kappa})<_{T}j(b)_{\kappa}$ (by regressiveness). Let
$\alpha_{b}<\kappa$ be such that

$f(j(b)_{\kappa})<_{T}j(b)_{\alpha_{b}}=j(b_{\alpha_{b}})=b_{\alpha_{b}}.$

Let us observe that the map $b\mapsto b_{\alpha_{b}}$ is one-to-one: Indeed,
if $b\neq b^{\prime}$ are cofinal branches of $T$ then the regressiveness of
$f$ ensures that $f(j(b)_{\kappa}),f(j(b^{\prime})_{\kappa})$ are
$<_{T}$-above $j(b)_{\kappa}\wedge j(b^{\prime})_{\kappa}=\sup_{<_{T}}\\{z\in
T\mathrel{|}\allowbreak z\in b\cap b^{\prime}\\}$. So
$b_{\alpha_{b}}\perp_{T}b_{\alpha_{b^{\prime}}}.$

This is a contradiction because $T$ has $\kappa^{+}$-many branches and
$|T|=\kappa$. ∎

###  2.3. The coding poset

Given a regular uncountable cardinal $\chi$ a stationary set $S\subseteq\chi$
is called _fat_ if for every club $C\subseteq\chi$, $S\cap C$ contains closed
sets of arbitrarily large order-type below $\chi$. Given a fat stationary set
$S\subseteq\chi$, Abraham and Shelah [AS83] defined the poset $\mathbb{C}(S)$
which consists of closed bounded sets $c\subseteq S$ and it is ordered by
$\sqsupseteq$-extension. Clearly $\mathbb{C}(S)$ adds a generic club
$\mathcal{C}\subseteq\chi$ contained in $S$. The poset is $\chi^{+}$-cc and
$\chi$-distributive – fatness is crucial to establish this latter fact (see
[AS83] or [Cum10, §6]).

The following poset will be employed in the forthcoming §4.1:

######  Definition 2.19 (Coding poset).

Let $\chi$ be a Mahlo cardinal and $\varepsilon<\chi$ be a regular cardinal.
By $\mathbb{C}(\varepsilon,\chi)$ we denote the two-step iteration

$\textstyle\mathbb{C}(S_{\chi,\varepsilon})\ast(\mathop{\mathrm{Col}}(\varepsilon,{\dot{c}}_{0}^{++})\times\prod_{\eta<\chi}\mathop{\mathrm{Col}}(\dot{c}_{\eta}^{++},\dot{c}_{\eta+1}^{++})),$

where
$S_{\chi,\varepsilon}:=\\{\sigma<\chi\mathrel{|}\allowbreak\sigma\in\mathrm{Sing}\,\wedge\,\sigma>\varepsilon\\}$
and $\langle\dot{c}_{\eta}\mathrel{|}\allowbreak\eta<\chi\rangle$ is a
$\mathbb{C}(S_{\chi,\varepsilon})$-name for the corresponding generic club.

The poset $\mathbb{C}(\varepsilon,\chi)$ makes $\chi$ non-Mahlo by introducing
a club consisting of singular cardinals and then collapses the gaps in between
the members of the generic club. Thus, $\mathbb{C}(\varepsilon,\chi)$ makes
$\chi$ be the first inaccessible cardinal above $\varepsilon$. In addition,
$\mathbb{C}(\varepsilon,\chi)$ preserves all cardinals ${\leq}\varepsilon$ and
collapses $\varepsilon^{+}.$

Forcing with $\mathbb{C}(\varepsilon,\chi)$ will be instrumental to make the
stationary set $\mathcal{S}$ of Lemma 4.2 definable in our final generic
extension. Although $\mathbb{C}(\varepsilon,\chi)$ is not $\chi$-closed it has
been posed to be $\varepsilon$-directed-closed.

###  2.4. Further coding

In the proof of Theorem 4.1 and Theorem 4.3 we will have to begin with a model
carrying an easily-definable well-ordering. This will be used to ensure that
the our forcing iteration is shifted correctly by the relevant extender
embeddings $j_{E^{*}}$ (see Clause (1) in page 1).

Consider the following minor variation of Hamkins-Reitz CCA [Rei07, §3]:

######  Definition 2.20.

$\mathrm{CCA}^{*}$ is the following assertion: For each ordinal $\alpha$ and
every set $a\subseteq\alpha$ there is yet another ordinal $\theta$ below the
first $\beth$-fixed point above $\alpha$ such that
$a=\mathsf{Code}(\alpha,\theta):=\\{\beta<\alpha\mathrel{|}\allowbreak
2^{\aleph_{\theta+\beta+1}}=\aleph_{\theta+\beta+2}\\}.$

Under $\mathrm{CCA}^{*}$ the set-theoretic universe carries a natural well-
ordering. Namely, for each set $a\subseteq\operatorname{Ord}$ denote
$\alpha_{a}:=\\{\beta\in\operatorname{Ord}\mathrel{|}\allowbreak
a\subseteq\beta\\}$ and
$\theta_{a}:=\min\\{\theta\in\operatorname{Ord}\mathrel{|}\allowbreak
a=\mathsf{Code}(a_{\alpha},\theta)\\}$. Given $a,b\subseteq\operatorname{Ord}$
we write

$a\prec b$ if and only if $\alpha_{a}<\alpha_{b}$ or ($\alpha_{a}=\alpha_{b}$
and $\theta_{a}<\theta_{b}$).

By coding sets as subsets of their corresponding ranks the above induces a
well-ordering over the set-theoretic universe. This well-ordering is quite
absolute; specifically, given an inaccessible cardinal $\lambda$ and sets
$a,b\in V_{\lambda}$,

$a\prec b$ if and only if $V_{\lambda}\models``a\prec b$”.

In [Rei07, §3], it is showed that if the GCH holds then $\mathrm{CCA}^{*}$
holds after forcing with McAloon’s class iteration $\mathbb{P}$ [McA71]. The
forcing $\mathbb{P}$ is fairly well-behaved and preserves (virtually) all
kinds of large cardinals. For instance forcing with $\mathbb{P}$ preserves
hugeness (recall Definition 2.12).

######  Lemma 2.21.

Assume the $\mathrm{GCH}$ holds. If $\kappa$ is a huge cardinal then

$\mathop{1\hskip-3.0pt{\rm l}}\Vdash_{\mathbb{P}}``\kappa$ is huge
$+\,\mathrm{CCA}^{*}$”.

###### Proof sketch.

Let $j\colon V\rightarrow M$ be a huge embedding. Without loss of generality
members of $M$ take the form $j(f)([\operatorname{id}]_{U})$ where
$\operatorname{id}\colon\mathcal{P}(j(\kappa))\rightarrow\mathcal{P}(j(\kappa))$
and $U$ is a normal measure on $\mathcal{P}(j(\kappa))$ ([Kan09, Theorem
24.8]). Let $G\subseteq\mathbb{P}$ a $V$-generic filter. Clearly $j$ lifts to
$j\colon V[G_{\kappa}]\rightarrow M[G_{j(\kappa)}]$. Note that
$M[G_{j(\kappa)}]$ is closed under $j(\kappa)$-sequences in $V[G_{j(\kappa)}]$
as $\mathbb{P}_{j(\kappa)}$ is $j(\kappa)$-cc. Next, note that
$j``G_{[\kappa,j(\kappa))}$ is a member of $M[G_{j(\kappa)}]$ and a directed
subset of $j(\mathbb{P}_{[\kappa,j(\kappa))})$, which is a
$j(\kappa)^{+}$-closed poset in $V[G_{j(\kappa)}]$. In particular, there is a
master condition $q$ for $j``G_{[\kappa,j(\kappa))}$. Using the
$\mathrm{GCH}_{\geq j(\kappa)}$ and the fact that each maximal antichain $A\in
M$ for $j(\mathbb{P}_{[\kappa,j(\kappa))})$ can be represented as
$j(f)([\operatorname{id}]_{U})$ for
$f\colon\mathcal{P}(j(\kappa))\rightarrow\mathcal{P}_{j(\kappa)}(\mathbb{P}_{[\kappa,j(\kappa))})$
we find $q\in H\in V[G_{j(\kappa)}]$ that is $M[G_{j(\kappa)}]$-generic for
$j(\mathbb{P}_{[\kappa,j(\kappa))})$. Thus, inside $V[G_{j(\kappa)}]$ we lift
our embedding to $j\colon V[G_{j(\kappa)}]\rightarrow M[G_{j(\kappa)}\ast H]$.
Finally, we use that $j$ has width555The _width_ of an elementary embedding
$j\colon M\rightarrow N$ between transitive models is said to be
${\leq}\theta$ if every member of $N$ can be expressed as $j(f)(a)$ for $a\in
N$ and a function $f\in M$ with $M\models|\operatorname{dom}(f)|\leq\theta$.
${\leq}j(\kappa)$ and that $\mathbb{P}_{[j(\kappa),\operatorname{Ord})}$ is
$j(\kappa)^{+}$-distributive to transfer the generic
$G_{[j(\kappa),\operatorname{Ord})}$ (see [Cum10, §15]). This yields a
$V[G]$-definable embedding $j\colon V[G]\rightarrow M[G_{j(\kappa)}\ast H\ast
j``G_{[j(\kappa),\operatorname{Ord})}]$ which witnesses hugeness of $\kappa$.
∎

Forcing with $\mathbb{P}$ preserves even stronger large cardinals; such as the
existence of an $I_{0}(\kappa)$ (or even an $I_{0}^{\sharp}(\kappa)$)
embedding (see e.g., [DF14]).

##  3\. Axiom $\mathcal{A}$ and its consequences

In this section we introduce axiom $\mathcal{A}$ and show that it yields the
sought equivalence between supercompactness and $C^{(1)}$-supercompactness. We
also show that $\mathcal{A}$ disproves Woodin’s EEA (see Definition 3.7).

######  Definition 3.1.

Axiom $\mathcal{A}$ is the conjunction of the following statements:

  1. (1)

There is a proper class of inaccessible cardinals.

  2. (2)

Suppose that $\delta$ is stationary-correct superstrong with inaccessible
target $\lambda$.666Recall Definition 2.7. Then, $\delta$ is tall with target
$\theta$ for all successor inaccessible cardinals $\theta\in(\delta,\lambda)$.

######  Remark 3.2.

Axiom $\mathcal{A}$ has been distilled as a result of several discussions
between the author and Woodin. The author wishes to thank Prof. Woodin for the
stimulating conversations on the topic and for his unvaluable insights.

The next provides a characterization of $\mathop{C^{(n)}}$-supercompact
cardinals:

######  Theorem 3.3.

Fix $n<\omega$. The following are equivalent:

  1. (1)

$\kappa$ is $C^{(n)}$-supercompact;

  2. (2)

$\kappa$ is both supercompact and enhanced $C^{(n)}$-tall.

###### Proof.

The implication $(1)\Rightarrow(2)$ is evident. As for $(2)\Rightarrow(1)$ one
argues as follows. Fix $\lambda>\kappa$ with $\lambda^{<\kappa}=\lambda$.
Since $\kappa$ is an enhanced $C^{(n)}$-tall cardinal there is $\theta\in
C^{(n)}$ with $\operatorname{cf}(\theta)>\lambda$ such that $\kappa$ is tall
with target $\theta$. Next, let $j\colon V\rightarrow M$ be the elementary
embedding induced by a supercompact measure on
$\mathcal{P}_{\kappa}(\lambda)$. Using that
$\operatorname{cf}(\theta)>\lambda=\lambda^{<\kappa}$ standard arguments yield
$j(\theta)=\theta$. Thus, by elementarity, $M$ thinks that $j(\kappa)$ is tall
with target $\theta$. Let $E\in M$ be a witnessing
$(j(\kappa),\theta)$-$M$-extender and let $i\colon M\rightarrow
N\simeq\mathrm{Ult}(M,E)$ be its induced ultrapower. Clearly, the composition
$\iota:=i\circ j$ yields an elementary embedding $\iota\colon V\rightarrow N$
with $\iota(\kappa)=\theta\in C^{(n)}$. It remains to show that
$N^{\lambda}\cap V\subseteq N$: Let $\langle
x_{\alpha}\mathrel{|}\allowbreak\alpha<\lambda\rangle\in N^{\lambda}\cap V$.
By definition of $N$, there are $s_{\alpha}\in[\theta]^{<\omega}$ and
functions $f_{\alpha}\colon[\zeta]^{|s_{\alpha}|}\rightarrow M$ in $M$ such
that $i(f_{\alpha})(s_{\alpha})=x_{\alpha}$. Since $M^{\lambda}\cap V\subseteq
M$ it follows that $\langle
f_{\alpha}\mathrel{|}\allowbreak\alpha<\lambda\rangle\in M$ and thus
$i(\langle f_{\alpha}\mathrel{|}\allowbreak\alpha<\lambda\rangle)=\langle
i(f_{\alpha})\mathrel{|}\allowbreak\alpha<\lambda\rangle\in N$. Similarly,
$\langle
s_{\alpha}\mathrel{|}\allowbreak\alpha<\lambda\rangle\in(H_{\theta}^{M})^{<\lambda}\subseteq
H^{M}_{\theta}=H^{N}_{\theta}$ in that $\theta$ is a strong limit cardinal
with cofinality ${>}\lambda$. From these two facts we conclude that $\langle
x_{\alpha}\mathrel{|}\allowbreak\alpha<\lambda\rangle\in N$, as needed. ∎

Part of our interest in $\mathcal{A}$ rests on the following key fact:

######  Theorem 3.4.

Assume $\mathcal{A}$ holds. Every supercompact cardinal is enhanced
$C^{(1)}$-tall. In particular, every supercompact cardinal is
$C^{(1)}$-supercompact.

Moreover, under $\mathcal{A}$, if $\delta$ is supercompact and
$\delta<\lambda$ then

$\text{$\delta$ is $\lambda$-supercompact if and only if $\delta$ is
$\lambda$-$C^{(1)}$-supercompact}.$

###### Proof.

Let $\delta$ be a supercompact cardinal and fix $\lambda>\delta$. Let $\theta$
be a successor inaccessible above $\lambda$ – this choice is possible by
Clause (1) of axiom $\mathcal{A}$.

Let $j\colon V\rightarrow M$ be a witness for the
$\theta^{+}$-supercompactness of $\delta$. Then,

$M\models``\text{$\mathcal{A}\,\wedge\,\delta$ is $(\delta+1)$-extendible with
target $j(\delta)$''.}$

The first property follows from elementarity of $j$; the second from
$j\mathbin{\upharpoonright}V_{\delta+1}$ being a member of $M$ (by
$\theta^{+}$-closure of $M$). In particular Proposition 2.6 implies that
$\delta$ is (inside $M$) stationary-correct superstrong with target
$j(\delta)$.

From the perspective of $M$, $\theta$ is also a successor inaccessible.
Invoking $\mathcal{A}$ inside $M$ the model thinks that $\delta$ is tall with
target $\theta$ and the same property holds true in $V$: Indeed, let $E\in M$
be a $(\delta,\theta)$-$M$-extender witnessing this. Since
$M^{\theta^{+}}\subseteq M$ it follows that $E$ is a
$(\delta,\theta)$-$V$-extender and its $V$-ultrapower embedding $j_{E}\colon
V\rightarrow M_{E}$ witnesses that $\delta$ is tall with target $\theta$. The
argument in Theorem 3.3 shows that $\delta$ is
$\lambda$-$C^{(1)}$-supercompact. ∎

######  Remark 3.5.

Note that $\mathcal{A}$ actually implies that every supercompact cardinal is
$C^{(1)}$-supercompact with inaccessible targets.

The next observation shows that under $\mathcal{A}$ every supercompact is
$C^{(n)}$-supercompact, for each $n\geq 1$, in certain inner model.

######  Proposition 3.6.

Assume $\mathcal{A}$ holds. If $\kappa$ is supercompact then there is $M$ an
inner model of $\mathrm{ZFC}$ where, for each $n\geq 1$, $M\models\kappa$ is
$C^{(n)}$-supercompact.

