# Separating Subversion Forcing Principles

Hiroshi Sakai  Graduate School of System Informatics, Kobe University, Rokko-
dai 1-1, Nada, Kobe, 657-8501, Japan<EMAIL_ADDRESS>and  Corey
Bacal Switzer  Institute of Discrete Mathematics and Geometry, Vienna
University of Technology. Wiedner Hauptstrasse 8-10, 1040 Vienna, Austria
<EMAIL_ADDRESS>
###### Abstract.

We study a family of variants of Jensen’s _subcomplete forcing axiom_ ,
$\mathsf{SCFA}$ and _subproper forcing axiom_ , $\mathsf{SubPFA}$. Using these
we develop a general technique for proving non-implications of
$\mathsf{SCFA}$, $\mathsf{SubPFA}$ and their relatives and give several
applications. For instance we show that $\mathsf{SCFA}$ does not imply
$\mathsf{MA}^{+}(\sigma$-closed$)$ and $\mathsf{SubPFA}$ does not imply
Martin’s Maximum.

###### Key words and phrases:

###### 2010 Mathematics Subject Classification:

03E17, 03E35, 03E50

_Acknowledgments:_ The first author would like to thank JSPS for the support
through grant numbers 18K03397 and 21K03338. The second author would like to
thank the Austrian Science Fund (FWF) for the generous support through grant
number Y1012-N35.

##  1\. Introduction

In this paper we study variants of subcomplete and subproper forcing classes
with an eye towards investigating and distinguishing their forcing principles.
Subcomplete and subproper forcing are two classes of forcing notions
introduced by Jensen in [13] in connection with the extended Namba problem,
see [14, Section 6.4] 111See Definition 1.4 below for precise definitions..
Both are iterable with revised countable support and generalize significantly
$\sigma$-closed and proper forcing notions respectively while allowing, under
some circumstances, new $\omega$-sequences of ordinals to be added to
uncountably cofinal cardinals. As such each comes with a forcing axiom
(consistent relative to a supercompact cardinal). The forcing axiom for
subcomplete forcing in particular, dubbed $\mathsf{SCFA}$ by Jensen in [11,
14] is especially interesting as it is consistent with $\diamondsuit$ while
implying some of the strong, structural consequences of $\mathsf{MM}$ such as
$\mathrm{SCH}$ and strong reflection principles including the failure of
square principles, see [14, Section 4]. Since their initial introduction
subcomplete and subproper forcing have been tied to several applications and
received further treatment, see for instance, [6, 8, 9, 10].

Already in [10] the second author and Fuchs found (seemingly) more general
classes, dubbed “$\infty$-subcomplete” and “$\infty$-subproper” each
containing their non “$\infty$” version respectively, and proved a variety of
iteration and preservation theorems. The main theorem in that work was that
the forcing axiom for $\infty$-subcomplete forcing notions,
$\infty$-$\mathsf{SCFA}$, is compatible with a large variety of behavior on
$\aleph_{1}$ when $\mathsf{CH}$ fails. For instance
$\aleph_{1}=\mathfrak{d}<\mathfrak{c}=\aleph_{2}$ and the existence of Souslin
trees are both consistent with $\infty$-$\mathsf{SCFA}$ $+\neg\mathsf{CH}$.

In this paper we combine the $\infty$-versions of these forcing classes with
further parametrization “above $\mu$” for cardinals $\mu$, initially
investigated, somewhat sparingly, by Jensen in [12, Section 3]. This leads to
a large family of forcing axioms $\infty$-$\mathsf{SubPFA}\upharpoonright\mu$
and $\infty$-$\mathsf{SCFA}\upharpoonright\mu$, where
$\infty$-$\mathsf{SubPFA}$ and $\infty$-$\mathsf{SCFA}$ coincide with
$\infty$-$\mathsf{SubPFA}\upharpoonright 2^{\aleph_{0}}$ and
$\infty$-$\mathsf{SCFA}\upharpoonright 2^{\aleph_{0}}$ respectively. The main
motivation of this work is to investigate how these axioms relate to one
another and to other, more well known axioms such as $\mathsf{MM}$ and
$\mathsf{MA}^{+}(\sigma\mbox{-closed})$. Formal definitions will be given in
the second part of this introduction and Section 2 but the definitions of
these axioms alongside well known results provides almost immediately that the
following diagram of implications holds with $2^{\aleph_{0}}\leq\nu<\mu$
cardinals.

$\mathsf{MM}$$\mathsf{MA}^{+}(\sigma{\rm-
closed})$$\infty$-$\mathsf{SubPFA}$$\infty$-$\mathsf{SubPFA}\upharpoonright\nu$$\infty$-$\mathsf{SubPFA}\upharpoonright\mu$$\mathsf{PFA}$$\forall\kappa\neg\square_{\kappa}$$\infty$-$\mathsf{SCFA}$$\infty$-$\mathsf{SCFA}\upharpoonright\nu$$\infty$-$\mathsf{SCFA}\upharpoonright\mu$$\forall\kappa\geq
2^{\aleph_{0}}\neg\square_{\kappa}$$\forall\kappa\geq\nu^{\aleph_{0}}\neg\square_{\kappa}$$\forall\kappa\geq\mu^{\aleph_{0}}\neg\square_{\kappa}$
Figure 1. Subversion forcing principles and then some

The main result of this work is that essentially no arrows are missing from
Figure 1 above.

######  Main Theorem 1.1.

Let $2^{\aleph_{0}}\leq\nu\leq\lambda<\mu=\lambda^{+}$ be cardinals with
$\nu^{\omega}<\mu$. Assuming the consistency of a supercompact cardinal, the
implications given in Figure 1 are complete in the sense that if no
composition of arrows exists from one axiom to another then there is a model
of $\mathsf{ZFC}$ in which the implication fails222Except for the trivial
$\forall\kappa\neg\square_{\kappa}\to\forall\kappa\geq
2^{\aleph_{0}}\neg\square_{\kappa}$ which did not fit aesthetically into the
picture..

As a corollary of this theorem and its proof we obtain separations of several
“subversion” forcing principles from other, more well-studied reflection
principles and forcing axioms. For instance we show the following.

######  Theorem 1.1 (See Theorem 3.1).

Assuming the consistency of a supercompact cardinal, $\mathsf{SCFA}$ does not
imply the failure of $\square_{\aleph_{1}}$ when $\mathsf{CH}$ fails.

######  Corollary 1.2.

Assuming the consistency of a supercompact cardinal, $\mathsf{SCFA}$ does not
imply $\mathsf{MA}^{+}(\sigma{\mbox{\rm-closed}})$.

The rest of this paper is organized as follows. In the next subsection of this
introduction we give relevant background and terminology. In the next Section
we introduce the variants $\infty$-subcompleteness and $\infty$-subproperness
above $\mu$ and discuss some of their properties. In Section 3 we study the
forcing axioms associated to these classes and show, amongst other things,
that they are distinct as well as the fact $\infty$-$\mathsf{SCFA}$ implies
neither $\mathsf{MA}^{+}(\sigma{\rm-closed})$ nor $\neg\square_{\kappa}$ for
any $\kappa<2^{\omega}$. In Section 4 we continue this investigation and show
that $\infty$-$\mathsf{SubPFA}$ does not imply $\mathsf{MM}$. Section 5
concludes with some final remarks and open problems.

###  1.1. Preliminaries

We conclude this introduction with the key definitions we will use throughout,
beginning with that of subproperness and subcompleteness. These are these two
classes of forcing notions defined by Jensen in [14] which have found several
applications, see e.g [17, 13, 6, 9]. More discussion of these concepts can be
found in [14] or [10]. Before beginning with the definition we will need one
preliminary definition. Below we denote by $\mathsf{ZFC}^{-}$ the axioms of
$\mathsf{ZFC}$ without the power set axiom.

######  Definition 1.3.

A transitive set $N$ (usually a model of $\mathsf{ZFC}^{-}$) is _full_ if
there is an ordinal $\gamma$ so that $L_{\gamma}(N)\models\mathsf{ZFC}^{-}$
and $N$ is regular in $L_{\gamma}(N)$ i.e. for all $x\in N$ and $f\in
L_{\gamma}(N)$ if $f:x\to N$ then ${\rm ran}(f)\in N$.

######  Definition 1.4.

Let $\mathbb{P}$ be a forcing notion and let $\delta(\mathbb{P})$ be the least
size of a dense subset of $\mathbb{P}$.