###### Proof.

Under $\mathcal{A}$ if $\kappa$ is supercompact then it is
$C^{(1)}$-supercompact with inaccessible target $j(\kappa)$ (Theorem 3.4). By
Tsaprounis [Tsa14, Proposition 2.8],

$C:=\\{h(\kappa)<j(\kappa)\mathrel{|}\allowbreak\text{$\kappa$ is tall with
target $h(\kappa)$}\\}$

contains a ${{\geq}\kappa^{+}}$-club. Since for each $\lambda<j(\kappa)$ and
$n<\omega$ the set

$C\cap C^{(n)}\cap E^{j(\kappa)}_{>\operatorname{cf}(\lambda)}$

is non-empty it follows that

$\text{$V_{j(\kappa)}\models``\mathrm{ZFC}+\kappa$ is supercompact and
enhanced $\mathop{C^{(n)}}$-tall''}.$

By Theorem 3.3 $V_{j(\kappa)}$ models that $\kappa$ is $C^{(n)}$-supercompact.
∎

Another interesting aspect of axiom $\mathcal{A}$ finds its roots in the inner
model program at the level of supercompactness [Woo24]. In [FS21] Fernandes
and Schindler showed that if there is no inner model with a Woodin cardinal
and $L[E]$ is iterable then, in $L[E]$, a cardinal is tall if and only if it
is either strong or a measurable limit of strong cardinals. A related question
is whether a similar equivalence is available between the classes of
superstrong cardinals and tall cardinals with strong limit targets. Especially
interersting is whether this equivalence is compatible with the existence of a
supercompact. The said equivalence is postulated by the _Extender Embedding
Axiom_ :

######  Definition 3.7 (Woodin, 2018).

The _Extender Embedding Axiom_ (EEA) says that every cardinal $\kappa$
carrying an elementary embedding $j\colon V\rightarrow M$ with
$\operatorname{crit}(j)=\kappa$, $M^{\kappa}\subseteq M$ and $j(\kappa)\in
C^{(1)}$ is superstrong with target $j(\kappa).$

Observe that the superstrong embeddings postulated by EEA might be different
from the departing $j\colon V\rightarrow M$. In fact, a version of EEA saying
that each $j\colon V\rightarrow M$ as above witnesses superstrongness is
inconsistent. For instance, this version of EEA would imply that every
$C^{(n)}$-supercompact cardinal is $C^{(n)}$-extendible, which is not possible
by virtue of [Pov24].

At the present time is unclear to the author whether EEA is consistent with a
supercompact cardinal – nevertheless, Woodin has conjectured that it will
follow from his axiom $V=\mathrm{Ultimate-}L$ (see [Woo17]):

######  Conjecture (Woodin, 2018).

$``\mathrm{ZFC}+V=\mathrm{Ultimate-}L$” proves $\mathrm{EEA}$.

Contingent to Woodin’s conjecture, the next shows that under
$V=\mathrm{Ultimate-}L$, $C^{(n)}$-supercompacts are
almost-$C^{(n)}$-extendibles in disguise.

######  Theorem 3.8 ($\mathrm{EEA}$).

Every $C^{(n)}$-supercompact is almost-$C^{(n)}$-extendible.

In particular, $\mathcal{A}+``$There is a supercompact cardinal” disproves
$\mathrm{EEA}$.

###### Proof.

Let $n<\omega$ and $\delta$ be a $C^{(n)}$-supercompact cardinal. Fix an
arbitrary $\lambda>\delta$. Clearly, $\delta$ is $\lambda$-supercompact. Also,
since $\delta$ is $\lambda$-$C^{(n)}$-supercompact, EEA yields an embedding
$j\colon V\rightarrow M$ with $\operatorname{crit}(j)=\delta$, $j(\delta)\in
C^{(n)}$, $\operatorname{cf}(j(\delta))>\lambda$ and $V_{j(\delta)}\subseteq
M$. Combining these two facts we conclude that $\delta$ is
almost-$C^{(n)}$-extendible. For the second claim invoke [Pov24, Proposition
2.3] to infer that every almost-$C^{(1)}$-extendible is a limit of
supercompacts. Under EEA this is equivalent to saying that every
$C^{(1)}$-supercompact is a limit of supercompacts, which is not the case
under $\mathcal{A}$ (by Theorem 3.4). ∎

In spite $V=\mathrm{Ultimate-}L$ may entail EEA –which in turn rules out
$\mathcal{A}$– in the next section we prove that $\mathcal{A}$ is consistent
with a proper class of supercompact cardinals – and in fact with $I_{0}$
cardinals.

##  4\. The proof of the main theorem

In this section we prove the consistency of $\mathcal{A}$ with strong large
cardinals.

######  Theorem 4.1.

Assume the $\mathrm{GCH}$ holds and that $\kappa$ is a huge cardinal. Then,
the theory $``\mathrm{ZFC}+\mathcal{A}+$Vopěnka’s Principle” is consistent.

Recall that _Vopěnka’s Principle_ is the schema asserting that every class of
structures $\mathscr{C}$ in the same language does carry different
$\mathcal{M},\mathcal{N}\in\mathscr{C}$ and an elementary embedding
$j\colon\mathcal{M}\rightarrow\mathcal{N}.$ For an inaccessible cardinal
$\kappa$, $V_{\kappa}\models\mathrm{VP}$ is equivalent to $\kappa$ being a
Vopěnka cardinal (see p. 2.1). By virtue of Bagaria’s [Bag12, Corollary 4.15]
the model of Theorem 4.1 is plenty of supercompact cardinals – in fact, it
contains a proper class of $C^{(n)}$-extendible cardinals, for all $n\geq 1$.
Thus, the consistency of $\mathcal{A}$ is not precluded by the strongest known
large cardinals. In other words, axiom $\mathcal{A}$ is consistent with a
mathematical universe rich and intricate (and this is very far from
$\mathrm{Ultimate-}L$).

###  4.1. Forcing $\mathcal{A}$

By forcing preliminary with McAloon iteration (Lemma 2.21) we may assume that
our ground model carries a well-ordering $\prec$ satisfying

($\dagger$) $a\prec b\;\Longleftrightarrow\;V_{\lambda}\models\text{$``a\prec
b$'',}$

for each $a,b\in V_{\lambda}$ and $\lambda$ inaccessible. The absoluteness of
$\prec$ will be crucial in the design of the iteration leading to a model of
axiom $\mathcal{A}.$

The proof of Theorem 4.1 will be broken into a series of lemmas. Let us fix a
normal measure $\mathcal{U}$ on $\kappa$ as in the statement of Lemma 2.11.

######  Lemma 4.2.

There is $\mathcal{S}\in\mathcal{U}$ consisting of measurable cardinals whose
members are indiscernibles in the following strong sense: Let
$\Phi(x,x_{0},\dots,x_{n})$ be a first order formula in $\mathcal{L}_{\in}$,
$\alpha_{0}<\cdots<\alpha_{n}$ and $\beta_{0}<\cdots<\beta_{n}$ members of
$\mathcal{S}$ and $a\in V_{\kappa}$ whose rank is less than
$\min(\\{\alpha_{i}\mathrel{|}\allowbreak i\leq
n\\}\cup\\{\beta_{i}\mathrel{|}\allowbreak i\leq n\\}).$ Then,

$\langle
V_{\kappa},\in\rangle\models\Phi(a,\alpha_{0},\dots,\alpha_{n})\,\Longleftrightarrow\,\langle
V_{\kappa},\in\rangle\models\Phi(a,\beta_{0},\dots,\beta_{n}).$

Moreover, $\\{\delta<\kappa\mathrel{|}\allowbreak\text{$\mathcal{S}\cap\delta$
is stationary in $\delta$}\\}\in\mathcal{U}$.

###### Proof.

Let $\langle\Phi_{n}\mathrel{|}\allowbreak n<\omega\rangle$ be an enumeration
of the first order formulae in the language of set theory. For each $a\in
V_{\kappa}$ define a coloring $c_{a}\colon[\kappa]^{<\omega}\rightarrow 3$ as

$c_{a}(\vec{\alpha}):=\begin{cases}0,&\text{if
$\operatorname{rank}(a)\geq\min(\vec{\alpha})$};\\\ 1,&\text{if
$\operatorname{rank}(a)<\min(\vec{\alpha})$ and $\langle
V_{\kappa},\in\rangle\models\Phi(a,\vec{\alpha})$};\\\ 2,&\text{if
$\operatorname{rank}(a)<\min(\vec{\alpha})$ and $\langle
V_{\kappa},\in\rangle\models\neg\Phi(a,\vec{\alpha})$};\end{cases}$

By Rowbottom’s theorem [Kan09, Theorem 7.17], for each $a\in V_{\kappa}$,
there is a $c_{a}$-homogeneous set $H_{a}\in\mathcal{U}$; to wit, for each
$n<\omega$, $c_{a}\mathbin{\upharpoonright}[H_{a}]^{n}$ is constant. Since
$\mathcal{U}$ is uniform it is evident that $c_{a}``[H_{a}]^{n}\neq\\{0\\}$
for all $n<\omega$.

Define $H:=\\{\alpha<\kappa\mathrel{|}\allowbreak\forall a\in
V_{\kappa}\,(\operatorname{rank}(a)<\alpha\,\Rightarrow\,\alpha\in H_{a})\\}.$
It is routine to check that $H\in\mathcal{U}$ and that members of $H$ are
indiscernibles in the desired sense. Finally, the moreover part follows from
normality of $\mathcal{U}$: If $j\colon V\rightarrow M$ is the ultrapower
induced by $\mathcal{U}$ then $M\models``j(\mathcal{S})\cap\kappa$ is
stationary in $\kappa$”. ∎

For the rest of this section we adopt the following notations:

######  Notation 4.3.

For each inaccessible $\varepsilon<\kappa$ let us denote:

$\blacktriangleright$ $\lambda_{\varepsilon}$ the next member of $\mathcal{S}$
past $\varepsilon$;

$\blacktriangleright$ $\mathscr{E}_{\varepsilon,\mathcal{S}}:=\\{E\in
V_{\kappa}\mathrel{|}\allowbreak\exists\lambda\in\mathcal{S}\,(V_{\kappa}\models\Phi(E,\varepsilon,\mathcal{S}\cap\varepsilon,\lambda))\\}$
where $\Phi(E,\varepsilon,\mathcal{S}\cap\varepsilon,\lambda)$ is the
conjunction of the following two sentences:

  1. (1)

$E$ is an extender with length $\varepsilon$ and strength $\varepsilon$.

  2. (2)

There is a superstrong extender $F$ such that

     * •

$F\mathbin{\upharpoonright}\varepsilon=E$ (in particular,
$\operatorname{crit}(E)=\operatorname{crit}(F)$);

     * •

$j_{F}(\operatorname{crit}(E))=\lambda;$

     * •

$(M_{F})^{\operatorname{crit}(F)}\subseteq M_{F}$;

     * •

$j_{F}(\mathcal{S})\cap(\varepsilon+1)=\mathcal{S}\cap(\varepsilon+1)$.777Note
that $j_{F}(\mathcal{S})\cap(\varepsilon+1)$ is the same as
$j_{F}(\mathcal{S}\cap\varepsilon)\cap(\varepsilon+1)$. As a result, the above
statement can be easily stated in terms of the parameters $\varepsilon$ and
$\mathcal{S}\cap\varepsilon.$

$\blacktriangleright$ For $i\in\\{0,1\\}$, $``\sigma\in^{1}\mathcal{S}$” means
$``\sigma\in\mathcal{S}$” and $``\sigma\in^{0}\mathcal{S}$” means
$``\sigma\notin\mathcal{S}$”

The next lemma reveals the role of indiscernibility of cardinals in
$\mathcal{S}$:

######  Lemma 4.4.

Let $\varepsilon<\kappa$ inaccessible and $E\in V_{\kappa}$ an extender with
length and strength $\varepsilon$. Then,

$E\in\mathscr{E}_{\varepsilon,\mathcal{S}}\;\Longleftrightarrow\;V_{\kappa}\models\Phi(E,\varepsilon,\mathcal{S}\cap\varepsilon,\lambda_{\varepsilon}).$

Moreover, for each $E\in\mathscr{E}_{\varepsilon,\mathcal{S}}$ there is a
$\prec$-minimal extender $E^{*}$ witnessing it

###### Proof.

Suppose that $E$ is an extender in $\mathscr{E}_{\varepsilon,\mathcal{S}}$. By
definition, there is a target $\lambda\in\mathcal{S}$ witnessing
$V_{\kappa}\models\Phi(E,\varepsilon,\mathcal{S}\cap\varepsilon,\lambda)$.
Since members of $\mathcal{S}$ are indiscernibles and
$\operatorname{rank}\\{E,\varepsilon,\mathcal{S}\cap\varepsilon\\}<\lambda_{\varepsilon}\leq\lambda$
it follows that
$V_{\kappa}\models\Phi(E,\varepsilon,\mathcal{S}\cap\varepsilon,\lambda_{\varepsilon}).$
Let $E^{*}$ be the $\prec$-least extender witnessing
$E\in\mathscr{E}_{\varepsilon,\mathcal{S}}$. ∎

We will define our main iteration $\mathbb{P}_{\kappa}$ in a prelimminary
generic extension by Cohen forcing $\mathrm{Add}(\omega,1)$. This will ensure
that $\mathrm{Add}(\omega,1)\ast\dot{\mathbb{P}}_{\kappa}$ admits a _gap below
$\omega_{1}$_. As a result, superstrong embeddings in the generic extension by
$\mathrm{Add}(\omega,1)\ast\dot{\mathbb{P}}_{\kappa}$ will be liftings of
superstrong extenders in $V$. This can be seen as a consequence of _Hamkin’s
Gap Forcing_ theorem [Ham01].

Working in a generic extension by ${\mathrm{Add}(\omega,1)}$ (for simplicity,
call it $V$) let

$\mathbb{P}_{\kappa}:=\varinjlim\langle\mathbb{P}_{\varepsilon};\dot{\mathbb{Q}}_{\varepsilon}\mathrel{|}\allowbreak\varepsilon<\kappa\rangle$

be the Easton-supported iteration defined by induction as follows:

######  Induction Hypothesis.

For each inaccessible $\sigma<\varepsilon$ with $\sigma\in^{i}\mathcal{S}$ and
a $V$-generic filter $G_{\sigma}\subseteq\mathbb{P}_{\sigma}$ we suppose the
following hold:

Trivial case: If $V[G_{\sigma}]\models``\sigma$ is not inaccessible” then
$(\dot{\mathbb{Q}}_{\sigma})_{G_{\sigma}}:=\\{\mathop{1\hskip-3.0pt{\rm
l}}\\}.$

Non-trivial case: If $V[G_{\sigma}]\models``\sigma$ is inaccessible” then we
distinguish among two cases seeking to define
$(\dot{\mathbb{Q}}_{\sigma})_{G_{\sigma}}.$ Namely,

Case $(\aleph)$: If $\mathcal{S}\cap\sigma$ is non-stationary (in $V$) then:

$\blacktriangleright$ If $\mathscr{E}_{\sigma,\mathcal{S}}=\emptyset$ then we
set
$(\dot{\mathbb{Q}}_{\sigma})_{G_{\sigma}}:=\mathbb{K}_{\sigma}\ast\dot{\mathbb{C}}(\sigma^{+2+i}_{V},\lambda_{\sigma}).$

$\blacktriangleright$ If $\mathscr{E}_{\sigma,\mathcal{S}}$ is non-empty then
we demand:

  1. (1)

For each extender $F\in\mathscr{E}_{\sigma,\mathcal{S}}$,
$j_{F^{*}}(\mathbb{P}_{\operatorname{crit}(F)})_{\sigma}=\mathbb{P}_{\sigma}$.

  2. (2)

For each extender $F\in\mathscr{E}_{\sigma,\mathcal{S}}$,
$j_{F^{*}}(\mathbb{P}_{\operatorname{crit}(F)})/{G}_{\sigma}\;\;\text{projects
onto ${\mathbb{K}}_{\sigma}$;}$

  3. (3)

The $\sigma$th-stage of the iteration is defined as

$(\dot{\mathbb{Q}}_{\sigma})_{G_{\sigma}}={\mathbb{K}}_{\sigma}\ast\dot{\mathbb{T}}^{\mathrm{NS}}_{\sigma}\ast\dot{\mathbb{C}}(\sigma^{+2+i}_{V},\lambda_{\sigma})$

where

$\textstyle\dot{\mathbb{T}}^{\mathrm{NS}}_{\sigma}=\prod_{F\in\mathscr{E}_{\sigma,\mathcal{S}}}j_{F^{*}}(\mathbb{P}_{\operatorname{crit}(F)})/({G}_{\sigma}\ast\dot{K}_{\sigma}).$

  4. (4)

$(\dot{\mathbb{Q}}_{\sigma})_{G_{\sigma}}$ is $\sigma$-closed and
$\text{$(\dot{\mathbb{Q}}_{\sigma})_{G_{\sigma}}/{K}_{\sigma}$ is
$\sigma^{+2+i}_{V}$-closed in $V[G_{\sigma}\ast K_{\sigma}]$}.$

Case $(\beth)$: If $\mathcal{S}\cap\sigma$ is stationary (in $V$) then:

$\blacktriangleright$ If $\mathscr{E}_{\sigma,\mathcal{S}}=\emptyset$ then we
set
$(\dot{\mathbb{Q}}_{\sigma})_{G_{\sigma}}:={\mathbb{C}}(\sigma^{+2+i}_{V},\lambda_{\sigma}).$

$\blacktriangleright$ If $\mathscr{E}_{\sigma,\mathcal{S}}$ is non-empty then
we demand:

  1. (1)

For each $F\in\mathscr{E}_{\sigma,\mathcal{S}}$,
$j_{F^{*}}(\mathbb{P}_{\operatorname{crit}(F)})_{\sigma}=\mathbb{P}_{\sigma}.$

  2. (2)

The $\sigma$th-stage of the iteration is defined as

$(\dot{\mathbb{Q}}_{\sigma})_{G_{\sigma}}={\mathbb{T}}^{\mathrm{NS}^{+}}_{\sigma}\ast\dot{\mathbb{C}}(\sigma^{+2+i}_{V},\lambda_{\sigma})$

where

$\textstyle\dot{\mathbb{T}}^{\mathrm{NS}^{+}}_{\sigma}=\prod_{F\in\mathscr{E}_{\sigma,\mathcal{S}}}j_{F^{*}}(\mathbb{P}_{\operatorname{crit}(F)})/{G}_{\sigma}.$

  3. (3)

$\text{$(\dot{\mathbb{Q}}_{\sigma})_{G_{\sigma}}$ is
$\sigma^{+2+i}_{V}$-closed}.$ ∎

Let $\mathbb{P}_{\varepsilon}$ be the direct limit of
$\langle\mathbb{P}_{\sigma};\dot{\mathbb{Q}}_{\sigma}\mathrel{|}\allowbreak\sigma<\varepsilon\rangle$
and $G_{\varepsilon}\subseteq\mathbb{P}_{\varepsilon}$ a $V$-generic filter.
Let us define $(\dot{\mathbb{Q}}_{\varepsilon})_{G_{\varepsilon}}$ complying
with the induction hypothesis. As before, if
$V[G_{\varepsilon}]\models``\varepsilon$ is non-inaccessible” then we continue
and declare $(\dot{\mathbb{Q}}_{\varepsilon})_{G_{\varepsilon}}$ to be
trivial. Otherwise $V[G_{\varepsilon}]\models``\varepsilon$ is inaccessible”
and we distinguish among two cases according to whether
$\mathcal{S}\cap\varepsilon$ is $(V$-) stationary or not.

Case $(\aleph)$: Suppose that $\mathcal{S}\cap\varepsilon$ is non-stationary
in $V$.

$\blacktriangleright$ If $\mathscr{E}_{\varepsilon,\mathcal{S}}=\emptyset$
then complying with the Induction Hypothesis we let

($\aleph$1)
$(\dot{\mathbb{Q}}_{\varepsilon})_{G_{\varepsilon}}:={\mathbb{K}}_{\varepsilon}\ast\dot{\mathbb{C}}(\varepsilon^{+2+i}_{V},\lambda_{\varepsilon})\;\text{
provided $\varepsilon\in^{i}\mathcal{S}$.}$

$\blacktriangleright$ Otherwise,
$\mathscr{E}_{\varepsilon,\mathcal{S}}\neq\emptyset$ and we proceed as
follows. For each extender $E$ in $\mathscr{E}_{\varepsilon,\mathcal{S}}$ let
$E^{*}$ be the $\prec$-least completion of $E$ to an enhanced superstrong
extender given by Lemma 4.4. Let $j_{E^{*}}\colon V\rightarrow M_{E^{*}}$ be
the corresponding ultrapower embedding. Say $\delta$ is the critical point of
$j_{E^{*}}$. This embedding witnesses

$j_{E^{*}}(\delta)=\lambda_{\varepsilon}$, $V_{\lambda_{\varepsilon}}\subseteq
M_{E^{*}}$, $(M_{E^{*}})^{\delta}\subseteq M_{E^{*}}$ and
$j_{E^{*}}(\mathcal{S})\cap(\varepsilon+1)=\mathcal{S}\cap(\varepsilon+1)$.

######  Lemma 4.5.

$j_{E^{*}}(\mathbb{P}_{\delta})_{\varepsilon}=\mathbb{P}_{\varepsilon}$. Thus,
(1) of the Induction Hypothesis.

###### Proof.

We argue by induction on $\sigma<\varepsilon$. Suppose that
$\mathbb{P}_{\sigma}=j_{E^{*}}(\mathbb{P}_{\delta})_{\sigma}$ and respectively
denote by $\mathbb{Q}_{\sigma}$ and $\mathbb{R}_{\sigma}$ the $\sigma$th-
stages of these iterations. If
$V[G_{\varepsilon}\mathbin{\upharpoonright}\sigma]\models``\sigma$ is non-
inaccessible” then we are done as
$\mathbb{Q}_{\sigma}=\\{\mathop{1\hskip-3.0pt{\rm l}}\\}=\mathbb{R}_{\sigma}.$

Let us assume that $V[G_{\sigma}]\models``\sigma$ is inaccessible”. Suppose
that $\mathcal{S}\cap\sigma$ is non-stationary (the case where
$\mathcal{S}\cap\sigma$ is stationary is treated analogously). Then,
$j_{E^{*}}(S)\cap\sigma$ is also non-stationary in $M_{E^{*}}$ because
$V_{\lambda_{\varepsilon}}\subseteq M_{E^{*}}$ and

$j_{E^{*}}(\mathcal{S})\cap\sigma=\mathcal{S}\cap\sigma.$

Thus, both iterations define $\mathbb{Q}_{\sigma}$ and $\mathbb{R}_{\sigma}$
according to Case $(\aleph)$.

######  Claim 4.5.1.

$\lambda_{\sigma}=(\lambda_{\sigma})^{M_{E^{*}}}$ and
$\lambda_{\sigma}<\lambda_{\varepsilon}$

###### Proof of claim.

Suppose that $\lambda_{\sigma}>\varepsilon$. Since
$V[G_{\varepsilon}\mathbin{\upharpoonright}\sigma]\models``\sigma$ is
inaccessible” it follows that $\lambda_{\sigma}$ is the first inaccessible
past $\sigma$ in $V[G_{\varepsilon}\mathbin{\upharpoonright}(\sigma+1)]$.
However, we were assuming that $V[G_{\varepsilon}]\models``\varepsilon$ is
inaccessible”, which yields a contradiction.

Therefore, $\lambda_{\sigma}\leq\varepsilon<\lambda_{\varepsilon}$. Since
$j_{E^{*}}(\mathcal{S})\cap(\varepsilon+1)=\mathcal{S}\cap(\varepsilon+1)$ we
have: on the one hand, $\lambda_{\sigma}\in
j_{E^{*}}(\mathcal{S})\cap(\varepsilon+1)$, hence
$(\lambda_{\sigma})^{M_{E^{*}}}\leq\lambda_{\sigma}$; on the other hand,
$(\lambda_{\sigma})^{M_{E^{*}}}\in\mathcal{S}\cap(\varepsilon+1)$, hence
$\lambda_{\sigma}\leq(\lambda_{\sigma})^{M_{E^{*}}}$ ∎

######  Claim 4.5.2.

$\mathscr{E}_{\sigma,\mathcal{S}}=(\mathscr{E}_{\sigma,j_{E^{*}}(\mathcal{S})})^{M_{E^{*}}}$.

###### Proof of claim.

Let $F$ be an extender with length and strength $\sigma$. Then,

$F\in\mathscr{E}_{\sigma,\mathcal{S}}\,\Leftrightarrow\,V_{\kappa}\models\Phi(\sigma,\mathcal{S}\cap\sigma,F,\lambda_{\sigma})\,\Leftrightarrow\,V_{\lambda_{\varepsilon}}\models\Phi(\sigma,\mathcal{S}\cap\sigma,F,\lambda_{\sigma}).$

Since $V_{\lambda_{\varepsilon}}\subseteq M$,
$j_{E^{*}}(\mathcal{S})\cap(\sigma+1)=\mathcal{S}\cap(\sigma+1)$ and
$\lambda_{\sigma}=(\lambda_{\sigma})^{M_{E^{*}}}$ the above is equivalent to
saying $F\in(\mathscr{E}_{\sigma,j_{E^{*}}(\mathcal{S})})^{M_{E^{*}}}$, which
concludes the proof. ∎

Suppose first $\mathscr{E}_{\sigma,\mathcal{S}}=\emptyset$. In that case the
above claims yield

$\mathbb{Q}_{\sigma}=\mathbb{K}_{\sigma}\ast\dot{\mathbb{C}}(\sigma^{+2+i}_{V},\lambda_{\sigma})=\mathbb{R}_{\sigma}.$

Suppose that $\mathscr{E}_{\sigma,\mathcal{S}}$ is non-empty. We have to check
that
$\mathbb{T}^{\mathrm{NS}}_{\sigma}=(\mathbb{T}^{\mathrm{NS}}_{\sigma})^{M_{E^{*}}}$
which in turn is equivalent to checking that $F^{*}$ (the $\prec$-least
extender witnessing $F\in\mathscr{E}_{\sigma,\mathcal{S}}$) is the same as
$(F^{*})^{M_{E^{*}}}$ (the $\prec$-$M_{E^{*}}$-least extender witnessing
$F\in(\mathscr{E}_{\sigma,j_{E^{*}}(\mathcal{S})})^{M_{E^{*}}}$). This follows
from the fact that $F^{*},(F^{*})^{M_{E^{*}}}\in
V_{\lambda_{\varepsilon}}\subseteq M_{E^{*}}$ and the absoluteness of the
well-ordering $\prec$ (see ($\dagger$ ‣ 4.1) in page $\dagger$ ‣ 4.1). ∎

######  Lemma 4.6.

$j_{E^{*}}(\mathbb{P}_{\delta})/G_{\varepsilon}$ projects onto
$\mathbb{K}_{\varepsilon}$. Thus (2) of the Induction Hypothesis.

###### Proof.

Note that our Induction Hypothesis holds up to the critical point of $E^{*}$
because $\delta<\varepsilon$. By elementarity of $j_{E^{*}}$ the same
induction hypothesis holds in $M_{E^{*}}$ at least up to
$j_{E^{*}}(\delta)=\lambda_{\varepsilon}$, which is bigger than $\varepsilon$.

Working in $M_{E^{*}}$ we have that $j_{E^{*}}(\mathcal{S})\cap\varepsilon$ is
non-stationary. This is because $V_{\lambda_{\varepsilon}}\subseteq
M_{E^{*}}$, $j_{E^{*}}(\mathcal{S})\cap\varepsilon=\mathcal{S}\cap\varepsilon$
and the latter is non-stationary in $V$. Therefore we enter the casusitic
given by Case $(\aleph)$. Invoking Clause (3) we get, denoting
$(\dot{\mathbb{R}}_{\varepsilon})_{G_{\varepsilon}}$ the $\varepsilon$th-stage
of the iteration $j_{E^{*}}(\mathbb{P}_{\delta})$,

$M_{E^{*}}[G_{\varepsilon}]\models(\dot{\mathbb{R}}_{\varepsilon})_{G_{\varepsilon}}=\mathbb{K}_{\varepsilon}\ast\dot{\mathbb{T}}^{\mathrm{NS}}_{\varepsilon}\ast\dot{\mathbb{C}}(\varepsilon^{+2+i}_{V},\lambda_{\varepsilon}).$

Thus, $j_{E^{*}}(\mathbb{P}_{\delta})/G_{\varepsilon}$ projects to
$\mathbb{K}_{\varepsilon}$. ∎

Let us now define the $\varepsilon$th-stage of the main iteration
$\mathbb{P}_{\kappa}$.

######  Definition 4.7 (Tail forcing in Case $(\aleph)$).

Let $\dot{\mathbb{T}}^{\mathrm{NS}}_{\varepsilon}$ be a
$\mathbb{K}_{\varepsilon}$-name for the full-support product of the tail
forcings
$\prod_{E\in\mathscr{E}_{\varepsilon,\mathcal{S}}}j_{E^{*}}(\mathbb{P}_{\operatorname{crit}(E^{*})})/({G}_{\varepsilon}\ast\dot{K}_{\varepsilon}).$

Working in $V[G_{\varepsilon}]$ define:

($\aleph$2)
$(\dot{\mathbb{Q}}_{\varepsilon})_{G_{\varepsilon}}:=\mathbb{K}_{\varepsilon}\ast\dot{\mathbb{T}}^{\mathrm{NS}}_{\varepsilon}\ast\dot{\mathbb{C}}(\varepsilon^{+2+i}_{V},\lambda_{\varepsilon})\;\text{
provided $\varepsilon\in^{i}\mathcal{S}$.}$

Let us check that this definition of
$(\dot{\mathbb{Q}}_{\varepsilon})_{G_{\varepsilon}}$ complies with the
remaining clauses of the Induction Hypothesis. Clearly, by the way we posed
$(\dot{\mathbb{Q}}_{\varepsilon})_{G_{\varepsilon}}$ Clause (3) holds. Let us
dispose with Clause (4).

######  Lemma 4.8.

$(\dot{\mathbb{Q}}_{\varepsilon})_{G_{\varepsilon}}$ is $\varepsilon$-closed
and $(\dot{\mathbb{Q}}_{\varepsilon})_{G_{\varepsilon}}/K_{\varepsilon}$ is
$\varepsilon^{+2+i}_{V}$-closed in the model $V[G_{\varepsilon}\ast
K_{\varepsilon}].$ Thus Clause (4) of the Induction Hypothesis holds.

###### Proof.

Fix $E\in\mathscr{E}_{\varepsilon,\mathcal{S}}$. Since
$V_{\lambda_{\varepsilon}}\subseteq M_{E^{*}}$,
$j_{E^{*}}(\mathcal{S})\cap\varepsilon$ is non-stationary in $M_{E^{*}}$ as
well. As a result, we can invoke Clause (4) of our Induction Hypothesis inside
$M_{E^{*}}$ for the $\varepsilon$th-stage of the iteration
$j_{E^{*}}(\mathbb{P}_{\delta})$. Thus we infer that

$M_{E^{*}}[G_{\varepsilon}]\models\text{$``(\dot{\mathbb{R}}_{\varepsilon})_{G_{\varepsilon}}$
is $\varepsilon$-closed and projects to $\mathbb{K}_{\varepsilon}$''}$

and since $\varepsilon\in^{i}\mathcal{S}$,

$M_{E^{*}}[G_{\varepsilon}]\models\text{$``(\dot{\mathbb{R}}_{\varepsilon})_{G_{\varepsilon}}/\dot{K}_{\varepsilon}$
is $\varepsilon^{+2+i}_{V}$-closed''}.$

In fact, applying the Induction Hypothesis at all the stages
$\sigma\in[\varepsilon,\lambda_{\varepsilon})$

($\star$) $M_{E^{*}}[G_{\varepsilon}\ast
K_{\varepsilon}]\models\text{$``j_{E^{*}}(\mathbb{P}_{\delta})/({G}_{\varepsilon}\ast{K}_{\varepsilon})$
is $\varepsilon^{+2+i}_{V}$-closed''}.$

We cannot outright infer that $(\star)$ is absolute between $M_{E^{*}}$ and
$V$ – after all, $M_{E^{*}}$ is closed only under $\delta$-sequences in $V$.

######  Claim 4.8.1.

$V[G_{\varepsilon}\ast
K_{\varepsilon}]\models\text{$``j_{E^{*}}(\mathbb{P}_{\delta})/({G}_{\varepsilon}\ast{K}_{\varepsilon})$
is $\varepsilon^{+2+i}_{V}$-closed''}.$

###### Proof of claim.

Let $\langle p_{\alpha}\mathrel{|}\allowbreak\alpha<\beta\rangle\in
V[G_{\varepsilon}\ast K_{\varepsilon}]$ be a decreasing sequence of conditions
in the poset with $\beta<\varepsilon^{+2+i}_{V}$. Let us note a few things:

  1. ($\alpha$)

$V[G_{\varepsilon}\ast
K_{\varepsilon}]_{\lambda_{\varepsilon}}=V_{\lambda_{\varepsilon}}[G_{\varepsilon}\ast
K_{\varepsilon}]\subseteq M_{E^{*}}[G_{\varepsilon}\ast K_{\varepsilon}]$;

  2. ($\beta$)

$j_{E^{*}}(\mathbb{P}_{\delta})/(G_{\varepsilon}\ast K_{\varepsilon})\subseteq
V[G_{\varepsilon}\ast K_{\varepsilon}]_{\lambda_{\varepsilon}}$;

  3. ($\gamma$)

$V[G_{\varepsilon}\ast K_{\varepsilon}]_{\lambda_{\varepsilon}}$ is closed
under $\lambda_{\varepsilon}$-sequences in $V[G_{\varepsilon}\ast
K_{\varepsilon}]$.