  1. (1)

We say that $\mathbb{P}$ is _subcomplete_ if for all sufficiently large
$\theta$, $\tau>\theta$ so that $H_{\theta}\subseteq
N:=L_{\tau}[A]\models\mathsf{ZFC}^{-}$, $s\in N$, $\sigma:\bar{N}\prec N$
countable, transitive and full with
$\sigma(\bar{\mathbb{P}},\bar{s},\bar{\theta})=\mathbb{P},s,\theta$ , if
$\bar{G}\subseteq\bar{\mathbb{P}}\cap\bar{N}$ is generic then there is a
$p\in\mathbb{P}$ so that if $p\in G$ is $\mathbb{P}$-generic over $V$ then in
$V[G]$ there is a $\sigma^{\prime}:\bar{N}\prec N$ so that

1\.
$\sigma^{\prime}(\bar{\mathbb{P}},\bar{s},\bar{\theta},\bar{\mu})=\mathbb{P},s,\theta,\mu$

2\. $\sigma^{\prime}``\bar{G}\subseteq G$

3\. ${\rm Hull}^{N}(\delta(\mathbb{P})\cup{\rm ran}(\sigma))={\rm
Hull}^{N}(\delta(\mathbb{P})\cup{\rm ran}(\sigma^{\prime}))$

  2. (2)

We say that $\mathbb{P}$ is _subproper_ if for all sufficiently large
$\theta$, $\tau>\theta$ so that $H_{\theta}\subseteq
N:=L_{\tau}[A]\models\mathsf{ZFC}^{-}$, $s\in N$, $p\in N\cap\mathbb{P}$,
$\sigma:\bar{N}\prec N$ countable, transitive and full with
$\sigma(\bar{p},\bar{\mathbb{P}},\bar{s},\bar{\theta})=p,\mathbb{P},s,\theta$,
there is a $q\in\mathbb{P}$ so that $q\leq p$ and if $q\in G$ is
$\mathbb{P}$-generic over $V$ then in $V[G]$ there is a
$\sigma^{\prime}:\bar{N}\prec N$ so that

1\.
$\sigma^{\prime}(\bar{p},\bar{\mathbb{P}},\bar{s},\bar{\theta})=p,\mathbb{P},s,\theta$

2\. $(\sigma^{\prime})^{-1}``G$ is $\bar{\mathbb{P}}$-generic over $\bar{N}$

3\. ${\rm Hull}^{N}(\delta(\mathbb{P})\cup{\rm ran}(\sigma))={\rm
Hull}^{N}(\delta(\mathbb{P})\cup{\rm ran}(\sigma^{\prime}))$

Note that the special case where $\sigma=\sigma^{\prime}$ is properness (for
subproperness) and (up to forcing equivalence) $\sigma$-closedness (for
subcomplete). It was pointed out in [10] that the “Hulls” condition 3) in both
definitions is somewhat unnatural. Indeed it is never used in applications and
appears solely for the purpose of proving the iteration theorem, [14, Theorem
3]. In [10] Fuchs and the second author showed that by iterating with
Miyamoto’s _nice iterations_ this condition could be avoided. As such it makes
sense to define the following.

######  Definition 1.5.

Let $\mathbb{P}$ be a forcing notion.

  1. (1)

We say that $\mathbb{P}$ is $\infty$-_subcomplete_ if for all sufficiently
large $\theta$, $\tau>\theta$ so that $H_{\theta}\subseteq
N:=L_{\tau}[A]\models\mathsf{ZFC}^{-}$, $s\in N$, $\sigma:\bar{N}\prec N$
countable, transitive and full with
$\sigma(\bar{\mathbb{P}},\bar{s},\bar{\theta})=\mathbb{P},s,\theta$ , if
$\bar{G}\subseteq\bar{\mathbb{P}}\cap\bar{N}$ is generic then there is a
$p\in\mathbb{P}$ so that if $p\in G$ is $\mathbb{P}$-generic over $V$ then in
$V[G]$ there is a $\sigma^{\prime}:\bar{N}\prec N$ so that

1\.
$\sigma^{\prime}(\bar{\mathbb{P}},\bar{s},\bar{\theta},\bar{\mu})=\mathbb{P},s,\theta,\mu$

2\. $\sigma^{\prime}``\bar{G}\subseteq G$

  2. (2)

We say that $\mathbb{P}$ is $\infty$-_subproper_ if for all sufficiently large
$\theta$, $\tau>\theta$ so that $H_{\theta}\subseteq
N:=L_{\tau}[A]\models\mathsf{ZFC}^{-}$, $s\in N$, $p\in N\cap\mathbb{P}$,
$\sigma:\bar{N}\prec N$ countable, transitive and full with
$\sigma(\bar{p},\bar{\mathbb{P}},\bar{s},\bar{\theta})=p,\mathbb{P},s,\theta$,
there is a $q\in\mathbb{P}$ so that $q\leq p$ and if $q\in G$ is
$\mathbb{P}$-generic over $V$ then in $V[G]$ there is a
$\sigma^{\prime}:\bar{N}\prec N$ so that

1\.
$\sigma^{\prime}(\bar{p},\bar{\mathbb{P}},\bar{s},\bar{\theta})=p,\mathbb{P},s,\theta$

2\. $(\sigma^{\prime})^{-1}``G$ is $\bar{\mathbb{P}}$-generic over $\bar{N}$

To be clear this is just the same as the definitions of the “non-$\infty$”
versions, simply with the additional “Hulls” condition removed. As mentioned
these classes come with an iteration theorem.

######  Theorem 1.6 (Theorem 3.19 (for Subcomplete) and Theorem 3.20 (for
Subproper) of [10]).

Let $\gamma$ be an ordinal and
$\langle\mathbb{P}_{\alpha},\dot{\mathbb{Q}}_{\alpha}\;|\;\alpha<\gamma\rangle$
be a nice iteration in the sense of Miyamoto so that for all $\alpha<\gamma$
we have $\Vdash_{\mathbb{P}_{\alpha}}$“$\dot{\mathbb{Q}}_{\alpha}$ is
$\infty$-subproper (respectively $\infty$-subcomplete). Then
$\mathbb{P}_{\gamma}$ is $\infty$-subproper (respectively
$\infty$-subcomplete).

We note that the above theorem in the case of $\infty$-subproper forcing was
originally proved first independently by Miyamoto in [16]. A consequence of
this theorem (initially observed for the non $\infty$-versions by Jensen) is
that, modulo a supercompact cardinal, these classes have a consistent forcing
axiom.

######  Definition 1.7.

Let $\Gamma$ be a class of forcing notions. The _forcing axiom for_ $\Gamma$,
denoted $\mathsf{FA}(\Gamma)$ is the statement that for all $\mathbb{P}$ in
$\Gamma$ and any $\omega_{1}$-sequence of dense subsets of $\mathbb{P}$, say
$\\{D_{i}\;|\;i<\omega_{1}\\}$ there is a filter $G\subseteq\mathbb{P}$ which
intersects every $D_{i}$.

If $\Gamma$ is the class of ($\infty$-)subproper forcing notions we denote
$\mathsf{FA}(\Gamma)$ by ($\infty$-)$\mathsf{SubPFA}$. Similarly if $\Gamma$
is the class of ($\infty$-)subcomplete forcing notions we denote
$\mathsf{FA}(\Gamma)$ by ($\infty$-)$\mathsf{SCFA}$.

It is not known whether up to forcing equivalence each class is simply equal
to its “$\infty$”-version or if their corresponding forcing axioms are
equivalent. However, since the “$\infty$” versions are more general (or appear
to be) and avoid the unnecessary technicality of computing hulls we will work
with them in this paper. Nearly everything written here however could be
formulated for the “non-$\infty$” versions equally well, though we leave the
translation to the oddly interested reader.

If $\Gamma\subseteq\Delta$ then $\mathsf{FA}(\Delta)$ implies
$\mathsf{FA}(\Gamma)$ so we get the following collection of implications,
which are part of Figure 1.

######  Proposition 1.8.

$\mathsf{MM}\to\infty\mbox{-}\mathsf{SubPFA}\to\mathsf{PFA}$ and
$\mathsf{MM}\to\infty\mbox{-}\mathsf{SubPFA}\to\infty\mbox{-}\mathsf{SCFA}$

Here $\mathsf{MM}$, known as _Martin’s Maximum_ and introduced in [5], is the
forcing axiom for forcing notions which preserve stationary subsets of
$\omega_{1}$ (all $\infty$-subproper forcing notions have this property) and
$\mathsf{PFA}$ is the forcing axiom for proper forcing notions. It is known
from the work of Jensen, see also [10] that none of the above implications can
be reversed with the exception of whether $\mathsf{SubPFA}$ implies
$\mathsf{MM}$. In this paper we will show the consistency of
$\mathsf{SubPFA}+\neg\mathsf{MM}$, see Theorem 4.1 below.

On that note we move to our last preliminary. Many of the theorems in this
paper involve showing that we can preserve some fragment of
$\infty$-$\mathsf{SCFA}$ (or $\infty$-$\mathsf{SubPFA}$) via a forcing killing
another fragment of it. Towards this end we will need an extremely useful
theorem due to Cox. Below recall that a class of forcing notions $\Gamma$ is
_closed under restrictions_ (see Definition 39 of [2]) if for all
$\mathbb{P}\in\Gamma$ and all $p\in\mathbb{P}$ the lower cone
$\mathbb{P}\upharpoonright p:=\\{q\in\mathbb{P}\;|\;q\leq p\\}\in\Gamma$. One
can check that both the classes of $\infty$-subcomplete and $\infty$-subproper
forcing notions (as well as the restrictions “above $\mu$” defined in Section
2) have this property.