$(\alpha)$ is true in that
$\mathbb{P}_{\varepsilon}\ast\mathbb{K}_{\varepsilon}$ is small compared to
$\lambda_{\varepsilon}$ and $V_{\lambda_{\varepsilon}}\subseteq M_{E^{*}}$;
$(\beta)$ is evident; $(\gamma)$ is an immediate consequence of the
inaccessibility of $\lambda_{\varepsilon}$ in $V[G_{\varepsilon}\ast
K_{\varepsilon}]$. Combining these items we have that $\langle
p_{\alpha}\mathrel{|}\allowbreak\alpha<\beta\rangle\in
M_{E^{*}}[G_{\varepsilon}\ast K_{\varepsilon}]$ so by equation ($\star$ ‣ 4.1)
above there is $p_{\beta}\in M_{E^{*}}[H]$ a lower bound for it. ∎

Since the extender $E\in\mathscr{E}_{\varepsilon,\mathcal{S}}$ was arbitrary
we conclude that

$\textstyle\mathbb{T}^{\mathrm{NS}}_{\varepsilon}:=\prod_{E\in\mathscr{E}_{\varepsilon,\mathcal{S}}}j_{E^{*}}(\mathbb{P}_{\delta})/(G_{\varepsilon}\ast
K_{\varepsilon})$

is $\varepsilon^{+2+i}_{V}$-closed in $V[G_{\varepsilon}\ast
K_{\varepsilon}]$. This combined with the definition of
$(\dot{\mathbb{Q}}_{\varepsilon})_{G_{\varepsilon}}$ given in equation
($\aleph$2) above implies that
$(\dot{\mathbb{Q}}_{\varepsilon})_{G_{\varepsilon}}$ is $\varepsilon$-closed
and $(\dot{\mathbb{Q}}_{\varepsilon})_{G_{\varepsilon}}/K_{\varepsilon}$ is
$\varepsilon^{+2+i}_{V}$-closed. Thus we are done with the lemma. ∎

The above discussion completes the inductive definition of the iteration
$\mathbb{P}_{\kappa}$ should we fell in Case $(\aleph)$. Let us next discuss
the other possible case.

Case $(\beth)$: Suppose that $\mathcal{S}\cap\varepsilon$ is stationary in
$V$.

$\blacktriangleright$ If $\mathscr{E}_{\varepsilon,\mathcal{S}}=\emptyset$
then complying with our Induction Hypothesis we define

($\beth$1)
$(\dot{\mathbb{Q}}_{\varepsilon})_{G_{\varepsilon}}:={\mathbb{C}}(\varepsilon^{+2+i}_{V},\lambda_{\varepsilon})\;\text{
provided $\varepsilon\in^{i}\mathcal{S}$.}$

$\blacktriangleright$ Otherwise,
$\mathscr{E}_{\varepsilon,\mathcal{S}}\neq\emptyset$ and we proceed as
follows. As before, for each $E\in\mathscr{E}_{\varepsilon,\mathcal{S}}$ we
let $E^{*}$ be the $\prec$-least extension of $E$ provided by Lemma 4.4. Since
$\mathbb{P}_{\varepsilon}$ is definable using $\mathcal{S}\cap\varepsilon$ and
the well-ordering $\prec$, $j_{E^{*}}$ shifts $\mathcal{S}\cap\varepsilon$
correctly and $\prec$ is sufficiently absolute we get
$j_{E^{*}}(\mathbb{P}_{\operatorname{crit}(E^{*})})_{\varepsilon}=\mathbb{P}_{\varepsilon}$.

######  Definition 4.9 (Tail forcing in Case $(\beth)$).

Denote by ${\mathbb{T}}^{\mathrm{NS}^{+}}_{\varepsilon}$ the full-support
product of the tail forcings
$\textstyle\prod_{E\in\mathscr{E}_{\varepsilon,\mathcal{S}}}j_{E^{*}}(\mathbb{P}_{\operatorname{crit}(E^{*})})/{G}_{\varepsilon}.$

Complying with our Induction Hypothesis we define

($\beth$2)
$(\dot{\mathbb{Q}}_{\varepsilon})_{G_{\varepsilon}}:={\mathbb{T}}^{\mathrm{NS}^{+}}_{\varepsilon}\ast\dot{\mathbb{C}}(\varepsilon^{+2+i}_{V},\lambda_{\varepsilon})\;\text{
provided $\varepsilon\in^{i}\mathcal{S}$.}$

Arguing exactly as in the previous lemmas one checks that this definition
complies with the requirements of the Induction Hypothesis.

For the reader’s benefit we summarize the definition of
$(\dot{\mathbb{Q}}_{\varepsilon})_{G_{\varepsilon}}$:

######  Definition 4.10 (Main iteration).

Let $i\in\\{0,1\\}$ be such that $\varepsilon\in^{i}\mathcal{S}$.

Case $(\aleph)$: Suppose that $\mathcal{S}\cap\varepsilon$ is non-stationary
in $V$. Then,

$({\mathbb{Q}}_{\varepsilon})_{G_{\varepsilon}}:=\begin{cases}{\mathbb{K}}_{\varepsilon}\ast\dot{\mathbb{C}}(\varepsilon^{+2+i}_{V},\lambda_{\varepsilon}),&\text{if
$\mathscr{E}_{\varepsilon,\mathcal{S}}=\emptyset$;}\\\
{\mathbb{K}}_{\varepsilon}\ast\dot{\mathbb{T}}^{\mathrm{NS}}_{\varepsilon}\ast\dot{\mathbb{C}}(\varepsilon^{+2+i}_{V},\lambda_{\varepsilon}),&\text{if
$\mathscr{E}_{\varepsilon,\mathcal{S}}\neq\emptyset$;}\end{cases}$

Case $(\beth)$: Suppose that $\mathcal{S}\cap\varepsilon$ is stationary in
$V$. Then,

$({\mathbb{Q}}_{\varepsilon})_{G_{\varepsilon}}:=\begin{cases}\mathbb{C}(\varepsilon^{+2+i}_{V},\lambda_{\varepsilon}),&\text{if
$\mathscr{E}_{\varepsilon,\mathcal{S}}=\emptyset$;}\\\
{\mathbb{T}}^{\mathrm{NS}^{+}}_{\varepsilon}\ast\dot{\mathbb{C}}(\varepsilon^{+2+i}_{V},\lambda_{\varepsilon}),&\text{if
$\mathscr{E}_{\varepsilon,\mathcal{S}}\neq\emptyset$;}\end{cases}$

This completes the inductive construction and yields $\mathbb{P}_{\kappa}$.

A few explanations about the design of the iteration $\mathbb{P}_{\kappa}$ are
in order:

######  Remark 4.11.

Case ($\aleph$) has been posed so as to ensure that $\mathbb{Q}_{\varepsilon}$
kills the measurability of $\varepsilon$ whenever $\mathcal{S}\cap\varepsilon$
is non-stationary. Therefore if $\delta$ happens to be measurable in a generic
extension by $\mathrm{Add}(\omega,1)\ast{\mathbb{P}_{\kappa}}$ then it must be
that $\mathcal{S}\cap\delta$ is stationary in $V$. Also, note that if
$\mathcal{S}\cap\varepsilon$ is stationary in $V$ then
$\mathbb{P}_{\varepsilon}$ will be $\varepsilon$-cc (see e.g. [Cum10,
Proposition 7.13]) and as a result $\mathbb{P}_{\kappa}$ itself will preserve
the stationarity of $\mathcal{S}\cap\varepsilon$ (because
$\mathbb{P}_{\kappa}/G_{\varepsilon}$ is $\varepsilon$-closed).

######  Remark 4.12.

The definition of the posets $\mathbb{T}^{\mathrm{NS}}_{\varepsilon}$ and
$\mathbb{T}^{\mathrm{NS^{+}}}_{\varepsilon}$ have been posed to anticipate the
generics for the tail forcings
$j_{E^{*}}(\mathbb{P}_{\operatorname{crit}(E^{*})})/G_{\varepsilon}$. Here
$j_{E^{*}}$ is an extender embedding that we would eventually like to lift.
This is akin to the usual bookkeeping procedure involved in classical proofs;
such as the consistency of PFA. We invite our readers to jump to Theorem 4.15
where we show how to use this to get $\mathcal{A}$ in the generic extension by
$\mathrm{Add}(\omega,1)\ast\mathbb{P}_{\kappa}.$ The next pages will be
instead devoted to prove the main properties of $\mathbb{P}_{\kappa}$.

The next lemma shows that we can indeed force with the coding poset
$\mathbb{C}(\varepsilon^{+2+i}_{V},\lambda_{\varepsilon})$ after both
$\mathbb{T}^{\mathrm{NS}}_{\varepsilon}$ and
$\mathbb{T}^{\mathrm{NS}^{+}}_{\varepsilon}$. We just give details when the
iteration chooses the former forcing – for the latter the argument is
analogous.

######  Lemma 4.13.

$\mathop{1\hskip-3.0pt{\rm
l}}\Vdash_{\mathbb{P}_{\varepsilon}\ast\dot{\mathbb{K}}_{\varepsilon}}``\dot{\mathbb{T}}^{\mathrm{NS}}_{\varepsilon}$
preserves the Mahloness of $\lambda_{\varepsilon}$”.

###### Proof.

For the scope of this argument our ground model (denoted $W$) will be $V[G]$
being $G\subseteq\mathbb{P}_{\varepsilon}\ast\mathbb{K}_{\varepsilon}$ a
$V$-generic filter. Recall that members of $\mathcal{S}$ were measurable in
$V$. So let us fix $U_{\varepsilon}$ a normal measure (in $W$) on
$\lambda_{\varepsilon}$ and set

$A:=\\{\theta<\lambda_{\varepsilon}\mathrel{|}\allowbreak\text{$\theta$ is
weakly compact and $\theta>\varepsilon$}\\}\in U_{\varepsilon}.$

We may assume that for each $E\in\mathscr{E}_{\varepsilon,S}$ the set
$\Sigma_{E^{*}}:=j_{E^{*}}(\mathcal{S})\cap\lambda_{\varepsilon}$ is unbounded
in $\lambda_{\varepsilon}$ for otherwise the posets
$\mathbb{T}^{\mathrm{NS}}_{\varepsilon,E^{*}}:=j_{E^{*}}(\mathbb{P}_{\operatorname{crit}(E)})/G$
would be small relative to $\lambda_{\varepsilon}$ and therefore they will
preserve the Mahloness of $\lambda_{\varepsilon}$.

Said this let us denote

$C:=\bigcap_{E\in\mathscr{E}_{\varepsilon,\mathcal{S}}}\operatorname{acc}(\Sigma_{E^{*}})$,

where $\operatorname{acc}(\Sigma_{E^{*}})$ are the accumulation points of
$\Sigma_{E^{*}}$. Note that $C$ is a club in $\lambda_{\varepsilon}$ in that
$|\mathscr{E}_{\varepsilon,\mathcal{S}}|<\lambda_{\varepsilon}$. Let us show
that $\lambda_{\varepsilon}$ remains Mahlo in
$W[{\mathbb{T}}_{\varepsilon}^{\mathrm{NS}}]$:

$\blacktriangleright$ First, since $\lambda_{\varepsilon}$ is weakly compact
in $W$, the poset ${\mathbb{T}}^{\mathrm{NS}}_{\varepsilon}$ is
$\lambda_{\varepsilon}$-cc: Each factor is $\lambda_{\varepsilon}$-cc and the
support of the product (i.e., $|\mathscr{E}_{\varepsilon,\mathcal{S}}|$) is
smaller than $\lambda_{\varepsilon}$. Thus, by weak compactness of
$\lambda_{\varepsilon}$, $\mathbb{T}_{\varepsilon}^{\mathrm{NS}}$ is
$\lambda_{\varepsilon}$-cc (see [CL17, Proposition 1.1]). Thus,
$\lambda_{\varepsilon}$ remains regular in any extension by
$\mathbb{T}^{\mathrm{NS}}_{\varepsilon}:=(\dot{\mathbb{T}}^{\mathrm{NS}}_{\varepsilon})_{G}$.

$\blacktriangleright$ Let $H\subseteq\mathbb{T}^{\mathrm{NS}}_{\varepsilon}$
be generic and let us argue that, in $W[H]$, $A\cap C$ is a stationary set of
regular cardinals below $\lambda_{\varepsilon}$. Clearly, $A\cap C$ is
stationary in $W[H]$ because it belongs to the normal measure
$U_{\varepsilon}$ and ${\mathbb{T}}^{\mathrm{NS}}_{\varepsilon}$ is
$\lambda_{\varepsilon}$-cc.

Let $\theta\in A\cap C$. Since $\theta$ is a weakly compact cardinal and an
accumulation point of each $\Sigma_{E^{*}}$ it follows that both
$(\mathbb{T}^{\mathrm{NS}}_{E^{*},\varepsilon}\mathbin{\upharpoonright}\theta)$
and
$\prod_{E}(\mathbb{T}^{\mathrm{NS}}_{E^{*},\varepsilon}\mathbin{\upharpoonright}\theta)$
are $\theta$-cc.

Let us denote the lower and upper parts of
$\mathbb{T}^{\mathrm{NS}}_{\varepsilon}$ as follows:

  * •

$\mathbb{L}^{E^{*}}_{\theta}:=(\mathbb{T}^{\mathrm{NS}}_{E^{*},\varepsilon}\mathbin{\upharpoonright}\theta)$
and
$\mathbb{L}_{\theta}:=\prod_{E\in\mathscr{E}_{\varepsilon,\mathcal{S}}}\mathbb{L}^{E^{*}}_{\theta}$;

  * •

$\dot{\mathbb{U}}^{E^{*}}_{\theta}:=(\dot{\mathbb{T}}^{\mathrm{NS}}_{E^{*},\varepsilon}\setminus\theta)$
and
$\dot{\mathbb{U}}_{\theta}:=\prod_{E\in\mathscr{E}_{\varepsilon,\mathcal{S}}}\dot{\mathbb{U}}^{E^{*}}_{\theta}.$

Following Foreman [For09], let $\mathbb{A}^{E^{*}}_{\theta}$ denote the term-
space forcing
$\mathbb{A}(\mathbb{L}^{E^{*}}_{\theta},\dot{\mathbb{U}}^{E^{*}}_{\theta})$.
Namely, conditions in $\mathbb{A}^{E^{*}}_{\theta}$ are
$\mathbb{L}^{E^{*}}_{\theta}$-names $\sigma$ ordered as follows

$\sigma\leq_{\mathbb{A}^{E^{*}}_{\theta}}\tau\;\Longleftrightarrow\;\mathop{1\hskip-3.0pt{\rm
l}}\Vdash_{\mathbb{L}^{E^{*}}_{\theta}}\sigma\leq_{\dot{\mathbb{U}}^{E^{*}}_{\theta}}\tau.$

Note that $\mathbb{A}^{E^{*}}_{\theta}$ is a $\theta$-closed partial order in
$W$ – this being an immediate consequence of the _Forcing Maximality
Principle_. Thus, the full-support product
$\mathbb{A}_{\theta}:=\prod_{E^{*}}\mathbb{A}^{E^{*}}_{\theta}$ is
$\theta$-closed in $W$. Since $\mathbb{L}_{\theta}$ is $\theta$-cc and
$\mathbb{A}_{\theta}$ is $\theta$-closed, Easton’s lemma (see [Cum10, Fact
5.10]) yields $\mathop{1\hskip-3.0pt{\rm l}}\Vdash_{\mathbb{L}_{\theta}}$
$``\mathbb{A}_{\theta}$ is $\theta$-distributive”. In particular,
$\mathbb{L}_{\theta}\times\mathbb{A}_{\theta}$ preserves the regularity of
$\theta$.

To complete the proof note the following:

  1. $(\alpha)$

$\mathbb{T}^{\mathrm{NS}}_{\varepsilon}$ projects onto $\mathbb{L}_{\theta}$;

  2. $(\beta)$

$\mathop{1\hskip-3.0pt{\rm
l}}\Vdash_{\mathbb{L}_{\theta}}\mathbb{T}^{\mathrm{NS}}_{\varepsilon}/\dot{G}_{\mathbb{L}_{\theta}}\simeq\dot{\mathbb{U}}_{\theta}$;

  3. $(\gamma)$

$\mathop{1\hskip-3.0pt{\rm l}}\Vdash_{\mathbb{L}_{\theta}}\mathbb{A}_{\theta}$
projects to $\dot{\mathbb{U}}_{\theta}.$

$(\alpha)$ is witnessed by $\langle
p_{\alpha}\mathrel{|}\allowbreak\alpha<|\mathscr{E}_{\varepsilon,\mathcal{S}}|\rangle\mapsto\langle
p_{\alpha}\mathbin{\upharpoonright}\theta\mathrel{|}\allowbreak\alpha<|\mathscr{E}_{\varepsilon,\mathcal{S}}|\rangle$;
$(\beta)$ is evident; and $(\gamma)$ is a consequence of the following general
fact when applied with respect to
$\langle\mathbb{P}_{E^{*}}\ast\dot{\mathbb{Q}}_{E^{*}}\mathrel{|}\allowbreak
E\in\mathscr{E}_{\varepsilon,S}\rangle$ where
$\mathbb{P}_{E^{*}}:=\mathbb{L}^{E^{*}}_{\theta}$ and
$\dot{\mathbb{Q}}_{E^{*}}:=\dot{\mathbb{U}}^{E^{*}}_{\theta}$.

######  Claim 4.13.1.

For a family of posets
$\langle\mathbb{P}_{\gamma}\ast\dot{\mathbb{Q}}_{\gamma}\mathrel{|}\allowbreak\gamma<\Gamma\rangle$
the trivial condition of $\prod_{\gamma<\Gamma}\mathbb{P}_{\gamma}$ forces
that
$\prod_{\gamma<\Gamma}\mathbb{A}(\mathbb{P}_{\gamma},\dot{\mathbb{Q}}_{\gamma})$
projects to $\prod_{\gamma<\Gamma}\dot{\mathbb{Q}}_{\gamma}$.

###### Proof of claim.

Let $K\subseteq\prod_{\gamma<\Gamma}\mathbb{P}_{\gamma}$ be $V$-generic and
for each $\gamma<\Gamma$ denote by $K_{\gamma}$ the induced
$\mathbb{P}_{\gamma}$-generic. Working in $V[K]$, let us consider the map

$\pi\colon\langle\sigma_{\gamma}\mathrel{|}\allowbreak\gamma<\Gamma\rangle\mapsto\langle(\sigma_{\gamma})_{K_{\gamma}}\mathrel{|}\allowbreak\gamma<\Gamma\rangle.$

Clearly, this is well-defined. It is also order-preserving as, for each
$\gamma<\Gamma$, if $\mathop{1\hskip-3.0pt{\rm
l}}\Vdash_{\mathbb{P}_{\gamma}}\sigma_{\gamma}\leq_{\dot{\mathbb{Q}}_{\gamma}}\rho_{\gamma}$
then $(\sigma_{\gamma})_{K_{\gamma}}\leq(\rho_{\gamma})_{K_{\gamma}}.$
Finally, let us show that $\pi$ is a projection. Suppose that $\langle
q_{\gamma}\mathrel{|}\allowbreak\gamma<\Gamma\rangle\leq\langle(\sigma_{\gamma})_{K_{\gamma}}\mathrel{|}\allowbreak\gamma<\Gamma\rangle$.
Since this is a true statement in $V[K]$ there is $p\in K$ forcing
$``\langle\dot{q}_{\gamma}\mathrel{|}\allowbreak\gamma<\Gamma\rangle\leq\langle\sigma_{\gamma}\mathrel{|}\allowbreak\gamma<\Gamma\rangle$”.
For each $\gamma<\Gamma$ let us define an auxiliary $\mathbb{P}_{\gamma}$-name
$\tau_{\gamma}$ as follows:

$\tau_{\gamma}:=\\{\langle\sigma,r\rangle\mathrel{|}\allowbreak(r\parallel
p_{\gamma}\,\Rightarrow\,\sigma=\dot{q}_{\gamma})\,\vee\,(r\perp
p_{\gamma}\,\Rightarrow\,\sigma=\sigma_{\gamma})\\}.$

By design,
$p_{\gamma}\Vdash_{\mathbb{P}_{\gamma}}\tau_{\gamma}=\dot{q}_{\gamma}$ and
$\mathop{1\hskip-3.0pt{\rm
l}}\Vdash_{\mathbb{P}_{\gamma}}\tau_{\gamma}\leq\sigma_{\gamma}$. Thus,
$\langle\tau_{\gamma}\mathrel{|}\allowbreak\gamma<\Gamma\rangle$ is stronger
than $\langle\sigma_{\gamma}\mathrel{|}\allowbreak\gamma<\Gamma\rangle$ and
its image under $\pi$ is exactly $\langle
q_{\gamma}\mathrel{|}\allowbreak\gamma<\Gamma\rangle$. ∎

Combining $(\alpha)$–$(\gamma)$ we conclude that
$\mathbb{T}^{\mathrm{NS}}_{\varepsilon}$ preserves the regularity of $\theta$.
This completes the proof of the lemma. ∎

Let us fix $G\subseteq\mathrm{Add}(\omega,1)\ast\dot{\mathbb{P}}_{\kappa}$ a
$V$-generic filter.

######  Lemma 4.14 (Properties of the generic extension).

  1. (1)

Each member of $\mathcal{S}\cup\\{\kappa\\}$ is inaccessible in $V[G]$.
Moreover, every successor inaccessible in $V[G]$ is a successor member of
$\mathcal{S}$;

  2. (2)

$\mathcal{S}$ is definable with ordinal parameters in $V[G]$. Specifically,

$\mathcal{S}=\\{\varepsilon<\kappa\mathrel{|}\allowbreak
V[G]\models\varphi(\varepsilon,\varepsilon^{+3}_{V},\varepsilon^{+4}_{V})\\}$

where $\varphi(\varepsilon,\varepsilon^{+3}_{V},\varepsilon^{+4}_{V})$ is the
conjunction of

     * •

$``\varepsilon$ is inaccessible”,

     * •

$``\varepsilon^{+3}_{V}$ is a cardinal”,

     * •

and $``\varepsilon^{+4}_{V}$ is not a cardinal”.

  3. (3)

If $\varepsilon<\kappa$ is a $V[G]$-measurable cardinal then
$\mathcal{S}\cap\varepsilon$ is stationary in

$V$, $V[G_{\varepsilon}]$ and $V[G]$;

###### Proof.

(1) Since $\mathcal{S}\subseteq\kappa$ is stationary $\mathbb{P}_{\kappa}$ is
$\kappa$-cc. Thus, $\kappa$ is regular in $V[G]$. The next arguments will show
that $\kappa$ is strong limit in $V[G]$.

Let $\langle\theta_{\alpha}\mathrel{|}\allowbreak\alpha<\kappa\rangle$ be an
enumeration of $\mathcal{S}$. If $\varepsilon<\kappa$ is the first
$V$-inaccessible, $V[G_{\varepsilon}]\models``\varepsilon$ is inaccessible”
and $\lambda_{\varepsilon}=\theta_{0}$. Thus,
$\mathbb{P}_{\varepsilon}\ast\dot{\mathbb{Q}}_{\varepsilon}$ makes
$\theta_{0}$ be the first inaccessible past $\varepsilon$, which is preserved
by the rest of the iteration $\mathbb{P}_{\kappa}/G_{\varepsilon+1}$. Next,
suppose that $\langle\theta_{\alpha}\mathrel{|}\allowbreak\alpha<\beta\rangle$
remain inaccessible in $V[G]$. If $\beta$ is successor one argues exactly as
before. So suppose $\beta$ is limit.

$\blacktriangleright$ Suppose that
$\theta_{\beta}=\sup_{\alpha<\beta}\theta_{\alpha}$. In this case it suffices
to check that

$V[G_{\theta_{\beta}}]\models``\theta_{\beta}$ is regular”.

Towards a contradiction, suppose there is $\lambda<\theta_{\beta}$ and a
cofinal increasing map $f\colon\lambda\rightarrow\theta_{\beta}$. Letting
$\alpha<\beta$ with $\lambda<\theta_{\alpha}$ it follows that $f\in
V[G_{\theta_{\alpha}}]$. This is no possible for it is a generic extension by
a forcing of size ${<}\theta_{\beta}$ and $\theta_{\beta}$ is regular (in fact
measurable) in $V$.

$\blacktriangleright$ Suppose that $\theta_{\beta}$ is greater than
$\sigma:=\sup_{\alpha<\beta}\theta_{\alpha}$. Since $V[G_{\sigma}]$ thinks
that $\theta_{\beta}$ is measurable we can let
$\varepsilon\in(\sigma,\theta_{\beta})$ be the first
$V[G_{\sigma}]$-inaccessible. Clearly $\varepsilon$ is inaccessible in
$V[G_{\varepsilon}]$ and $\theta_{\beta}=\lambda_{\varepsilon}$ so the tail
poset $\mathbb{P}_{\kappa}/G_{\varepsilon}$ makes $\theta_{\beta}$ be the
first inaccessible past $\varepsilon$. This completes the discussion.

For the moreover part of Clause (1) suppose that $\theta$ is a successor
inaccessible in $V[G]$ and let $\varepsilon$ be its inaccessible predecessor.
If $\theta<\lambda_{\varepsilon}$ then
$(\dot{\mathbb{Q}}_{\varepsilon})_{G_{\varepsilon}}$ would collapse $\theta$.
Similarly, $\theta$ cannot be larger than $\lambda_{\varepsilon}$ because this
latter remains inaccessible in $V[G]$. Therefore,
$\theta=\lambda_{\varepsilon}\in\mathcal{S}$ and we are done.

(2) Suppose that $\varepsilon\in\mathcal{S}$. By the argument in (1),
$V[G_{\varepsilon}]$ thinks that $\varepsilon$ is inaccessible. Thus we fall
either in Case $(\aleph)$ or Case $(\beth)$ and the design of the iteration
$\mathbb{P}_{\kappa}$ ensures that $\varepsilon^{+3}_{V}$ is preserved and
$\varepsilon^{+4}_{V}$ is collapsed. Conversely, suppose that
$\varphi(\varepsilon,\varepsilon^{+3}_{V},\varepsilon^{+4}_{V})$ holds in
$V[G]$. We claim that $V[G_{\varepsilon}]$ thinks that $\varepsilon$ is
inaccessible: Otherwise, $\mathbb{Q}_{\varepsilon}$ would be the trivial
forcing and $\mathbb{P}_{\kappa}/G_{\varepsilon+1}$ would be (more than)
$\varepsilon^{+4}_{V}$-closed, hence
$\varphi(\varepsilon,\varepsilon^{+3}_{V},\varepsilon^{+4}_{V})$ would fail.
Thus, $V[G_{\varepsilon}]$ thinks that $\varepsilon$ is inaccessible and we
fall either in Case $(\aleph)$ or Case $(\beth)$. Since $\varepsilon^{+3}_{V}$
is preserved it must be the case that $\varepsilon\in\mathcal{S}$, as sought.

(3) Towards a contradiction, suppose that $\mathcal{S}\cap\varepsilon$ is non-
stationary in $V[G_{\varepsilon}]$. Either $\mathcal{S}\cap\varepsilon$ is
stationary in $V$ or it is not. In the first scenario,
$\mathbb{P}_{\varepsilon}$ is a $\varepsilon$-cc iteration and as a result
$\mathcal{S}\cap\varepsilon$ remains $V[G_{\varepsilon}]$-stationary – a
contradiction. In the second scenario, $\mathbb{Q}_{\varepsilon}$ prevents
$\varepsilon$ from being measurable in $V[G]$ – this is because
$\mathbb{Q}_{\varepsilon}$ forces with $\mathbb{K}_{\varepsilon}$ (see
Proposition 2.17). This contradicts our departing assumption and establishes
that $\mathcal{S}\cap\varepsilon$ is stationary in $V[G_{\varepsilon}]$.

Finally, note that $\mathbb{P}_{\kappa}/G_{\varepsilon}$ is an
$\varepsilon$-closed iteration (see Clause $(1)_{\varepsilon}$ in p.Induction
Hypothesis) so it preserves the stationarity of $\mathcal{S}\cap\varepsilon$
in $V[G]$. ∎

We are now in conditions to show that axiom $\mathcal{A}$ holds after forcing.

######  Theorem 4.15.

$\mathcal{A}$ holds in $V[G]_{\kappa}$.

###### Proof.

First note that $V[G]_{\kappa}=W[G]$ where $W=V_{\kappa}$. Working in $W[G]$,
let $\delta$ be a stationary-correct superstrong cardinal with inaccessible
target. Fix a witnessing extender ultrapower $j\colon W[G]\rightarrow M_{E}$
and let $\theta\in(\delta,j(\delta))$ be a successor inaccessible. We denote
by $\varepsilon$ the inaccessible predecessor of $\theta$.111111Note that
$\varepsilon$ may be $\delta$.

######  Lemma 4.16.

  1. (1)

$\theta\in\mathcal{S}$ and $\theta=\lambda_{\varepsilon}$;

  2. (2)

$\mathcal{S}\cap\delta$ is stationary in $W$;

###### Proof.

(1) By Lemma 4.14(1) $\theta\in\mathcal{S}$. It cannot be that
$\lambda_{\varepsilon}<\theta$ for then $\theta$ would not be the successor
inaccessible of $\varepsilon$. Likewise, it can either be that
$\theta<\lambda_{\varepsilon}$ as $\varepsilon$ is inaccessible in
$W[G_{\varepsilon}]$ and in that case the poset $\mathbb{Q}_{\varepsilon}$
would collapse $\theta$. Clause (2) is a consequence of Lemma 4.14(3). ∎

######  Lemma 4.17.

$\delta$ is enhanced superstrong with target
$\lambda\in\mathcal{S}\setminus(\theta+1)$ in $W[G]$.

###### Proof.

First, note that the poset
$\mathrm{Add}(\omega,1)\ast\dot{\mathbb{P}}_{\kappa}$ admits a _gap below
$\omega_{1}$_; to wit, $|\mathrm{Add}(\omega,1)|<\aleph_{1}$ and
$\mathop{1\hskip-3.0pt{\rm
l}}\Vdash_{\mathrm{Add}(\omega,1)}``\dot{\mathbb{P}}_{\kappa}$ is
$\aleph_{1}$-strategically-closed”. Thus, by _Hamkins’ Gap Forcing Theorem_
[Ham01], $j\mathbin{\upharpoonright}W\colon W\rightarrow M:=M_{E}\cap W$ is
amenable to $W$ (i.e., $j\mathbin{\upharpoonright}x\in W$ for all $x\in W$)
and it is definable within $W$.

Let us observe that $j\mathbin{\upharpoonright}W$ is a superstrong embedding
(in $W$) with target $j(\delta)$. Clearly,
$\operatorname{crit}(j\mathbin{\upharpoonright}W)=\delta$. In addition we
claim that $W_{j(\delta)}\subseteq M$: First, since $j(\delta)$ is a
$W[G]$-inaccessible limit of inaccessibles it is an inaccessible limit of
members of $\mathcal{S}$ (by Lemma 4.14(1)). Since Easton-supports are taken
at stage $j(\delta)$ this yields $\mathbb{P}_{j(\delta)}\subseteq
W_{j(\delta)}$. In particular, $W_{j(\delta)}[G_{j(\delta)}]$ makes sense. By
our assumption upon $M_{E}$, $W[G]_{j(\delta)}\subseteq M_{E}$ but since the
model $W_{j(\delta)}[G_{j(\delta)}]$ is always contained in $W[G]_{j(\delta)}$
we infer that

$W_{j(\delta)}\subseteq W_{j(\delta)}[G_{j(\delta)}]\cap W\subseteq M_{E}\cap
W=M.$

######  Claim 4.17.1.

$j(\mathcal{S}\cap\delta)=\mathcal{S}\cap j(\delta)$.

###### Proof of claim.

By Lemma 4.14(2),

$\mathcal{S}\cap\delta=\\{\varepsilon<\delta\mathrel{|}\allowbreak
W[G]\models\text{$\varphi(\varepsilon,\varepsilon^{+3}_{W},\varepsilon^{+4}_{W})$}\\}.$

By _Laver’s Ground Model Definability theorem_ (see [Lav07]), $W$ is a class
definable within $W[G]$. Thus, since $j$ moves $W$ to $M$,

$j(\mathcal{S}\cap\delta)=\\{\varepsilon<j(\delta)\mathrel{|}\allowbreak
M_{E}\models\text{$\varphi(\varepsilon,\varepsilon^{+3}_{M},\varepsilon^{+4}_{M})$}\\}.$

Since $W[G]_{j(\delta)}\subseteq M_{E}$ it follows that $\varepsilon^{+3}_{M}$
(resp. $\varepsilon^{+4}_{M}$) is a cardinal in $M_{E}$ if and only if it is
so in $W[G]$. Similarly, since $W_{j(\delta)}\subseteq M$ necessarily
$\varepsilon^{+3+i}_{M}=\varepsilon^{+3+i}_{W}$ for $i\in\\{0,1\\}$. These two
facts combined yield

$j(\mathcal{S}\cap\delta)=\\{\varepsilon<j(\delta)\mathrel{|}\allowbreak
W[G]\models\text{$\varphi(\varepsilon,\varepsilon^{+3}_{W},\varepsilon^{+4}_{W})$}\\}=\mathcal{S}\cap
j(\delta).\qed$

Note that $\mathcal{S}\cap\delta$ is $W[G]$-stationary as $\delta$ is
measurable in $W[G]$ (Lemma 4.14). Hence $j(\mathcal{S}\cap\delta)$ is
stationary in $M_{E}$ and since this latter model is stationary-correct at
$j(\delta)$, $\mathcal{S}\cap j(\delta)$ is stationary in $W[G]$, as well.
Finally, $\delta$ is superstrong with inaccessible target $j(\delta)$ so there
is a ${\geq}\delta^{+}$-club $C\subseteq j(\delta)$ of possible superstrong
targets witnessing that $\delta$ is enhanced superstrong (see Fact 2.4). In
particular, there must be a cardinal
$\lambda\in(C\cap\mathcal{S})\setminus\theta+1$ for which $\delta$ is enhanced
superstrong with target $\lambda.$ ∎

Let $\iota\colon W[G]\rightarrow M$ be a superstrong embedding with
$\operatorname{crit}(\iota)=\delta$,
$\iota(\delta)=\lambda\in\mathcal{S}\setminus(\theta+1)$ and
$M^{\delta}\subseteq M$. We may assume, without losing any generality, that
$\iota$ is the extender ultrapower by some $F^{*}\in W[G]$. Once again by
_Hamkins’ Gap Forcing theorem_ [Ham01], $F:=F^{*}\cap W$ is a supestrong
extender in $W$ with $j_{F}(\delta)=\lambda$ and $(M_{F})^{\delta}\cap
W\subseteq(M_{F}).$

The definability of $\mathcal{S}$ again implies that

$j_{F}(\mathcal{S}\cap\delta)=j_{F^{*}}(\mathcal{S}\cap\delta)=\mathcal{S}\cap\lambda.$

Letting $E:=F\mathbin{\upharpoonright}\varepsilon$ it is clear that
$E\in\mathscr{E}_{\varepsilon,\mathcal{S}}$ (recall Notation 4.3). By
indescernibility (i.e., by Lemma 4.4) there is an extender $E^{*}$ witnessing

$\langle
V_{\kappa},\in\rangle\models\Phi(E,\varepsilon,\mathcal{S}\cap\varepsilon,\lambda_{\varepsilon}).$

######  Lemma 4.18.

$W[G]\models``\delta$ is tall with target $\theta$”.