######  Theorem 1.9 (Cox, see Theorem 20 of [2]).

Let $\Gamma$ be a class of forcing notions closed under restrictions and
assume $\mathsf{FA}(\Gamma)$ holds. Let $\mathbb{P}$ be a forcing notion.
Suppose that for every $\mathbb{P}$-name $\dot{\mathbb{Q}}$ for a forcing
notion in $\Gamma$ there is a $\mathbb{P}*\dot{\mathbb{Q}}$-name
$\dot{\mathbb{R}}$ for a forcing notion so that the following hold:

  1. (1)

$\mathbb{P}*\dot{\mathbb{Q}}*\dot{\mathbb{R}}$ is in $\Gamma$

  2. (2)

If $j:V\to N$ is a generic elementary embedding,
$\theta\geq|\mathbb{P}*\dot{\mathbb{Q}}*\dot{\mathbb{R}}|^{+}$ is regular in
$V$ and

a) $H_{\theta}^{V}$ is in the wellfounded part of $N$;

b) $j``H_{\theta}^{V}\in N$ has size $\omega_{1}$ in $N$;

c) ${\rm crit}(j)=\omega_{2}^{V}$

d) There exists a $G*H*K$ in $N$ that is
$(H_{\theta}^{V},\mathbb{P}*\dot{\mathbb{Q}}*\dot{\mathbb{R}})$-generic

Then $N$ believes that $j``G$ has a lower bound in $j(\mathbb{P})$

Then $\Vdash_{\mathbb{P}}\mathsf{FA}(\Gamma)$ i.e. $\mathbb{P}$ preserves the
forcing axiom for $\Gamma$.

See [2] for more on strengthenings and generalizations of this wide ranging
theorem.

##  2\. $\infty$-Subcompleteness and $\infty$-Subproperness above $\mu$

Most theorems in this paper filter through the notions of
$\infty$-_subcompleteness (respectively $\infty$-subproperness) above_ $\mu$
for a cardinal $\mu$. These are technical strengthenings of
$\infty$-subcompleteness (respective $\infty$-subproperness). In this section
we define these strengthenings as well as make some elementary observations
which will be used in rest of the paper.

######  Definition 2.1.

Let $\mu$ be a cardinal and $\mathbb{P}$ a forcing notion.

  1. (1)

We say that $\mathbb{P}$ is $\infty$-_subcomplete above_ $\mu$ if for all
sufficiently large $\theta$, $\tau>\theta$ so that $H_{\theta}\subseteq
N:=L_{\tau}[A]\models\mathsf{ZFC}^{-}$, $s\in N$, $\sigma:\bar{N}\prec N$
countable, transitive and full with
$\sigma(\bar{\mathbb{P}},\bar{s},\bar{\theta},\bar{\mu})=\mathbb{P},s,\theta,\mu$,
if $\bar{G}\subseteq\bar{\mathbb{P}}\cap\bar{N}$ is generic then there is a
$p\in\mathbb{P}$ so that if $p\in G$ is $\mathbb{P}$-generic over $V$ then in
$V[G]$ there is a $\sigma^{\prime}:\bar{N}\prec N$ so that

1\.
$\sigma^{\prime}(\bar{\mathbb{P}},\bar{s},\bar{\theta},\bar{\mu})=\mathbb{P},s,\theta,\mu$

2\. $\sigma^{\prime}``\bar{G}\subseteq G$

3\. $\sigma^{\prime}\upharpoonright\bar{\mu}=\sigma\upharpoonright\bar{\mu}$

  2. (2)

We say that $\mathbb{P}$ is $\infty$-_subproper above_ $\mu$ if for all
sufficiently large $\theta$, $\tau>\theta$ so that $H_{\theta}\subseteq
N:=L_{\tau}[A]\models\mathsf{ZFC}^{-}$, $s\in N$, $p\in N\cap\mathbb{P}$,
$\sigma:\bar{N}\prec N$ countable, transitive and full with
$\sigma(\bar{p},\bar{\mathbb{P}},\bar{s},\bar{\theta},\bar{\mu})=p,\mathbb{P},s,\theta,\mu$,
there is a $q\in\mathbb{P}$ so that $q\leq p$ and if $q\in G$ is
$\mathbb{P}$-generic over $V$ then in $V[G]$ there is a
$\sigma^{\prime}:\bar{N}\prec N$ so that

1\.
$\sigma^{\prime}(\bar{p},\bar{\mathbb{P}},\bar{s},\bar{\theta},\bar{\mu})=p,\mathbb{P},s,\theta,\mu$

2\. $(\sigma^{\prime})^{-1}``G$ is $\bar{\mathbb{P}}$-generic over $\bar{N}$

3\. $\sigma^{\prime}\upharpoonright\bar{\mu}=\sigma\upharpoonright\bar{\mu}$

Concretely being $\infty$-subcomplete above $\mu$ simply means that
$\mathbb{P}$ is subcomplete and, moreover, for any $\sigma:\bar{N}\prec N$ the
corresponding $\sigma^{\prime}$ (in $V[G]$) witnessing the subcompleteness can
be arranged to agree with $\sigma$ “up to $\mu$” i.e. on the ordinals below
$\sigma^{-1}``\mu$ (and idem for $\infty$-subproperness). The “non-$\infty$”
versions of these classes was first introduced by Jensen in [13] under the
name “$\mu$-subcompleteness” and “$\mu$-subproperness”. They were investigated
further by Fuchs in [7] who introduced the terminology “above $\mu$” and made
several of the elementary observations we repeat below. Following Fuchs (as
opposed to Jensen) have moved the parameter $\mu$ to the end to avoid the
awkwardness of “$\mu$-$\infty$-subcomplete/$\mu$-$\infty$-subproper”. The
following is immediate from the definitions.

######  Observation 2.2.

Let $\mu<\nu$ be cardinals. If $\mathbb{P}$ is $\infty$-subcomplete
(respectively $\infty$-subproper) above $\nu$ then it is $\infty$-subcomplete
(respectively subproper) above $\mu$ and it is $\infty$-subcomplete
(respectively $\infty$-subproper) (without any restriction).

It is easy to see that being $\infty$-subcomplete (respectively,
$\infty$-subproper) is equivalent to being $\infty$-subcomplete (respectively
$\infty$-subproper) above $\omega_{1}$, however more is true, an observation
due independently to the first author and Fuchs (see [7, Observation 4.2]).

######  Proposition 2.3.

Let $\mathbb{P}$ be a forcing notion. $\mathbb{P}$ is $\infty$-subcomplete
(respectively $\infty$-subproper) if and only if $\mathbb{P}$ is
$\infty$-subcomplete above $2^{\aleph_{0}}$ (respectively $\infty$-subproper
above $2^{\aleph_{0}}$).

As noted above, this proposition (in the case of subcompleteness) is proved as
Observation 4.2 of [7] but we give a detailed proof in order to help the
reader get accustomed to $\infty$-subversion forcing as well as to include the
mild difference of subproperness. However, let us note that essentially the
point is that, using the definable well order in $L_{\tau}[A]$, the reals of
$\bar{N}$ code the cardinality of the continuum.

###### Proof.

We prove the case of $\infty$-subcomplete and leave the reader to check the
case of $\infty$-subproper. Let $\mathbb{P}$ be a forcing notion. It is
immediate as noted above that if $\mathbb{P}$ is $\infty$-subcomplete above
$2^{\omega}$ then it is $\infty$-subcomplete so we need to check just the
reverse direction. Thus assume that $\mathbb{P}$ is $\infty$-subcomplete and
let $\tau>\theta$ be cardinals so that $\sigma:\bar{N}\prec N:=L_{\tau}[A]$
with $H_{\theta}\subseteq N$ be as in the definition of
$\infty$-subcompleteness. Let $\bar{G}$ be a $\bar{\mathbb{P}}$-generic filter
over $\bar{N}$ with $\bar{\mathbb{P}}$ the preimage of $\mathbb{P}$ under
$\sigma$. Finally let $p\in\mathbb{P}$ force that there is a
$\sigma^{\prime}:\bar{N}\prec N$ so that
$\sigma^{\prime}(\bar{\mathbb{P}})=\mathbb{P}$ and
$\sigma^{\prime}``\bar{G}\subseteq G$ for any generic $G\ni p$ (the existence
of such a condition is the heart of the definition of $\infty$-subcompleteness
of course). We need to show that $p$ forces that
$\sigma^{\prime}\upharpoonright 2^{\aleph_{0}}=\sigma\upharpoonright
2^{\aleph_{0}}$, where, to be clear, $2^{\aleph_{0}}$ denotes cardinal (as
computed in $\bar{N}$) which bijects onto the continuum (as defined in
$\bar{N}$). To avoid confusion let us denote the cardinal
$2^{\aleph_{0}}=\kappa$ (in $V$ and hence $N$) and the preimage of $\kappa$ in
$\bar{N}$ under $\sigma$ as $\bar{\kappa}$. First note that by the
absoluteness of $\omega$ we have that for all reals $x\in\bar{N}$ it must be
the case that $\sigma(x)=\sigma^{\prime}(x)=x$ (and being a real is absolute
between $\bar{N}$ and $V$). Moreover, since $N=L_{\tau}[A]$ there is a
definable well order of the universe, and in particular there is a definable
bijection of the reals onto $\kappa$, say $f:2^{\omega}\to\kappa$. By
elementarity in $\bar{N}$ there is a definable bijection
$\bar{f}:2^{\omega}\cap\bar{N}\to\bar{\kappa}$. But since $f$ is definable we
have $\sigma(\bar{f})=\sigma^{\prime}(\bar{f})=f$ and hence for all
$\alpha\in\bar{\kappa}$ we get
$\sigma(\alpha)=\sigma(\bar{f}(\bar{f}^{-1}(\alpha)))=\sigma^{\prime}(\bar{f}(\bar{f}^{-1}(\alpha)))=\sigma^{\prime}(\alpha)$,
as needed. ∎

In fact $2^{\aleph_{0}}$ is the best possible in $\mathsf{ZFC}$. Jensen showed
that Namba forcing is $\infty$-subcomplete above $\omega_{1}$ assuming
$\mathsf{CH}$ while it is not even $\infty$-subproper above $\omega_{2}$ in
$\mathsf{ZFC}$, a consequence of the next observation.