###### Proof.

By Lemma 4.16, $\lambda_{\varepsilon}=\theta.$ So $j_{E^{*}}\colon
W\rightarrow M_{E^{*}}$ is such that $j_{E^{*}}(\delta)=\theta$,
$W_{\theta}\subseteq M_{E^{*}}$ and $(M_{E^{*}})^{\delta}\cap W\subseteq
M_{E^{*}}$. By our Induction Hypothesis,
$j_{E^{*}}(\mathbb{P}_{\delta})_{\varepsilon}=\mathbb{P}_{\varepsilon}$ (see
also Lemma 4.5). In particular, $G_{\varepsilon}$ is generic for
$j_{E^{*}}(\mathbb{P}_{\delta})_{\varepsilon}$ over $M_{E^{*}}.$ Clearly, for
each $H\subseteq j_{E^{*}}(\mathbb{P}_{\delta})/G_{\varepsilon}$ generic over
$W[G_{\varepsilon}]$ the embedding $j_{E^{*}}$ lifts (in
$W[G_{\varepsilon}\ast H]$) to

$j_{E^{*}}\colon W[G_{\delta}]\rightarrow M_{E^{*}}[G_{\varepsilon}\ast H].$

We argue that there is $H$ which is internal to our generic extension $W[G]$.

In $W[G_{\varepsilon}]$, $\varepsilon$ is inaccessible so we fall in one of
the cases of Definition 4.10. Suppose for instance that we fall in Case
($\aleph$) – the other case is treated analogously. The poset
$\mathbb{Q}_{\varepsilon}$ forces over $W[G_{\varepsilon}\ast
K_{\varepsilon}]$ with $\mathbb{T}^{\mathrm{NS}}_{\varepsilon}$. By design,
this latter poset projects to
$\mathbb{T}^{\mathrm{NS}}_{\varepsilon,E^{*}}:=j_{E^{*}}(\mathbb{P}_{\delta})/(G_{\varepsilon}\ast
K_{\varepsilon})$. Thus, any $W[G_{\varepsilon}\ast K_{\varepsilon}]$-generic
for $(\mathbb{Q}_{\varepsilon})_{G_{\varepsilon}\ast K_{\varepsilon}}$ yields
a $W[G_{\varepsilon}\ast K_{\varepsilon}]$-generic for
$j_{E^{*}}(\mathbb{P}_{\delta})/(G_{\varepsilon}\ast K_{\varepsilon})$. Let
$H_{\varepsilon}$ be the induced generic. Clearly, $j_{E^{*}}$ lifts to

$j_{E^{*}}\colon W[G_{\delta}]\rightarrow M_{E^{*}}[G_{\varepsilon}\ast
K_{\varepsilon}\ast H_{\varepsilon}]$

and the latter is definable in the generic extension $W[G_{\varepsilon+1}]$.

Let us now complete the lifting argument. First,
$(\dot{\mathbb{Q}}_{\delta})_{G_{\delta}}$ is $\delta^{+}$-closed as
$\mathcal{S}\cap\delta$ is stationary in $W$ (see Case ($\beth$) of Definition
4.10). Since the width121212For the definition of the _width_ of an embedding
see Footnote 5. of the embedding $j_{E^{*}}$ is ${\leq}\delta$ and
$\mathbb{P}_{\kappa}/G_{\delta}$ is $\delta^{+}$-closed, $j_{E^{*}}$ lifts to

$j_{E^{*}}\colon W[G]\rightarrow M_{E^{*}}[G_{\varepsilon}\ast
K_{\varepsilon}\ast H_{\varepsilon}][j_{E^{*}}``G_{[\delta,\kappa)}].$

For details, see [Cum10, §15]. This lifting is completely internal to $W[G]$.

Since $M_{E^{*}}$ was closed under $\delta$-sequences in $W$ standard
arguments show that $M_{E^{*}}[G_{\varepsilon}\ast K_{\varepsilon}\ast
H_{\varepsilon}][j_{E^{*}}``G_{[\delta,\kappa)}]$ is closed under
$\delta$-sequences in $W[G]$, as well. Altogether, $\delta$ is a tall cardinal
with target $\theta$ in $W[G]$. ∎

This completes the proof of axiom $\mathcal{A}$ in $W[G]=V[G]_{\kappa}$. ∎

###  4.2. $\mathcal{A}$ is consistent with Vopěnka’s Principle

The main result of this section is Theorem 4.21 where we show that $\kappa$
remains a Vopěnka cardinal after forcing with $\mathbb{P}_{\kappa}$. Recall
that we were assuming that $\kappa$ is almost huge and that $\prec$ is a
absolute-enough well-ordering of the set-theoretic universe.

One could be tempted to invoke Brooke-Taylor’s preservation theorem of Vopěnka
cardinals [BT11, Theorem 15] but there is a technical caveat – our iteration
is not _progressively-directed-closed_. This is because at a number of stages
we are forcing with the Kurepa-type forcing $\mathbb{K}_{\varepsilon}$ which
is not $\varepsilon$-directed-closed. Fortunately, for a $\mathcal{U}$-large
set of $\delta<\kappa$ (recall that $\mathcal{F}\subseteq\mathcal{U}$ by Lemma
2.11) $\mathbb{P}_{\kappa}$ contains a dense sub-iteration
$\mathbb{D}^{\delta}$ such that:

  1. ($\alpha$)

$\mathbb{D}^{\delta}$ is definable via $\mathcal{S}$, $\delta$ and $\prec$;

  2. ($\beta$)

$(\mathbb{D}^{\delta})_{\delta}$ forces the tail forcing
$(\mathbb{D}^{\delta})_{[\delta,\kappa)}$ to be $\delta^{+}$-directed-closed.

######  Lemma 4.19.

$\\{\delta<\kappa\mathrel{|}\allowbreak\text{There is
$\mathbb{D}^{\delta}\subseteq\mathbb{P}_{\kappa}$ with $(\alpha)$ and
$(\beta)$ above}\\}\in\mathcal{U}.$

###### Proof.

We show that
$\\{\delta<\kappa\mathrel{|}\allowbreak\text{$\mathcal{S}\cap\delta$ is
stationary}\\}\in\mathcal{U}$ is included in the above-displayed set. First we
prove a prelimminary claim:

######  Claim 4.19.1.

For each $\varepsilon<\kappa$ inaccessible and $\sigma<\varepsilon$ there is a
$\mathbb{P}_{\varepsilon}$-name for a poset
$\dot{\mathbb{D}}_{\varepsilon,\sigma}$ such that the trivial condition of
$\mathbb{P}_{\varepsilon}$ forces

$\text{$``\dot{\mathbb{D}}_{\varepsilon,\sigma}\in\dot{\operatorname{HOD}}_{\\{\sigma,\varepsilon,\mathcal{S},\prec\\}}\;\wedge\;\dot{\mathbb{D}}_{\varepsilon,\sigma}\subseteq\dot{\mathbb{Q}}_{\varepsilon}\;\text{is
dense and $|\sigma|^{+}$-directed-closed}$''}.$

###### Proof of claim.

We prove this by induction on $\varepsilon<\kappa$. If $\varepsilon$ happens
to be the first inaccessible cardinal we fall in Case $(\aleph)$ of the
definition of $\mathbb{P}_{\kappa}$ (i.e., $\mathcal{S}\cap\varepsilon$ is not
stationary). In those circumstances $\mathbb{P}_{\varepsilon}$ is trivial and
$\mathscr{E}_{\varepsilon,\mathcal{S}}=\emptyset$. Thus, for each
$\sigma<\varepsilon$ it suffices to take

$\mathbb{D}_{\varepsilon,\sigma}:=\\{(p,\dot{c})\in\mathbb{K}_{\varepsilon}\ast\mathbb{C}(\varepsilon^{+2+i}_{V},\lambda_{\varepsilon})\mathrel{|}\allowbreak\mathrm{ht}(t^{p})>\sigma\\}.$

Assume the claim holds for all inaccessibles $\epsilon<\varepsilon.$ Suppose
that $\mathcal{S}\cap(\varepsilon+1)$ is non-stationary (the other case is
treated analogously). Then, we fall in Case $(\aleph)$ of the definition of
$\mathbb{P}_{\kappa}$ (see Definition 4.10). There are two cases to consider;
namely, either $\mathscr{E}_{\varepsilon,S}=\emptyset$ or
$\mathscr{E}_{\varepsilon,S}\neq\emptyset$. In the former scenario it suffices
to take $\mathbb{D}_{\varepsilon,\sigma}$ as in the above-displayed equation.
Otherwise, $\dot{\mathbb{Q}}_{\varepsilon}$ takes the form
$\dot{\mathbb{K}}_{\varepsilon}\ast\dot{\mathbb{T}^{\mathrm{NS}}_{\varepsilon}}\ast\dot{\mathbb{C}}(\varepsilon^{+2+i}_{V},\lambda_{\varepsilon}).$
Let $E\in\mathscr{E}_{\varepsilon,S}$. Set $\delta:=\operatorname{crit}(E)$.
Since $\delta<\varepsilon$ our induction hypothesis applied within $M_{E^{*}}$
yields, for each $\varepsilon\leq\rho<\lambda_{\varepsilon}$

$\mathop{1\hskip-3.0pt{\rm
l}}\Vdash^{M_{E^{*}}}_{j_{E^{*}}(\mathbb{P}_{\delta})_{\rho}}``\text{There is
$\dot{\mathbb{D}}_{\rho,\sigma}\subseteq\mathbb{Q}_{\rho}$ witnessing the
claim''}.$

Let $\dot{\mathbb{D}}^{E^{*}}_{\varepsilon,\sigma}$ a
$j_{E^{*}}(\mathbb{P}_{\delta})_{\varepsilon}=\mathbb{P}_{\varepsilon}$-name
for the iteration resulting from replacing each $\mathbb{Q}_{\rho}$ by
$\mathbb{D}_{\rho,\sigma}.$ It turns out that this iteration is now definable
using

$\varepsilon,\sigma,\prec\;\text{and}\;j_{E^{*}}(\mathcal{S})\cap\lambda_{\varepsilon}.$

In turn,

  * •

$E^{*}$ is definable using $\mathcal{S}\cap\varepsilon$, $\varepsilon$ and
$\lambda_{\varepsilon}$;

  * •

and $\lambda_{\varepsilon}$ is definable using $\mathcal{S}$ and
$\varepsilon$.

Therefore $\dot{\mathbb{D}}^{E^{*}}_{\varepsilon,\sigma}$ is
$\mathbb{P}_{\varepsilon}$-forced to be definable via
$\varepsilon,\sigma,\prec$ and $\mathcal{S}$. By the way the iteration
$\dot{\mathbb{D}}^{E^{*}}_{\varepsilon,\sigma}$ is defined at stage
$\varepsilon$ it factors as

$\\{p\in\dot{\mathbb{K}}_{\varepsilon}\mathrel{|}\allowbreak\mathrm{ht}(t^{p})>\sigma\\}\ast(\dot{\mathbb{D}}^{E^{*}}_{\varepsilon,\sigma})^{\mathrm{tail}}.$

The full-support product of the family
$\langle(\dot{\mathbb{D}}^{E^{*}}_{\varepsilon,\sigma})^{\mathrm{tail}}\mathrel{|}\allowbreak{E\in\mathscr{E}_{\varepsilon,\mathcal{S}}}\rangle$
is forced to be a dense subforcing of
$\dot{\mathbb{T}}^{\mathrm{NS}}_{\varepsilon}$ which is definable via
$\varepsilon,\sigma$ and $\mathcal{S}$. Finally, let
$\dot{\mathbb{D}}_{\varepsilon,\sigma}$ be a $\mathbb{P}_{\varepsilon}$-name
for the poset

$\textstyle\\{p\in\dot{\mathbb{K}}_{\varepsilon}\mathrel{|}\allowbreak\mathrm{ht}(t^{p})>\sigma\\}\ast(\prod_{E\in\mathscr{E}_{\varepsilon,\mathcal{S}}}(\dot{\mathbb{D}}^{E^{*}}_{\varepsilon,\sigma})^{\mathrm{tail}})\ast\mathbb{C}(\varepsilon^{+2+i}_{V},\lambda_{\varepsilon}).$

This poset is forced to be dense in $\dot{\mathbb{Q}}_{\varepsilon}$,
$|\sigma|^{+}$-directed-closed and definable using
$\\{\sigma,\varepsilon,\mathcal{S}\\}$. This completes the verification of the
induction step. ∎

For each $\delta$ in the set displayed in the lemma, let $\mathbb{D}^{\delta}$
be the iteration which up to $\delta+1$ is defined as $\mathbb{P}$ and for all
other inaccessible stages $\delta<\varepsilon$ it forces with
$\dot{\mathbb{D}}_{\varepsilon,\delta}.$ It follows that $\mathbb{D}^{\delta}$
is dense in $\mathbb{P}_{\kappa}$, it is definable via $\mathcal{S},\prec$ and
$\delta$ and $(\mathbb{D}^{\delta})_{\delta}$ forces the tail poset
$\dot{\mathbb{D}}^{\delta}_{[\delta,\kappa)}$ be $\delta^{+}$-directed closed.
For this latter claim we use that $\mathcal{S}\cap\delta$ is stationary as
then we fall in Case ($\beth$) of Definition 4.10, which does not involve the
Kurepa-type forcing. ∎

A second important fact is next lemma. We refer our readers to §2, Definition
2.8, where we supplied the relevant definitions.

######  Lemma 4.20.

$\\{\delta<\kappa\mathrel{|}\allowbreak\langle
V_{\kappa},\in,\mathbb{P}_{\kappa},\mathcal{S}\rangle\models\text{$``\delta$
is
$C^{(n)}_{\mathbb{P}_{\kappa}}$-$\mathcal{S}$-extendible''}\\}\in\mathcal{U}$.

###### Proof.

Let $\mathfrak{M}:=\langle
V_{\kappa},\in,\mathbb{P}_{\kappa},\mathcal{S}\rangle$ and
$f\colon\kappa\rightarrow\kappa$ be the function defined as

$f(\delta):=\begin{cases}\delta,&\text{if $\mathfrak{M}\models\text{$``\delta$
is $C^{(n)}_{\mathbb{P}_{\kappa}}$-$\mathcal{S}$-extendible''}$};\\\
\delta+\sigma,&\text{$\sigma$ is least with
$\mathfrak{M}\models\text{$``\delta$ is not
$(\delta+\sigma)$-$C^{(n)}_{\mathbb{P}_{\kappa}}$-$\mathcal{S}$-extendible''}$}.\end{cases}$

The set

$C:=\\{\delta<\kappa\mathrel{|}\allowbreak
f``\delta\subseteq\delta\,\wedge\,\langle
V_{\delta},\in,\mathbb{P}_{\kappa}\cap
V_{\delta}\rangle\prec_{\Sigma_{n}}\langle
V_{\kappa},\in,\mathbb{P}_{\kappa}\rangle\\}$

is a club because it is the intersection of the closure points of $f$ with the
club $(C^{(n)}_{\mathbb{P}_{\kappa}})^{\langle
V_{\kappa},\in,\mathbb{P}_{\kappa}\rangle}$ (by inaccessibility of $\kappa$,
[Kan09, Proposition 6.2]).

By normality of the Vopěnka filter ([Kan09, Proposition 24.14])
$C\in\mathcal{F}$ so that it belongs to $\mathcal{U}$ as well. Therefore,
there is a natural sequence
$\langle\mathcal{M}_{\alpha}\mathrel{|}\allowbreak\alpha<\kappa\rangle$ such
that every elementary embedding
$j\colon\mathcal{M}_{\alpha}\rightarrow\mathcal{M}_{\beta}$ has
$\operatorname{crit}(j)\in C$.

Denote by $X$ the set displayed in the statement of the lemma. Our goal is to
show that $X\in\mathcal{F}$; equivalently, that $\kappa\setminus X$ is not
Vopěnka in $\kappa$. To show this we consider yet another natural sequence
$\langle\mathcal{N}_{\alpha}\mathrel{|}\allowbreak\alpha<\kappa\rangle$ given
by

$\mathcal{N}_{\alpha}:=\langle
V_{\gamma_{\alpha}},\in,\\{\alpha\\},\mathcal{M}_{\alpha},C\cap\gamma_{\alpha},\mathcal{S}\cap\gamma_{\alpha}\rangle$

where $\gamma_{\alpha}:=\min\\{\rho\in
C\mathrel{|}\allowbreak\rho>\sup(\mathcal{M}_{\alpha}\cap\mathrm{Ord})\\}$.

Since $\kappa$ is assumed to be almost huge (hence Vopěnka) there are
$\alpha<\beta<\kappa$ and non-trivial elementary embeddings
$j\colon\mathcal{N}_{\alpha}\rightarrow\mathcal{N}_{\beta}$ with
$\operatorname{crit}(j)<\kappa$. We will show that the critical point of any
$j\colon\mathcal{N}_{\alpha}\rightarrow\mathcal{N}_{\beta}$ as above must
belong to $X$ – this will establish that $\kappa\setminus X$ is not Vopěnka in
$\kappa$.

Let $j\colon\mathcal{N}_{\alpha}\rightarrow\mathcal{N}_{\beta}$ be with
$\operatorname{crit}(j)=\delta$. By way of contradiction, suppose

$\mathfrak{M}\models\text{$``\delta$ is not
$C^{(n)}_{\mathbb{P}_{\kappa}}$-$\mathcal{S}$-extendible''}.$

Thus $f(\delta)=\delta+\sigma$ for certain ordinal $\sigma$. Since
$j\mathbin{\upharpoonright}\mathcal{M}_{\alpha}\colon\mathcal{M}_{\alpha}\rightarrow\mathcal{M}_{\beta}$
is an elementary embedding our previous comments yield $\delta\in
C\cap\gamma_{\alpha}$. Also, $\gamma_{\alpha}$ is a closure point of $f$, so
$\delta+\sigma=f(\delta)<\gamma_{\alpha}$. Thus there is an elementary
embedding $j\colon V_{\delta+\sigma}\rightarrow V_{j(\delta+\sigma)}$ (namely,
the restriction of $j$) with $\operatorname{crit}(j)=\delta$.

Let us remark a few things:

  1. (1)

Since $\delta<j(\delta)$ and $j(\delta)\in C\cap\gamma_{\beta}$,

$\delta+\sigma=f(\delta)<j(\delta).$

  2. (2)

$j(\mathcal{S}\cap(\delta+\sigma))=j(\mathcal{S}\cap\gamma_{\alpha})\cap
j(\delta+\sigma)=(\mathcal{S}\cap\gamma_{\beta})\cap
j(\delta+\sigma)=\mathcal{S}\cap j(\delta+\sigma);$

  3. (3)

$\langle V_{j(\delta)},\in,\mathbb{P}_{\kappa}\cap
V_{j(\delta)}\rangle\prec_{\Sigma_{n}}\langle
V_{\kappa},\in,\mathbb{P}_{\kappa}\rangle$ because $j(\delta)\in C.$

From the above we infer that $\mathfrak{M}\models``\delta$ is
$(\delta+\sigma)$-$C^{(n)}_{\mathbb{P}_{\kappa}}$-$\mathcal{S}$-extendible” as
witnessed by $j\colon\langle
V_{\delta+\sigma},\in,\mathcal{S}\cap(\delta+\sigma)\rangle\rightarrow\langle
V_{j(\delta+\sigma)},\in,\mathcal{S}\cap j(\delta+\sigma)\rangle$. Nonetheless
this contradicts the fact that $f(\delta)=\delta+\sigma.$ ∎

We are now in conditions to prove the promissed theorem:

######  Theorem 4.21.

$V[G]_{\kappa}\models``\kappa$ is Vopěnka”.

###### Proof.

This is equivalent to saying $V[G]_{\kappa}\models\mathrm{VP}$. To show this
we will employ Bagaria’s characterization of $\mathrm{VP}$ (inside
$V[G]_{\kappa}$); namely, $\mathrm{VP}$ holds if and only if, for each
$n<\omega$, there is a $C^{(n)}$-extendible cardinal [Bag12]. First, recall
that $\mathcal{U}$ is a normal ultrafilter and that
$\mathcal{S}\in\mathcal{U}$ so that

$A_{0}:=\\{\delta<\kappa\mathrel{|}\allowbreak\text{$\mathcal{S}\cap\delta$ is
stationary}\\}\in\mathcal{U}.$

As argued in Lemmas 4.19 and 4.20, the following sets are also
$\mathcal{U}$-large:

$A_{1}:=\\{\delta<\kappa\mathrel{|}\allowbreak\text{There is
$\mathbb{D}^{\delta}\subseteq\mathbb{P}_{\kappa}$ with $(\alpha)$ and
$(\beta)$ in p.\ref{ClausesDirectedClosure}}\\},$
$A_{2}:=\\{\delta<\kappa\mathrel{|}\allowbreak\langle
V_{\kappa},\in,\mathbb{P}_{\kappa},\mathcal{S}\rangle\models\text{$``\delta$
is $C^{(n)}_{\mathbb{P}_{\kappa}}$-$\mathcal{S}$-extendible''}\\}.$

We shall argue that each $\delta\in\cap_{i<3}A_{i}$ remains
$C^{(n)}$-extendible in $V[G]_{\kappa}$. As in Theorem 4.15 we denote
$W=V_{\kappa}$ and note that $V[G]_{\kappa}=W[G]$.

Fix a $\delta\in\cap_{i<3}A_{i}$ and let $\lambda>\delta$ be such that
$\mathop{1\hskip-3.0pt{\rm
l}}\Vdash_{\mathbb{P}_{\kappa}}W[\dot{G}]_{\lambda}=W_{\lambda}[\dot{G}_{\lambda}]$.
A cardinal $\lambda$ with the above-described property always exists (in fact
there is a proper class of them below $\kappa$) by virtue of [BP, Lemma 5.7].

######  Claim 4.21.1.

Working in $W[G_{\delta}]$, the set defined by

$\mathcal{D}_{\lambda}:=\\{r\in\mathbb{P}_{\kappa}/G_{\delta}\mathrel{|}\allowbreak
r\Vdash_{\mathbb{P}_{\kappa}/G_{\delta}}\varphi(\kappa,\delta,\lambda,n)\\}$

is dense. Here $\varphi(\kappa,\delta,\lambda,n)$ denotes the assertion

$``\exists\theta\in\kappa\,\exists\tau\colon
W[G_{\delta}][\dot{G}_{[\delta,\kappa)}]_{\lambda}\rightarrow
W[G_{\delta}][\dot{G}_{[\delta,\kappa)}]_{\theta}\,\operatorname{crit}(\tau)=\delta\;\text{and}\;\tau(\delta)\in\dot{C}^{(n)}\setminus(\lambda+1)$”.

###### Proof of claim.

Let $r\in\mathbb{P}_{\kappa}/G_{\delta}$. Since $\mathbb{P}_{\kappa}$ has
Easton support there is $\mu\in A_{0}$ inaccessible above $\lambda$ such that
$r\in\mathbb{P}_{\mu}/G_{\delta}$. By our assumption that $\delta$ is
$C^{(n)}_{\mathbb{P}_{\kappa}}$-$\mathcal{S}$-extendible in $\langle
W,\in,\mathbb{P}_{\kappa},\mathcal{S}\rangle$ there is an elementary embedding

$j\colon\langle W_{\mu},\in,\mathcal{S}\cap\mu\rangle\rightarrow\langle
W_{\theta},\in,\mathcal{S}\cap\theta\rangle$

with $\theta<\kappa$, $\operatorname{crit}(j)=\delta$, $j(\delta)>\mu$ and
$j(\delta)\in{C}^{(n)}_{\mathbb{P}_{\kappa}}$.

Since the tail $\mathbb{P}_{[\mu,\kappa)}$ is $\mu^{+}$-closed the trivial
condition of $\mathbb{P}_{\mu}$ forces

$``W[\dot{G}]_{\lambda}=W_{\lambda}[\dot{G}_{\lambda}]$”.

Also, $\mathbb{P}_{\mu}$ is a definable class in $\langle
W_{\mu},\in,\mathcal{S}\cap\mu\rangle$, so that

$\langle
W_{\mu},\in,\mathcal{S}\cap\mu\rangle\models\text{$``\mathop{1\hskip-3.0pt{\rm
l}}\Vdash_{\mathbb{P}_{\mu}}W_{\lambda}[\dot{G}_{\lambda}]=W[\dot{G}]_{\lambda}$''}$

and by elementarity,

$\langle
W_{\theta},\in,\mathcal{S}\cap\theta\rangle\models\text{$``\mathop{1\hskip-3.0pt{\rm
l}}\Vdash_{\mathbb{P}_{\theta}}W_{j(\lambda)}[\dot{G}_{j(\lambda)}]=W[\dot{G}]_{j(\lambda)}$''.}$

Since we have chosen $\delta$ in $A_{1}$ there is a dense subiteration
$\mathbb{D}^{\delta}$ of $\mathbb{P}_{\kappa}$ witnessing $(\alpha)$ and
$(\beta)$ above. Since $\mathcal{S}\cap\mu$ is stationary (as $\mu\in A_{0}$)
one has

$(\mathbb{D}^{\delta})^{\langle
W_{\mu},\in,\mathcal{S}\cap\mu\rangle}=(\mathbb{D}^{\delta})_{\mu},$

and, by elementarity,

$(\mathbb{D}^{j(\delta)})^{\langle
W_{\theta},\in,\mathcal{S}\cap\theta\rangle}=(\mathbb{D}^{j(\delta)})_{\theta}.$

Since $\delta\in A_{1}$ we have that $\mathop{1\hskip-3.0pt{\rm
l}}\Vdash_{\mathbb{D}_{\delta}}``\dot{\mathbb{D}}_{[\delta,\lambda)}$ is
$\delta^{+}$-directed-closed” and thus,

$\mathop{1\hskip-3.0pt{\rm
l}}\Vdash_{\mathbb{D}_{j(\delta)}}``\dot{\mathbb{D}}_{[j(\delta),j(\lambda))}$
is $j(\delta)^{+}$-directed-closed”.

Now work in $W[G_{j(\delta)}]$. Clearly

$\\{j(p)\mathrel{|}\allowbreak p\in
G_{[\delta,\lambda)}\cap(\mathbb{D}^{\delta})\\}$

is a directed subset of the tail forcing

$(\mathbb{D}^{j(\delta)}_{[j(\delta),j(\lambda))})$

of cardinality ${<}j(\delta)$ so we can let
$r^{*}\in\mathbb{D}^{j(\delta)}_{[j(\delta),j(\lambda))}\subseteq\mathbb{P}_{j(\lambda)}/G_{j(\delta)}$
such that

$r^{*}\leq j(p)$ for all $p\in
G_{[\delta,\lambda)}\cap(\mathbb{D}^{\delta})_{[\delta,\lambda)}.$

Since $(\mathbb{D}^{\delta})_{\delta}$ is an Easton-supported iteration it is
easy to define a condition

$r\wedge r^{*}\leq_{\mathbb{P}_{j(\lambda)}/G_{\delta}}r,r^{*}$

(for details see [BP, p.319]). This condition serves as a master condition and
thus necessarily forces the existence of an elementary embedding

$\tau\colon W_{\mu}[G_{\delta}][\dot{G}_{[\delta,\lambda)}]\rightarrow
W_{\theta}[G_{\delta}][\dot{G}_{[\delta,j(\lambda))}].$

Note that the restriction of $\tau$ to $W_{\lambda}[G_{\lambda}]$ is forced to
be

$\tau\mathbin{\upharpoonright}W_{\lambda}[\dot{G}_{\lambda}]\colon
W_{\lambda}[\dot{G}_{\lambda}]\rightarrow
W_{j(\lambda)}[\dot{G}_{j(\lambda)}],$

but by our preliminary comments this is equivalent to saying that

$\tau\mathbin{\upharpoonright}W_{\lambda}[\dot{G}_{\lambda}]\colon
W[\dot{G}]_{\lambda}\rightarrow W[\dot{G}]_{j(\lambda)}.$

On a different note, Fact 2.9 shows that $\mathop{1\hskip-3.0pt{\rm
l}}_{\mathbb{P}_{\kappa}}$ forces $``j(\delta)\in C^{(n)}$”.

Putting everything into the same canopy one concludes that

$W[G_{\delta}]\models r\wedge
r^{*}\Vdash_{\mathbb{P}_{\kappa}/G_{\delta}}\varphi(\kappa,\delta,\lambda,n).$

This proves that, indeed, $\mathcal{D}_{\lambda}$ is a dense set in
$W[G_{\delta}].$ ∎

Invoking the claim we let $r\in G_{[\delta,\kappa)}\cap\mathcal{D}_{\lambda}$,
which yields an elementary embedding $j\colon W[G]_{\lambda}\rightarrow
W[G]_{\theta}$ witnessing that $\delta$ is $\lambda$-$C^{(n)}$-extendible in
$W[G]=V[G]_{\kappa}$. Thus we conclude that $\delta$ is $C^{(n)}$-extendible
in $V[G]_{\kappa}$. ∎

Putting together Theorems 4.15 and 4.21 we infer that $V[G]_{\kappa}$ is a
model of ZFC wittnessing the configuration claimed in Theorem 4.1.

######  Remark 4.22.

Recall that $``\mathrm{Ord}$ is Mahlo” is the scheme asserting that for each
first-order formula $\varphi(x,y)$, if for some parameter $a$,
$\\{\alpha\in\operatorname{Ord}\mathrel{|}\allowbreak\varphi(\alpha,a)\\}$ is
a closed unbounded class of ordinals then there is a regular cardinal $\theta$
for which $\varphi(\theta,a)$ holds. Note that $``\mathrm{Ord}$ is Mahlo”
holds in $V[G]_{\kappa}$. In fact, much more is true as Vopenka’s Principle
holds in this model. Therefore in $V[G]_{\kappa}$ we obtain the stronger
version of axiom $\mathcal{A}$ where a proper class of inaccessible cardinals
is replaced by $``\mathrm{Ord}$ is Mahlo”.

###  4.3. $\mathcal{A}$ and $I_{0}$ cardinals

In this section we show that $\mathcal{A}$ is compatible with the existence of
$I_{0}$-cardinals (see p.2.13 in §2 for definitions). Our departing hypothesis
is that $\kappa$ carries a normal measure $\mathcal{U}$ such that, for each
$X\subseteq\kappa$,

$\\{\delta<\kappa\mathrel{|}\allowbreak\exists\lambda\,I_{0}(\delta,\lambda,X\cap\lambda)\\}\in\mathcal{U}$

and that $\prec$ is a well-ordering of $V$ witnessing equation ($\dagger$ ‣
4.1) in p.$\dagger$ ‣ 4.1. Denote this assumption $\oplus_{\kappa}$. Woodin
has informed us that $\oplus_{\kappa}$ follows from the stronger hypothesis
$I_{0}^{\sharp}(\kappa,\lambda)$ employing Cramer’s results on inverse
reflection [Cra12]. Recall that $I^{\sharp}_{0}(\kappa,\lambda)$ is a
shorthand for “There is a an elementary embedding with $j\colon
L(V_{\lambda+1}^{\sharp})\rightarrow L(V_{\lambda+1}^{\sharp})$ with
$\operatorname{crit}(j)=\kappa<\lambda$”. Another argument yielding
$\oplus_{\kappa}$ can be found in [Woo11, Theorem 137].

######  Theorem 4.23.

Assume that $\oplus_{\kappa}$ holds. Then there is a model for the theory

$\mathrm{ZFC}+\mathcal{A}+\exists\lambda\,I_{0}(\lambda).$

Let $\mathcal{U}$ be a normal measure witnessing $\oplus_{\kappa}$. Arguing as
in Lemma 4.2 we find $\mathcal{S}\in\mathcal{U}$ consisting of indiscernibles
such that $\\{\delta<\kappa\mathrel{|}\allowbreak\text{$\mathcal{S}\cap\delta$
is stationary}\\}$ is $\mathcal{U}$-large. Thus, by virtue of
$\oplus_{\kappa}$, the set of all $\delta<\kappa$ such that

  1. $(\alpha)$

$\mathcal{S}\cap\delta$ is stationary,

  2. $(\beta)$

and $I_{0}(\delta,\lambda,\mathcal{S}\cap\lambda)$ holds for some $\lambda$

is $\mathcal{U}$-large as well. Let $\delta$ be witnessing $(\alpha)$ and
$(\beta)$ above and let us show that it witnesses $I_{0}(\delta,\lambda)$ in
certain generic extension by
$\mathrm{Add}(\omega,1)\ast\dot{\mathbb{P}}_{\kappa}$. To simplify notations
our ground model $V$ will be a generic extension by Cohen forcing
$\mathrm{Add}(\omega,1)$. The preservation of $I_{0}(\delta,\lambda)$ (as in
Theorem 4.21) requires passing to a dense subiteration of
$\mathbb{P}_{\kappa}$ which is nicely behaved.