######  Lemma 2.4.

Let $\mu$ be a cardinal.

  1. (1)

If $\mathbb{P}$ is $\infty$-subproper above $\mu$ then any new countable set
of ordinals less than $\mu$ added by $\mathbb{P}$ is covered by an old
countable set of ordinals (less than $\mu$). In particular if
$\Vdash_{\mathbb{P}}$“${\rm cf}(\mu)=\omega$” then ${\rm cf}(\mu)=\omega$ (in
$V$).

  2. (2)

If $\mathbb{P}$ is $\infty$-subcomplete above $\mu$ then $\mathbb{P}$ adds no
new countable sets of ordinals below $\mu$.

###### Proof.

The proofs of both are similar to the corresponding proofs that every new
countable set of ordinals added by a proper forcing notion is contained in an
old countable set of ordinals and $\sigma$-closed forcing notions do not add
new countable sets of ordinals at all respectively. The point is that to show
the corresponding fact “below $\mu$” one only needs $\infty$-subproperness
(respectively $\infty$-subcompleteness) above $\mu$.

Let us begin with the first item. Assume that $\mu$ is a cardinal,
$\mathbb{P}$ is $\infty$-subproper above $\mu$, $p\in\mathbb{P}$ and $\dot{x}$
is a $\mathbb{P}$-name so that $p\Vdash\dot{x}:\omega\to\mu$. We need to find
a $q\leq p$ and a countable $X\subseteq\mu$ so that $q\Vdash{\rm
im}(\dot{x})\subseteq\check{X}$. To this end, let $\sigma:\bar{N}\prec N$ be
as in the definition of $\infty$-subproperness with
$\sigma(\bar{p},\bar{\mathbb{P}},\bar{\mu},\dot{\bar{x}})=p,\mathbb{P},\mu,\dot{x}$.
We claim that $X:=\sigma``\bar{\mu}$ is as needed. Indeed, by the definition
of $\infty$-subproperness above $\mu$ there is a $q\leq p$ forcing that there
is an embedding $\sigma^{\prime}:\bar{N}\prec N$ so that
$\sigma^{\prime}(\bar{p},\bar{\mathbb{P}},\bar{\mu},\dot{\bar{x}})=p,\mathbb{P},\mu,\dot{x}$
and $\sigma^{\prime}\upharpoonright\bar{\mu}=\sigma\upharpoonright\bar{\mu}$.
Fix such a $q$ and let $q\in G$ be $\mathbb{P}$-generic over $V$. Note that by
elementarity we have that $\bar{p}\Vdash\dot{\bar{x}}(\check{n})\in\bar{\mu}$
for all $n<\omega$. Also since $\bar{G}:=(\sigma^{\prime})^{-1}``G$ is
$\bar{\mathbb{P}}$-generic over $\bar{N}$ we can find in $\bar{N}$ ordinals
$\bar{\mu}_{n}<\bar{\mu}$ so that
$\bar{N}[\bar{G}]\models\dot{\bar{x}}^{\bar{G}}(n)=\bar{\mu}_{n}$ for each
$n<\omega$. Finally we have that and
$\sigma^{\prime}(\bar{p},\bar{\mathbb{P}},\bar{\mu},\dot{\bar{x}})=p,\mathbb{P},\mu,\dot{x}$
and hence, by elementarity again alongside the fact that
$\sigma\upharpoonright\bar{\mu}=\sigma^{\prime}\upharpoonright\bar{\mu}$ we
get
$N[G]\models\dot{x}^{G}(n)=\sigma^{\prime}(\bar{\mu}_{n})=\sigma(\bar{\mu}_{n})\in
X$ for all $n<\omega$. This completes the proof of the first item.

The second case is nearly verbatim noting that since for $\infty$-subcomplete
forcing we can choose $\bar{G}$ in the ground model we can actually define
$\sigma``\dot{\bar{x}}^{\bar{G}}=(\sigma^{\prime})``\dot{\bar{x}}^{\bar{G}}$
in $V$. ∎

As mentioned before Lemma 2.4 an immediate consequence is the following.

######  Lemma 2.5.

Namba forcing is not $\infty$-subproper above $\omega_{2}$. In particular
Namba forcing is not $\infty$-subproper if $\mathsf{CH}$ fails.

Finally we end this section with some observations about the associated
forcing axioms for the classes we have been discussing.

######  Definition 2.6.

Let $\mu$ be a cardinal. Denote by
$\infty$-$\mathsf{SubPFA}\upharpoonright\mu$ the forcing axiom for forcing
notions $\mathbb{P}$ which are $\infty$-subproper above $\mu$ and
$\infty$-$\mathsf{SCFA}\upharpoonright\mu$ the same for $\mathbb{P}$ which are
$\infty$-subcomplete above $\mu$.

The following is immediate by Observation 2.2.

######  Proposition 2.7.

Let $\mu<\nu$ be cardinals. We have that $\infty$-$\mathsf{SCFA}$ implies
$\infty$-$\mathsf{SCFA}\upharpoonright\mu$ implies
$\infty$-$\mathsf{SCFA}\upharpoonright\nu$. Similarly for the variants of
$\infty$-$\mathsf{SubPFA}$.

In the next section we will show that (in many cases) the reverse implications
do not hold.

##  3\. Separating the $\infty$-$\mathsf{SCFA}\upharpoonright\mu$ Principles

In this section we show that under certain cardinal arithmetic assumptions
$\infty$-$\mathsf{SCFA}\upharpoonright\nu$ does not imply
$\infty$-$\mathsf{SCFA}\upharpoonright\mu$ for $\mu<\nu$. Before proving this
general theorem we introduce our technique with the simple example of
separating $\infty$-$\mathsf{SCFA}\upharpoonright\omega_{1}$ from
$\infty$-$\mathsf{SCFA}\upharpoonright\omega_{2}$. This involves showing that
adding a $\square_{\omega_{1}}$-sequence to a model of
$\infty$-$\mathsf{SCFA}$ preserves
$\infty$-$\mathsf{SCFA}\upharpoonright\omega_{2}$. However, it follows from a
result of Jensen, see [14] that $\mathsf{SCFA}+\mathsf{CH}$ implies the
failure of $\square_{\omega_{1}}$.

###  3.1. The Case of $\infty$-$\mathsf{SCFA}\upharpoonright\omega_{2}$:
Adding Non-Reflecting Structures of Size $\aleph_{2}$

Recall that for an uncountable cardinal $\lambda$ a
$\square_{\lambda}$-sequence is a sequence $\langle
C_{\alpha}\;|\;\alpha\in\lambda^{+}\cap{\rm Lim}\rangle$ so that for all
$\alpha$ the following hold:

  1. (1)

$C_{\alpha}$ is club in $\alpha$

  2. (2)

${\rm ot}(\alpha)\leq\lambda$

  3. (3)

For each $\beta\in{\rm lim}(C_{\alpha})$ we have that
$C_{\alpha}\cap\beta=C_{\beta}$

We recall the poset $\mathbb{P}_{0}$ from [3, Example 6.6] for adding a square
sequence. Conditions $p\in\mathbb{P}_{0}$ are functions so that the domain of
$p$ is $\beta+1\cap{\rm Lim}$ for some $\beta\in\lambda^{+}\cap{\rm Lim}$ and

  1. (1)

For all $\alpha\in{\rm dom}(p)$ we have that $p(\alpha)$ is club in $\alpha$
with order type $\leq\lambda$; and

  2. (2)

If $\alpha\in{\rm dom}(p)$ then for each $\beta\in{\rm lim}(p(\alpha))$ we
have $p(\alpha)\cap\beta=p(\beta)$

The order is end extension. We remark that a moment’s reflection confirms that
this poset is $\sigma$-closed. Moreover it is ${<}\lambda^{+}$-strategically
closed (see [3]). In particular it preserves cardinals up to $\lambda^{+}$.

######  Theorem 3.1.

Assume $\infty$-$\mathsf{SCFA}\upharpoonright\omega_{2}$ and let
$\mathbb{P}_{0}$ be the forcing notion defined above for adding a
$\square_{\omega_{1}}$-sequence. Then $\Vdash_{\mathbb{P}_{0}}$
$\infty$-$\mathsf{SCFA}\upharpoonright\omega_{2}$. In particular if the
existence of a supercompact cardinal is consistent with $\mathsf{ZFC}$ then
$\infty$-$\mathsf{SCFA}\upharpoonright\omega_{2}+\square_{\omega_{1}}$ is
consistent as well.

Before proving this theorem we need to define one more poset. Recall that if
$G\subseteq\mathbb{P}_{0}$ is generic and $\vec{\mathcal{C}}_{G}=\langle
C_{\alpha}\;|\;\alpha\in\lambda^{+}\cap{\rm Lim}\rangle$ is the generic
$\square_{\lambda}$-sequence added by $G$ then for any cardinal
$\gamma<\lambda$ we can _thread the square sequence_ via the following poset,
$\mathbb{T}_{G,\gamma}$. Conditions are closed, bounded subsets
$c\subseteq\lambda^{+}$ so that $c$ has order type $<\gamma$, and for all
limit points $\beta\in c$ we have that $\beta\cap c=C_{\beta}$. See [4, §6]
and [15, p.7] for more on this threading poset. The point is the following.

######  Fact 3.2 (Lemma 6.9 of [4]).

Let $\gamma<\lambda$ be cardinals, $\mathbb{P}_{0}$ the forcing notion
described above for adding a $\square_{\lambda}$-sequence and
$\dot{\mathbb{T}}_{\dot{G},\gamma}$ be the $\mathbb{P}_{0}$-name for the
forcing to thread the generic square sequence with conditions of size
$<\gamma$. Then $\mathbb{P}_{0}*\dot{\mathbb{T}}_{\dot{G},\gamma}$ has a dense
$<\gamma$-closed subset.

We can now prove Theorem 3.1.

###### Proof.

We let $\mathbb{P}_{0}$ be the forcing described above for adding a
$\square_{\omega_{1}}$-sequence (so $\lambda=\omega_{1}$). Let
$\gamma=\aleph_{1}$ so in $V^{\mathbb{P}_{0}}$ the threading poset
$\dot{\mathbb{T}}:=\dot{\mathbb{T}}_{\dot{G},\aleph_{1}}$ consists of
countable closed subsets of $\omega_{2}$. We want to apply Theorem 1.9 to
$\mathbb{P}_{0}$. Note that if $\dot{\mathbb{Q}}$ is a $\mathbb{P}_{0}$-name
for an $\infty$-subcomplete above $\omega_{2}$ forcing notion, then
$\dot{\mathbb{T}}=\dot{\mathbb{T}}_{\dot{G},\aleph_{1}}$ is absolute between
$V^{\mathbb{P}_{0}}$ and $V^{\mathbb{P}_{0}*\dot{\mathbb{Q}}}$ by Lemma 2.4
(2).

######  Claim 3.3.

It is enough to show that for any $\mathbb{P}_{0}$-name $\dot{\mathbb{Q}}$ for
a forcing notion which is $\infty$-subcomplete above $\omega_{2}$, the three
step $\mathbb{P}_{0}*\dot{\mathbb{Q}}*\dot{\mathbb{T}}$ is
$\infty$-subcomplete above $\omega_{2}$.

###### Proof of Claim.

This is because $\mathbb{T}$ adds a lower bound to $j``G$ as described in the
statement of Theorem 1.9. In more detail, let $\dot{\mathbb{Q}}$ be a
$\mathbb{P}_{0}$-name for a forcing notion which is $\infty$-subcomplete above
$\omega_{2}$, we want to show that for $\dot{\mathbb{R}}=\dot{\mathbb{T}}$ the
hypotheses of Theorem 1.9 are satisfied assuming that
$\mathbb{P}_{0}*\dot{\mathbb{Q}}*\dot{\mathbb{T}}$ is $\infty$-subcomplete
above $\omega_{2}$. Since this is exactly the first clause we only need to
concern ourselves with the second one. Recall that, relativized to this
situation, this says that if $j:V\to N$ is a generic elementary embedding,
$\theta\geq|\mathbb{P}_{0}*\dot{\mathbb{Q}}*\dot{\mathbb{T}}|^{+}$ is regular
in $V$ and

a) $H_{\theta}^{V}$ is in the wellfounded part of $N$;

b) $j``H_{\theta}^{V}\in N$ has size $\omega_{1}$ in $N$;

c) ${\rm crit}(j)=\omega_{2}^{V}$

d) There exists a $G*H*K$ in $N$ that is
$(H_{\theta}^{V},\mathbb{P}_{0}*\dot{\mathbb{Q}}*\dot{\mathbb{T}})$-generic

Then $N$ believes that $j``G$ has a lower bound in $j(\mathbb{P}_{0})$.

So fix some $\theta$ and $j:V\to N$ as described in a) to d). Note that
$j``G=G$ by c) and the fact that $G$ is coded as a subset of $\omega_{2}^{V}$.
Thus it suffices to find a lower bound of $G$ in $j(\mathbb{P}_{0})$. The
point is now though that since $G*H*K\in N$ we can in particular form $\bigcup
K\in N$ which is a club subset of $\omega_{2}^{V}=\sup_{p\in G}{\rm dom}(p)$
and coheres with all of the elements of $G$, and hence $(\bigcup
G)\cup\langle\omega_{2}^{V},\bigcup K\rangle$ is as needed. ∎

Let us now show that $\mathbb{P}_{0}*\dot{\mathbb{Q}}*\dot{\mathbb{T}}$ is
$\infty$-subcomplete above $\omega_{2}$. Let $\tau>\theta$ be sufficiently
large cardinals and $\sigma:\bar{N}\prec N=L_{\tau}[A]\supseteq H_{\theta}$ be
as in the definition of $\infty$-subcompleteness above $\omega_{2}$. Let
$\sigma(\bar{\mathbb{P}}_{0},\dot{\bar{\mathbb{Q}}},\dot{\bar{\mathbb{T}}})=\mathbb{P}_{0},\dot{\mathbb{Q}},\dot{\mathbb{T}}$.
Let $\bar{G}*\bar{H}*\bar{K}$ be
$\bar{\mathbb{P}}_{0}*\dot{\bar{\mathbb{Q}}}*\dot{\bar{\mathbb{T}}}$-generic
over $\bar{N}$. There are few things to note. First let us point out that
$\bar{G}$ and $\bar{K}$ are (coded as) subsets of $\bar{\omega}_{2}$, the
second uncountable cardinal from the point of view of $\bar{N}$ (so
$\sigma(\bar{\omega}_{2})=\omega_{2}$). Next note that
$\mathbb{P}_{0}*\dot{\mathbb{Q}}*\dot{\mathbb{T}}$ is forcing equivalent to
$\mathbb{P}_{0}*\dot{\mathbb{T}}*\dot{{\mathbb{Q}}}$ since both
$\dot{\mathbb{Q}}$ and $\dot{\mathbb{T}}$ are in $V^{\mathbb{P}_{0}}$, and the
same for the “bar” versions in $\bar{N}$. Now note that since
$\mathbb{P}_{0}*\dot{\mathbb{T}}$ has a $\sigma$-closed dense subset,
$\sigma``\bar{G}*\bar{K}$ has a lower bound (in $N$), say $(p,t)$ ($t$ is in
the ground model and the $\sigma$-closed dense subset is simply the collection
of conditions whose second coordinate is a check name decided by $p$). By
$\sigma$-closed-ness $(p,t)$ forces that there is a unique lift of
$\sigma:\bar{N}\prec N$ to some $\sigma_{0}:\bar{N}[\bar{G}]\prec N[G]$ with
$\sigma_{0}(\bar{G})=G$ for any $\mathbb{P}_{0}$-generic $G\ni p$ (technically
we need to work in the extension by $\mathbb{P}_{0}*\dot{\mathbb{Q}}$, but we
only want to specify the embedding of the $\bar{\mathbb{P}}_{0}$ extension).
Fix such a $G$ (from which $\sigma_{0}$ is defined) and work in $V[G]$. Note
that $\sigma_{0}``\bar{K}=\sigma``\bar{K}$ has $t\in N$ as a lower bound.
Since $\mathbb{Q}:=\dot{\mathbb{Q}}^{G}$ is $\infty$-subcomplete above
$\omega_{2}$ here we can apply the definition of $\infty$-subcompleteness to
$\sigma_{0}:\bar{N}[\bar{G}]\prec N[G]$ to obtain a condition
$\dot{q}^{G}:=q\in\mathbb{Q}$ so that if $H\ni q$ is $\mathbb{Q}$-generic over
$V[G]$ then in $V[G][H]$ there is a $\sigma_{1}:\bar{N}[\bar{G}]\prec N[G]$ so
that
$\sigma_{1}(\bar{G},\bar{\mathbb{P}}_{0},\dot{\bar{\mathbb{Q}}}^{\bar{G}},\dot{\bar{\mathbb{T}}}^{\bar{G}})=G,\mathbb{P}_{0},\mathbb{Q},\mathbb{T}$
where $\mathbb{T}\in V[G]$ is $\dot{\mathbb{T}}^{G}$,
$\sigma_{1}``\bar{H}\subseteq H$ and
$\sigma_{1}\upharpoonright\bar{\omega}_{2}=\sigma\upharpoonright\bar{\omega}_{2}$.
Note also that by condensation we have that $\bar{N}=L_{\bar{\tau}}[\bar{A}]$
and hence we can ensure that $\sigma_{1}\upharpoonright\bar{N}:\bar{N}\prec
N$.