Let $j\colon L(V_{\lambda+1})\rightarrow L(V_{\lambda+1})$ be a witness for
$I_{0}(\delta,\lambda,\mathcal{S}\cap\lambda)$ and denote by
$\langle\delta_{n}\mathrel{|}\allowbreak n<\omega\rangle$ the corresponding
critical sequence. Since $\mathcal{S}\cap\delta$ is stationary and
$j(\mathcal{S}\cap\lambda)=\mathcal{S}\cap\lambda$ it follows that
$\mathcal{S}\cap\delta_{n}$ is stationary for all $n<\omega.$

Define an Easton-supported iteration
$\mathbb{D}_{\kappa}:=\mathbb{D}(\kappa,\mathcal{S},\langle\delta_{n}\mathrel{|}\allowbreak
n<\omega\rangle)$ as follows. Suppose that $\mathbb{D}_{\varepsilon}$ has been
defined and let $G_{\varepsilon}\subseteq\mathbb{D}_{\varepsilon}$ a
$V$-generic. If $V[G_{\varepsilon}]\models``\varepsilon$ is not inaccessible”,
$(\dot{\mathbb{Q}}^{*}_{\varepsilon})_{G_{\varepsilon}}$ is declared to be the
trivial poset. Otherwise we define it according to the following casuistics:

######  Definition 4.24.

$(\dot{\mathbb{Q}}^{*}_{\varepsilon})_{G_{\varepsilon}}$ is defined according
to the following casuistic:

Case $(\aleph)$: If $\mathcal{S}\cap\varepsilon$ is non-stationary in $V$ then

$(\dot{\mathbb{Q}}^{*}_{\varepsilon})_{G_{\varepsilon}}:=\begin{cases}(\dot{\mathbb{Q}}_{\varepsilon})_{G_{\varepsilon}},&\text{if
$\varepsilon<\delta_{0}$;}\\\
(\dot{\mathbb{D}}_{\varepsilon,\delta_{n}})_{G_{\varepsilon}},&\text{if
$\varepsilon>\delta_{0}$ and
$n=\max\\{m<\omega\mathrel{|}\allowbreak\delta_{m}<\varepsilon\\}$.}\end{cases}$

Case $(\beth)$: If $\mathcal{S}\cap\varepsilon$ is stationary in $V$ then
$(\dot{\mathbb{Q}}^{*}_{\varepsilon})_{G_{\varepsilon}}:=(\dot{\mathbb{Q}}_{\varepsilon})_{G_{\varepsilon}}.$

Here $(\dot{\mathbb{Q}}_{\varepsilon})_{G_{\varepsilon}}$ is as in Definition
4.10 and $(\dot{\mathbb{D}}_{\varepsilon,\delta_{n}})_{G_{\varepsilon}}$ is as
in Claim 4.19.1.

Finally, define

$\mathbb{D}_{\kappa}:=\varinjlim\langle\mathbb{D}_{\varepsilon};\dot{\mathbb{Q}}^{*}_{\varepsilon}\mathrel{|}\allowbreak\varepsilon<\kappa\rangle.$

A few datapoints in regards to $\mathbb{D}_{\kappa}$:

  1. (1)

$\mathbb{D}_{\kappa}$ (resp. $\mathbb{D}_{\lambda}$) is a dense subiteration
of $\mathbb{P}_{\kappa}$ (resp. $\mathbb{P}_{\lambda}$);

  2. (2)

$\mathbb{D}_{\lambda}$ is definable in $L(V_{\lambda+1})$ via
$\lambda,\mathcal{S}\cap\lambda,\prec$ and
$\langle\delta_{n}\mathrel{|}\allowbreak n<\omega\rangle$;

  3. (3)

Case ($\aleph$) above is well-posed as it precludes $\varepsilon$ be one of
the $\delta_{n}$’s.

  4. (4)

For each $n<\omega$, $\mathop{1\hskip-3.0pt{\rm
l}}\Vdash_{\mathbb{D}_{\delta_{n}}}\dot{\mathbb{D}}_{[\delta_{n},\delta_{n+1})}$
is $\delta_{n}^{+}$-directed-closed.

We remind our readers that $\mathbb{P}_{\kappa}$ (resp.
$\mathbb{P}_{\lambda}$) was the ($\lambda$-stage of) the main iteration of
§4.1; namely, the iteration given in Definition 4.10 using
$\mathcal{S}\cap\lambda.$

######  Lemma 4.25.

$I_{0}(\delta,\lambda)$ holds in certain generic extension by
$\mathbb{P}_{\kappa}$.

###### Proof.

First note that
$j(\mathbb{D}_{\lambda})=\mathbb{D}(\lambda,\mathcal{S}\cap\lambda,\langle\delta_{n}\mathrel{|}\allowbreak
n\geq 1\rangle)$. That is, $j(\mathbb{D}_{\lambda})$ is an iteration which
agrees with $\mathbb{P}_{\lambda}$ up to $(\delta_{1}+1)$ and beyond that it
agrees with $\mathbb{D}_{\lambda}$. Clearly, $\mathbb{D}_{\lambda}\subseteq
j(\mathbb{D}_{\lambda})$ and both are dense subiterations of
$\mathbb{P}_{\lambda}$.

######  Claim 4.25.1.

For each $n<\omega$, $|\mathbb{D}_{\gamma}|<\delta_{n}$ for all
$\gamma<\delta_{n}$.

###### Proof of claim.

Suppose towards a contradiction that
$|\mathbb{D}_{{\gamma_{*}}}|\geq\delta_{0}$ for some
${\gamma_{*}}<\delta_{0}$. By elementarity and since
$j({\gamma_{*}})={\gamma_{*}}$,

$|j(\mathbb{D}_{\gamma_{*}})|=|j(\mathbb{D}_{\lambda})_{j({\gamma_{*}})}|=|j(\mathbb{D}_{\lambda})_{\gamma_{*}}|=|\mathbb{P}_{\gamma_{*}}|\geq\delta_{1}.$

Since $j(\mathbb{P}_{\gamma_{*}})=\mathbb{P}_{\gamma_{*}}$ (because
$\mathbb{P}_{\lambda}$ is definable via $\mathcal{S}\cap\lambda$) one can
repeat this $\omega$-many times and conclude that
$|\mathbb{P}_{\gamma_{*}}|\geq\lambda$, which is certainly false.

Fix an arbitrary $1\leq n<\omega$ and let us show that
$|\mathbb{D}_{\gamma}|<\delta_{n}$ for all $\gamma<\delta_{n}$. By
elementarity and our base case,
$|j^{n}(\mathbb{D}_{\lambda})_{\gamma}|<\delta_{n}$ for all
$\gamma<\delta_{n}$.

For each $\gamma<\delta_{n}$ we have

$j^{n}(\mathbb{D}_{\lambda})_{\gamma}=\mathbb{D}(\lambda,\mathcal{S}\cap\lambda,\langle\delta_{m}\mathrel{|}\allowbreak
m\geq n\rangle)_{\gamma}=\mathbb{P}_{\gamma}\supseteq\mathbb{D}_{\gamma}.$

Therefore clearly $|\mathbb{D}_{\gamma}|<\delta_{n}$ for all
$\gamma<\delta_{n}$. This completes the claim. ∎

The next is the master condition argument needed to lift our embedding:

######  Claim 4.25.2.

There is $q\in j(\mathbb{D}_{\lambda})$ with
$q\Vdash_{j(\mathbb{D}_{\lambda})}``p\in\dot{G}\cap\mathbb{D}_{\lambda}\,\Rightarrow\,j(p)\in\dot{G}$”.

###### Proof of claim.

The construction of $q$ is divided in three cases.

$\blacktriangleright_{(\mathrm{I})}$ Let
$q\mathbin{\upharpoonright}\delta_{1}$ be the trivial condition in
$j(\mathbb{D}_{\lambda})_{\delta_{1}}$.

$\blacktriangleright_{(\mathrm{II})}$ For the construction of the
$j(\mathbb{D}_{\lambda})_{\delta_{n}}$-name $q(\delta_{n})$ notice that

  1. (1)

$\mathop{1\hskip-3.0pt{\rm
l}}\Vdash_{j(\mathbb{D}_{\lambda})_{\delta_{n}}}\\{j(p)(\delta_{n})\mathrel{|}\allowbreak
p\in\dot{G}\cap\mathbb{D}_{\lambda}\\}$ is a directed subset of size
${<\delta_{n}}$,

  2. (2)

and $\mathop{1\hskip-3.0pt{\rm
l}}\Vdash_{j(\mathbb{D}_{\lambda})_{\delta_{n}}}\dot{\mathbb{Q}}^{*}_{\delta_{n}}$
is $\delta_{n}^{+}$-directed-closed.

For Clause (1): The trivial condition of
$j(\mathbb{D}_{\lambda})_{\delta_{n}}$ forces

$\\{j(p)(\delta_{n})\mathrel{|}\allowbreak
p\in\dot{G}\cap\mathbb{D}_{\lambda}\\}\subseteq\\{j(p(\delta_{n-1}))\mathrel{|}\allowbreak(p\mathbin{\upharpoonright}\delta_{n-1}+1)\in\dot{G}\cap\mathbb{D}_{\delta_{n-1}+1}\\}$

and by the previous claim $|\mathbb{D}_{\delta_{n-1}+1}|<\delta_{n}$.

For Clause (2): Since $\mathcal{S}\cap\delta_{n}$ is stationary when defining
$\mathbb{Q}^{*}_{\delta_{n}}$ we fall in Case $(\beth)$ of Definition 4.24, so
the poset is $\delta_{n}^{+}$-directed-closed.

Finally, combine (1) and (2) to produce a
$j(\mathbb{D}_{\lambda})_{\delta_{n}}$-name $q(\delta_{n})$ such that

$\mathop{1\hskip-3.0pt{\rm l}}\Vdash_{\mathbb{P}_{\delta_{n}}}\forall
p\in\dot{G}\cap\mathbb{D}_{\lambda}\,(q(\delta_{n})\leq j(p)(\delta_{n})).$

$\blacktriangleright_{(\mathrm{III})}$ For the construction of the
$j(\mathbb{D}_{\lambda})_{\delta_{n}+1}$-name
$q\mathbin{\upharpoonright}(\delta_{n},\delta_{n+1})$ one argues similarly.
There are two key observations that one has to make; namely,

  1. (1)

$\mathop{1\hskip-3.0pt{\rm
l}}\Vdash_{j(\mathbb{D}_{\lambda})_{\delta_{n}+1}}|(j``\mathbb{D}_{\lambda})\mathbin{\upharpoonright}(\delta_{n},\delta_{n+1})|\leq\delta_{n}$;

  2. (2)

$\mathop{1\hskip-3.0pt{\rm
l}}\Vdash_{j(\mathbb{D}_{\lambda})_{\delta_{n}+1}}j(\mathbb{D}_{\lambda})\mathbin{\upharpoonright}(\delta_{n},\delta_{n+1})$
is $\delta_{n}^{+}$-directed-closed.

Once the above is established it is easy to find a master condition; to wit, a
$j(\mathbb{D}_{\lambda})_{\delta_{n}+1}$-name
$q\mathbin{\upharpoonright}(\delta_{n},\delta_{n+1})$ for a condition in
$j(\mathbb{D}_{\lambda})\mathbin{\upharpoonright}(\delta_{n},\delta_{n+1})$
such that

$\mathop{1\hskip-3.0pt{\rm
l}}\Vdash_{j(\mathbb{D}_{\lambda})_{\delta_{n}+1}}\forall
p\in\dot{G}\cap\mathbb{D}_{\lambda}\,(q\mathbin{\upharpoonright}(\delta_{n},\delta_{n+1})\leq
j(p)\mathbin{\upharpoonright}(\delta_{n},\delta_{n+1})).$

By design of $j(\mathbb{D}_{\lambda})$ the tail forcing
$j(\mathbb{D}_{\lambda})\mathbin{\upharpoonright}(\delta_{n},\delta_{n+1})$ is
$\delta_{n}^{+}$-directed-closed (see Clause (4) in p.4). For this we have
crucially used that $n\geq 1$ as then

$\mathop{1\hskip-3.0pt{\rm
l}}\Vdash_{j(\mathbb{D}_{\lambda})_{\delta_{n}+1}}j(\mathbb{D}_{\lambda})\mathbin{\upharpoonright}(\delta_{n},\delta_{n+1})=\mathbb{D}_{\lambda}\mathbin{\upharpoonright}(\delta_{n},\delta_{n+1}).$

Suppose towards a contradiction that (1) is false and let $H\subseteq
j(\mathbb{D}_{\lambda})_{\delta_{n}+1}$ a generic filter such that in
$L(V_{\lambda+1})[H]$ there is an injective sequence

$\langle
j(p_{\alpha})\mathbin{\upharpoonright}(\delta_{n},\delta_{n+1})\mathrel{|}\allowbreak\alpha<\delta_{n}^{+}\rangle.$

Since Easton-supports are taken at stage $\delta_{n}$, there is
$\sigma_{\alpha}<\delta_{n}$ such that
$\operatorname{supp}(p_{\alpha}\mathbin{\upharpoonright}\delta_{n})\subseteq\sigma_{\alpha}$.
Working in $L(V_{\lambda+1})$ let $I\subseteq\delta_{n}^{+}$ be with
$|I|=\delta_{n}^{+}$ such that $\alpha\mapsto\sigma_{\alpha}$ is constant, say
with value $\sigma_{*}$. Thus, in $L(V_{\lambda+1})[H]$,

$\langle
j(p_{\alpha})\mathbin{\upharpoonright}j(\sigma_{*})\mathrel{|}\allowbreak\alpha\in
I\rangle$

is still injective. By elementarity, this implies that

$\langle
p_{\alpha}\mathbin{\upharpoonright}\sigma_{*}\mathrel{|}\allowbreak\alpha\in
I\rangle$

is also injective and thus $|\mathbb{D}_{\sigma_{*}}|\geq\delta_{n}^{+}$. Note
that this latter assertion is precluded by Claim 4.25.1, so we obtain the
desired contradiction.

Let us check that $q$ forces $``p\in\dot{G}\cap\mathbb{D}_{\lambda}\Rightarrow
j(p)\in\dot{G}$”. Let $q\in G$ and $p\in G\cap\mathbb{D}_{\lambda}$. Then
$j(p)\mathbin{\upharpoonright}\delta_{1}\geq
p\mathbin{\upharpoonright}\delta_{1}\in G_{\delta_{1}}$. By construction, for
each $n\geq 1$, $q\mathbin{\upharpoonright}[\delta_{n},\delta_{n+1}]\leq
j(p)\mathbin{\upharpoonright}[\delta_{n},\delta_{n+1}]$, hence the latter
belongs to $G_{[\delta_{n},\delta_{n+1}]}$. ∎

We are now in conditions to lift $j\colon L(V_{\lambda+1})\rightarrow
L(V_{\lambda+1})$. Let $G\subseteq\mathbb{P}_{\kappa}$ be a $V$-generic with
$q\in G$ (recall that $\mathbb{P}_{\kappa}$ is the iteration of §4.1). Since
both $\mathbb{D}_{\lambda}$ and $j(\mathbb{D}_{\lambda})$ are dense in
$\mathbb{P}_{\lambda}$ we have that $G_{\lambda}\cap\mathbb{D}_{\lambda}$ and
$G_{\lambda}\cap j(\mathbb{D}_{\lambda})$ are $L(V_{\lambda+1})$-generics for
the corresponding posets.

By the previous claim it follows that $j$ lifts to

$j\colon L(V_{\lambda+1})[G_{\lambda}\cap\mathbb{D}_{\lambda}]\rightarrow
L(V_{\lambda+1})[G_{\lambda}\cap j(\mathbb{D}_{\lambda})].$

By density of $\mathbb{D}_{\lambda}$ in $\mathbb{P}_{\lambda}$,

$L(V_{\lambda+1})[G_{\lambda}\cap\mathbb{D}_{\lambda}]=L(V_{\lambda+1})[G_{\lambda}]$

and by $\aleph_{1}$-closure of $\mathbb{P}_{\lambda}$,

$L(V_{\lambda+1})[G_{\lambda}]=L(V[G_{\lambda}]_{\lambda+1}).$

Similar comments apply to $L(V_{\lambda+1})[G_{\lambda}\cap
j(\mathbb{D}_{\lambda})]$. Thereby, $j$ lifts to

$j\colon L(V[G_{\lambda}]_{\lambda+1})\rightarrow
L(V[G_{\lambda}]_{\lambda+1}).$