Now by the first observation above we know that since $\bar{G}$ and $\bar{K}$
are coded as subsets of $\bar{\omega}_{2}$ it must be the case that in fact
$\sigma_{1}\upharpoonright\bar{G}=\sigma\upharpoonright\bar{G}$ and idem for
$\bar{K}$. In particular $(p,t)$ is still a lower bound in
$\sigma_{1}``\bar{G}*\bar{K}$. But putting all of these observations together
now ensures that the triple
$(p,\dot{q},t)\in\mathbb{P}_{0}*\dot{\mathbb{Q}}*\dot{\mathbb{T}}$ forces that
$\sigma_{2}:=\sigma_{1}\upharpoonright\bar{N}$ is as needed to witness that
the three step is $\infty$-subcomplete above $\omega_{2}$ as needed. ∎

Note the following corollary of Theorem 3.1.

######  Corollary 3.4.

$\infty$-$\mathsf{SCFA}$ does not imply $\mathsf{MA}^{+}(\sigma{\mbox{\rm-
closed}})$ assuming the consistency of a supercompact cardinal. In particular
$\infty$-$\mathsf{SCFA}$ does not imply $\mathsf{SCFA}^{+}$.

###### Proof.

Begin with a model of
$\infty$-$\mathsf{SCFA}+2^{\aleph_{0}}=2^{\aleph_{1}}=\aleph_{2}$ (for
instance a model of $\mathsf{MM}$). Force with $\mathbb{P}_{0}$ to preserve
these axioms and add a $\square_{\omega_{1}}$-sequence. Then
$\infty$-$\mathsf{SCFA}\upharpoonright\omega_{2}$ and $\square_{\omega_{1}}$
hold in the extension by Theorem 3.1. But, since $\mathbb{P}_{0}$ does not
collapse cardinals (by $2^{\aleph_{1}}=\aleph_{2}$) or add reals, the
continuum is still $\aleph_{2}$ hence $\infty$-$\mathsf{SCFA}$ holds yet
$\mathsf{MA}^{+}(\sigma\mbox{-closed})$ fails since this axiom implies that
$\square_{\kappa}$ fails for all $\kappa$, see [5]. ∎

The proof of Theorem 3.1 can be generalized in many ways. Observe that very
little about $\mathbb{P}_{0}$ is used. For instance almost an analogous proof
gives the following.

######  Theorem 3.5.

Assume $\infty$-$\mathsf{SCFA}\upharpoonright\omega_{2}$. The forcing
$\mathbb{S}_{\omega_{2}}$ to add an $\omega_{2}$-Souslin tree preserves
$\infty$-$\mathsf{SCFA}\upharpoonright\omega_{2}$.

###### Sketch.

Let $\mathbb{S}_{\omega_{2}}$ be the standard forcing to add an
$\omega_{2}$-Souslin tree: conditions are binary trees $p\subseteq
2^{<\omega_{2}}$ of size $<\aleph_{2}$ ordered by end extension. This adds an
$\omega_{2}$-Souslin tree and is $\sigma$-closed. Let $\dot{T}_{\dot{G}}$ be
the canonical name for the tree added i.e. if
$G\subseteq\mathbb{S}_{\omega_{2}}$ is generic over $V$ then
$(\dot{T}_{\dot{G}})^{G}=\bigcup G$. Let $\dot{\mathbb{Q}}$ be a
$\mathbb{S}_{\omega_{2}}$-name for a forcing notion which is
$\infty$-subcomplete above $\omega_{2}$. As before it is enough to show that
$\mathbb{S}_{\omega_{2}}*\dot{\mathbb{Q}}*\dot{T}_{\dot{G}}$ is
$\infty$-subcomplete above $\omega_{2}$ where $\dot{T}_{\dot{G}}$ is the name
for the tree as a forcing notion. The proof is now though exactly as before
noting that, since $\dot{T}_{\dot{G}}\in V^{\mathbb{S}_{\omega_{2}}}$ we have
again that $\mathbb{S}_{\omega_{2}}*\dot{\mathbb{Q}}*\dot{T}_{\dot{G}}$ is
forcing equivalent to
$\mathbb{S}_{\omega_{2}}*\dot{T}_{\dot{G}}*\dot{\mathbb{Q}}$ and the generic
is coded by a subset of $\omega_{2}$ hence not moved by the new embedding
added by $\dot{\mathbb{Q}}$. ∎

We have the following corollary similar to Corollary 3.4 above by invoking a
model of $\infty$-$\mathsf{SCFA}+2^{\aleph_{0}}=\aleph_{2}$.

######  Corollary 3.6.

Assuming the consistency of a supercompact cardinal we have the consistency of
$\mathsf{SCFA}+\neg\mathsf{CH}+\neg{\rm TP}(\omega_{2})$.

Here ${\rm TP}(\omega_{2})$ is the tree property at $\omega_{2}$ i.e. no
$\omega_{2}$-Aronszajn trees. This result contrasts with [18, Corollary 4.1]
which shows that under Rado’s Conjecture, another forcing axiom-like statement
compatible with $\mathsf{CH}$, ${\rm TP}(\omega_{2})$ is equivalent to
$\neg\mathsf{CH}$.

###  3.2. The General Case

The proof of Theorem 3.1 can be easily generalized to establishe that for any
cardinal $\mu$ adding a $\square_{\mu}$ sequence via $\mathbb{P}_{0}$
preserves $\infty$-$\mathsf{SCFA}\upharpoonright\mu^{+}$.

######  Theorem 3.7.

Let $\mu$ be an uncountable cardinal and assume
$\infty$-$\mathsf{SCFA}\upharpoonright\mu^{+}$ holds. If $\mathbb{P}_{0}$ is
the forcing from the previous subsection to add a $\square_{\mu}$-sequence
then $\mathbb{P}_{0}$ preserves
$\infty$-$\mathsf{SCFA}\upharpoonright\mu^{+}$.

###### Proof.

In $V^{\mathbb{P}_{0}}$ let
$\dot{\mathbb{T}}:=\dot{\mathbb{T}}_{\dot{G},\aleph_{1}}$. We only give the
proof of the claim obtained from Claim 3.3 by replacing $\omega_{2}$ with
$\mu$. The rest of the proof is exactly the same as Theorem 3.1 (just replace
$\omega_{2}$ with $(\mu^{+})^{V}$).

Suppose $j:V\to N$, $\theta$ and $G*H*K$ are as in the proof of Claim 3.3. Let
$\beta:=(\mu^{+})^{V}=\sup_{p\in G}{\rm dom}(p)$. Then $\bigcup K\in N$ is a
club subset of $\beta$ and coheres with all of the elements of $G$. Note that
all initial segments of $\bigcup K$ are countable sets in $V$. So
$K^{*}:=j``\bigcup K$ is club in $\beta^{*}:=\sup(j``\beta)$ and coheres with
all of the elements of $G^{*}:=j``G$. Hence $(\bigcup
G^{*})\cup\langle\beta^{*},K^{*}\rangle$ is a lower bound of $j``G$ in
$j(\mathbb{P}_{0})$. ∎

Note that in particular $\mathsf{SCFA}$ does not imply $\square_{\mu}$ for any
$\mu<2^{\aleph_{0}}$. On the other hand, as in Figure 1 at the introduction,
we have the following theorem, which is essentially known.

######  Theorem 3.8.

Let $2^{\aleph_{0}}\leq\nu\leq\kappa<\mu=\kappa^{+}$ be cardinals with
$\nu^{\omega}<\mu$. Modulo the existence of a supercompact cardinal
$\infty\mbox{-}\mathsf{SCFA}\upharpoonright\mu+\neg\infty\mbox{-}\mathsf{SCFA}\upharpoonright\nu$
is consistent.

###### Proof.

By Theorem 3.7 we know that $\infty$-$\mathsf{SCFA}\upharpoonright\mu$ is
consistent with $\square_{\kappa}$ hence it suffices to show that
$\infty$-$\mathsf{SCFA}\upharpoonright\nu$ implies the failure of
$\square_{\kappa}$. This is essentially known though it needs to be pieced
together from a few sources. First, to get the failure of $\square_{\kappa}$
Jensen uses the forcing notion (at $\kappa$) from [14, Lemma 6.3]. Second, [8,
Lemma 3.5] implies that this forcing notion is indeed $\infty$-subcomplete
above $\nu$ under the cardinal arithmetic assumptions mentioned in the theorem
statement. See the proof of [8, Lemma 3.5] and the discussion therein for more
details. ∎

##  4\. Separating $\mathsf{MM}$ from $\mathsf{SubPFA}$

In this section we prove the following result.

######  Theorem 4.1.

Assume there is a supercompact cardinal. Then there is a forcing extension in
which $\infty$-$\mathsf{SubPFA}$ holds but $\mathsf{MM}$ fails. In particular,
modulo the large cardinal assumption, $\infty$-$\mathsf{SubPFA}$ does not
imply $\mathsf{MM}$.

The idea behind this theorem is a combination of the proof technique from [1,
Theorem 2.6] and the proof of Theorem 3.1. Starting from a model of
$\mathsf{MM}$ we will force to add a non-reflecting stationary set to
$2^{\aleph_{0}}$ ($=\aleph_{2}$ since $\mathsf{MM}$ holds). This kills
$\mathsf{MM}$ by the results of [5] but will preserve
$\infty$-$\mathsf{SubPFA}$ by an argument similar to that of [1, Theorem 2.6].
The interesting difference is that $\infty$-$\mathsf{SubPFA}$ (in fact
$\mathsf{SCFA}$) implies that there are no non-reflecting stationary sets
above the continuum (and more) so here $\aleph_{2}$ matters which is not true
for $\mathsf{PFA}$ (though the proof for $\aleph_{2}$ is the same for
$\mathsf{PFA}$ and its “subversion”). We begin by recalling the relevant
definitions.

######  Definition 4.2.

Let $\kappa$ be a cardinal of uncountable cofinality and $S\subseteq\kappa$.
For a limit ordinal $\alpha<\kappa$ of uncountable cofinality we say that $S$
reflects to $\alpha$ if $S\cap\alpha$ is stationary in $\alpha$. We say that
$S$ is non-reflecting if it does not reflect to any $\alpha<\kappa$ of
uncountable cofinality.

######  Fact 4.3 (See Theorem 9 [5]).

$\mathsf{MM}$ implies that for every regular $\kappa>\aleph_{1}$ every
stationary subset of $\kappa\cap{\rm Cof}(\omega)$ reflects.

Compare this with the following.

######  Fact 4.4 (See Lemma 6[14]).

$\mathsf{SCFA}$ implies that for every regular $\kappa>2^{\aleph_{0}}$ every
stationary subset of $\kappa\cap{\rm Cof}(\omega)$ reflects.

######  Remark 1.

Note that in [14] it is claimed that $\mathsf{SCFA}$ implies that the above
holds for all $\kappa>\aleph_{1}$, regardless of the size of the continuum.
However, Sean Cox observed333Private communication, see also the discussion in
[8] preceding Lemma 3.5. that the proof only works for
$\kappa>2^{\aleph_{0}}$. In light of Theorem 3.1 this bound is optimal.

There is a natural forcing notion to add a non-reflecting stationary subset
$S\subseteq\kappa\cap{\rm Cof}(\omega)$ for a fixed regular cardinal $\kappa$.
The definition and basic properties are given in Example 6.5 of [3]. We record
the basics here for reference.

######  Definition 4.5.

Fix a regular cardinal $\kappa>\aleph_{1}$. The forcing notion
$\mathbb{NR}_{\kappa}$ is defined as follows. Conditions are functions $p$
with domain the set of countably cofinal ordinals below some ordinal
$\alpha<\kappa$ mapping into $2$ with the property that if $\beta\leq{\rm
sup}({\rm dom}(p))$ has uncountable cofinality then there is a set
$c\subseteq\beta$ club in $\beta$ which is disjoint from
$p^{-1}(1)=\\{\alpha\in{\rm dom}(p)\mid p(\alpha)=1\\}$. The extension
relation is simply $q\leq_{\mathbb{NR}_{\kappa}}p$ if and only if $q\supseteq
p$.

Proofs of the following can be found in [3].

######  Proposition 4.6.

For any regular $\kappa>\aleph_{1}$ the forcing $\mathbb{NR}_{\kappa}$ has the
following properties.

  1. (1)

$\mathbb{NR}_{\kappa}$ is $\sigma$-closed.

  2. (2)

$\mathbb{NR}_{\kappa}$ is $\kappa$-strategically closed and in particular
preserves cardinals.

  3. (3)

If $G\subseteq\mathbb{NR}_{\kappa}$ is generic then $S_{G}:=\bigcup_{p\in
G}p^{-1}(1)$ is a non-reflecting stationary subset of $\kappa$.

We neglect to give the definition of strategic closure since we will not need
it beyond the fact stated above, see [4] or [3] for a definition.

Let $\kappa$ be as above, $G\subseteq\mathbb{NR}_{\kappa}$ be generic over $V$
and let $S_{G}:=\bigcup_{p\in G}p^{-1}(1)$ be the generic non-reflecting
stationary set. We want to define a forcing to kill $S_{G}$ (this will be the
“$\dot{\mathbb{R}}$” in our application of Theorem 1.9). Specifically we will
define a forcing notion $\mathbb{Q}_{S_{G}}$ so that forcing with
$\mathbb{Q}_{S_{G}}$ will add a club to $\kappa\setminus S_{G}$ and hence kill
the stationarity of $S_{G}$. Note that since $S_{G}$ is non-reflecting its
complement must also be stationary and indeed has to be fat i.e. contain
continuous sequences of arbitrary length $\alpha<\kappa$ cofinally high.

######  Definition 4.7.

Borrowing the notation from the previous paragraph define the forcing notion
$\mathbb{Q}_{S_{G}}$ as the set of closed, bounded subsets of $\kappa\setminus
S_{G}$ ordered by end extension.

Clearly the above forcing generically adds a club to the complement of $S_{G}$
thus killing its stationarity, see [3] Definition 6.10. It is also
$\omega$-distributive.

We are now ready to prove Theorem 4.1.

###### Proof of Theorem 4.1.

Assume $\infty$-$\mathsf{SubPFA}$ holds (the consistency of this is the only
application of the supercompact). Note that the continuum is $\aleph_{2}$ and
will remain so in any cardinal preserving forcing extension which adds no
reals. Let $\mathbb{P}=\mathbb{NR}_{\aleph_{2}}$, $G\subseteq\mathbb{P}$ be
generic over $V$ and work in $V[G]$. Obviously in this model we have “there is
a non-reflecting stationary subset of $\aleph_{2}$” and thus $\mathsf{MM}$
fails by Fact 4.3. We need to show that $\infty$-$\mathsf{SubPFA}$ holds.

We will apply Theorem 1.9 much as in the proof of Theorem 3.1. Let
$\dot{\mathbb{Q}}$ be a $\mathbb{P}$-name for an $\infty$-subproper forcing
notion and let $\dot{\mathbb{R}}$ name $\mathbb{Q}_{S_{\dot{G}}}$ in
$V^{\mathbb{P}*\dot{\mathbb{Q}}}$ (NOT just in $V^{\mathbb{P}}$ \- this is
different than the proof of Theorem 3.1 and crucial). By exactly the same
argument as in the proof of Theorem 3.1 it suffices to show that
$\mathbb{P}*\dot{\mathbb{Q}}*\dot{\mathbb{R}}$ is $\infty$-subproper (in $V$).
This is because (2) from Theorem 1.9 follows from the fact that, borrowing the
notation from that Theorem applied to our situation $\dot{\mathbb{R}}$ shoots
a club through the complement of $S_{G}$ hence $j``S_{G}=S_{G}$ is non-
stationary in its supremum and so has a lower bound in $N$.

So we show that $\mathbb{P}*\dot{\mathbb{Q}}*\dot{\mathbb{R}}$ is
$\infty$-subproper. This is very similar to the proof of Theorem 3.1 but
enough details are different to warrant repeating everything for completeness.
Let $\tau>\theta$ be sufficiently large cardinals and $\sigma:\bar{N}\prec
N=L_{\tau}[A]\supseteq H_{\theta}$ be as in the definition of
$\infty$-subproperness. Let
$\sigma(\bar{\mathbb{P}},\dot{\bar{\mathbb{Q}}},\dot{\bar{\mathbb{R}}},\bar{\omega_{2}})=\mathbb{P},\dot{\mathbb{Q}},\dot{\mathbb{R}},\omega_{2}$.
Let $(p_{0},\dot{q}_{0},\dot{r}_{0})$ be a condition in
$\mathbb{P}*\dot{\mathbb{Q}}*\dot{\mathbb{R}}$ with
$\sigma(\bar{p}_{0},\dot{\bar{q}}_{0},\dot{\bar{r}}_{0})=(p_{0},\dot{q}_{0},\dot{r}_{0})$.
Applying the $\sigma$-closure of $\mathbb{P}$ we can find a
$\bar{\mathbb{P}}$-generic $\bar{G}$ over $\bar{N}$ and a condition $p\leq
p_{0}$ so that $p$ is a lower bound on $\sigma``\bar{G}$ and, letting
$\alpha={\rm sup}(\sigma``\bar{\omega}_{2})$, we have $p(\alpha)=0$ (i.e. $p$
forces $\alpha$ to not be in the generic stationary set). Let us assume $p\in
G$ and note that this condition forces $\sigma``\bar{G}\subseteq G$ and hence
$\sigma$ lifts uniquely to a $\tilde{\sigma}:\bar{N}[\bar{G}]\prec N[G]$ that
$\tilde{\sigma}(\bar{G})=G$ and $\alpha:={\rm
sup}(\sigma``\bar{\omega}_{2})\notin S_{G}$. Let
$\bar{\mathbb{Q}}=\dot{\bar{\mathbb{Q}}}^{\bar{G}}$ as computed in
$\bar{N}[\bar{G}]$ and let
$\bar{q}_{0}=\dot{\bar{q}}_{0}^{\bar{G}}\in\bar{N}[\bar{G}]$. Applying the
fact that $\dot{\mathbb{Q}}$ is forced to be $\infty$-subproper let $q\leq
q_{0}=\tilde{\sigma}(\bar{q}_{0})$ be a condition forcing that if
$H\subseteq\mathbb{Q}$ is $V$-generic with $q\in H$ then there is a
$\sigma^{\prime}\in V[G][H]$ so that $\sigma^{\prime}:\bar{N}[\bar{G}]\prec
N[G]$ as in the definition of $\infty$-subproperness (with respect to
$\tilde{\sigma}$). Note that as in the proof of Theorem 3.1
$\sigma^{\prime}\upharpoonright\bar{N}:\bar{N}\prec N$ and
$\sigma^{\prime}\upharpoonright\bar{\omega}_{2}=\sigma\upharpoonright\bar{\omega}_{2}$.
Let $\tilde{\sigma}^{\prime}:\bar{N}[\bar{G}][\bar{H}]\to N[G][H]$ be the lift
of $\sigma^{\prime}$, where $\bar{H}=(\sigma^{\prime})^{-1}``H$.

######  Claim 4.8.

In $V[G][H]$ the set $S_{G}$ does not contain a club.

###### Proof of Claim.

Since $\aleph_{2}$ is the continuum in $V[G]$ note that $\omega_{2}^{V[G]}$
remains uncountably cofinal in $V[G][H]$ (though of course it can be collapsed
to $\omega_{1}$). Suppose towards a contradiction that $S_{G}$ contains a club
and note that since we chose $\theta$ etc sufficiently large we have
$N[G][H]\models$ “$\exists C$ which is club and $C\subseteq S_{G}$”. By
elementarity there is a $\bar{C}\in\bar{N}[\bar{G}][\bar{H}]$ so that

$\bar{N}[\bar{G}][\bar{H}]\models\bar{C}\subseteq\bar{S}_{G}\;{\rm is\,club}$

where $\bar{H}:=\sigma^{\prime}{}^{-1}H$ is $\bar{\mathbb{Q}}$-generic over
$\bar{N}[\bar{G}]$ by the definition of $\infty$-subcompleteness and the
choice of $q$. But now note that if $C=\tilde{\sigma}^{\prime}(\bar{C})$ then
$C\cap\alpha$ is cofinal in $\alpha$ by elementarity so $\alpha\in C$ but
$\alpha\notin S_{G}$ which is a contradiction. ∎

Given the claim we know that $\omega_{2}^{V}\setminus S_{G}$ is a stationary
set in $V[G][H]$ and hence $\mathbb{R}:=\dot{\mathbb{R}}^{G*H}$ is the forcing
to shoot a club through a stationary set. Let
$\bar{\mathbb{R}}\in\bar{N}[\bar{G}][\bar{H}]$ be
$\dot{\bar{\mathbb{R}}}^{\bar{G}*\bar{H}}$. Note that for each
$\beta\in\bar{N}\cap\bar{\omega}_{2}$ it is dense (in
$\bar{N}[\bar{G}][\bar{H}]$) that there is a condition
$\bar{r}\in\bar{\mathbb{R}}$ with $\beta\in{\rm dom}(\bar{r})$. It follows
that if $\bar{K}$ is generic for $\bar{\mathbb{R}}$ over
$\bar{N}[\bar{G}][\bar{H}]$ with
$\bar{K}\ni\bar{r}_{0}:=\dot{\bar{r}}^{\bar{G}*\bar{H}}$ then
$\tilde{\sigma^{\prime}}``\bar{K}$ unions to a club in $\alpha\setminus
S_{G}$. Since $\alpha\notin S_{G}$ we have that
$r:=\bigcup\tilde{\sigma^{\prime}}``\bar{K}\cup\\{\alpha\\}$ is a condition in
$\mathbb{R}$ which is a lower bound on $\tilde{\sigma^{\prime}}``\bar{K}$ and
hence $r\leq\dot{r}_{0}^{G*H}$. Finally let $K\ni r$ be $\mathbb{R}$-generic
over $V[G][H]$. It is now easy to check that the condition
$(p,\dot{q},\dot{r})$ and $\sigma^{\prime}\upharpoonright\bar{N}$ collectively
witness the $\infty$-subproperness of
$\mathbb{P}*\dot{\mathbb{Q}}*\dot{\mathbb{R}}$ so we are done. ∎

We note that by the same proof adding a nonreflecting stationary set of
$\mu\cap{\rm Cof}(\omega)$ for larger cardinals $\mu$ we can preserve
$\infty$-$\mathsf{SubPFA}\upharpoonright\mu$. The following therefore holds.

######  Theorem 4.9.

Let $2^{\aleph_{0}}\leq\mu\leq\lambda<\nu=\lambda^{+}$ be cardinals with
$\mu^{\omega}<\nu$. Modulo the existence of a supercompact cardinal
$\infty$-$\mathsf{SubPFA}\upharpoonright\nu+\neg\infty$-$\mathsf{SubPFA}\upharpoonright\mu$
is consistent.

The proof of this Theorem finishes the proof of all nonimplications involved
in Main Theorem 2.

##  5\. Conclusion and Open Questions

We view this paper, alongside its predecessor [10] as showing, amongst other
things, that the continuum forms an interesting dividing line for subversion
forcing: below the continuum the “sub” plays no roles as witnessed, as the
same non-implications can hold as those that hold for the non-sub versions.
Above, it adds considerable strength to the associated forcing axioms.
However, as of now we only know how to produce models of $\mathsf{SCFA}$ in
which the continuum is either $\aleph_{1}$ or $\aleph_{2}$. The most pressing
question in this area is therefore whether consistently $\mathsf{SCFA}$ can
co-exist with a larger continuum.

######  Question 1.

Is $\mathsf{SCFA}$ consistent with the continuum $\aleph_{3}$ or greater?

We note here that the most obvious attempt to address this question i.e.
starting with a model of $\mathsf{SCFA}$ and adding $\aleph_{3}$-many reals
with e.g. ccc forcing, does not work, an observation due to the first author.

######  Lemma 5.1.

Suppose $\mathbb{P}$ is a proper forcing notion adding a real. Then
$\mathsf{SCFA}$ fails in $V^{\mathbb{P}}$.

All that is needed about “properness” here is that being proper implies that
stationary subsets of $\kappa\cap{\rm Cof}(\omega)$ are preserved. The proof
of this is standard and generalizes the proof of Lemma 2.4 above (swapping
subproper for proper and removing the bound by the continuum).

###### Proof.

Assume $\mathbb{P}$ is proper. Let $G$ be a $\mathbb{P}$-generic filter over
$V$. For a contradiction, assume $\mathsf{SCFA}$ holds in $V[G]$.

Take a regular cardinal $\nu>2^{\omega}$ in $V[G]$. In $V$, take stationary
partitions $\langle A_{k}:k<\omega\rangle$ of $\nu\cap{\rm Cof}(\omega)$ and
$\langle D_{i}:i<\omega\rangle$ of $\omega_{1}$. In $V[G]$, take a subset $r$
of $\omega$ which is not in $V$. Let $\\{k(i)\\}_{i<\omega}$ be the increasing
enumeration of $r$.

By [14, Lemma 7.1], in $V[G]$, there is an increasing continuous function
$f:\omega_{1}\to\nu$ such that $f[D_{i}]\subseteq A_{k(i)}$ for all
$i<\omega$. Let $\alpha:={\rm sup}({\rm range}(f))$. Then, in $V[G]$ , we have
that $r=\\{k\in\omega:A_{k}\cap\alpha$ is stationary in $\alpha\\}$.

But the set $\\{k\in\omega:A_{k}\cap\alpha$ is stationary in $\alpha\\}$ is
absolute between $V$ and $V[G]$ since $\mathbb{P}$ is proper and hence
preserves stationary subsets of ${\rm Cof}(\omega)$ points. But then $r$ is in
$V$, which is a contradiction. ∎

This shows that either $\mathsf{SCFA}$ implies the continuum is at most
$\aleph_{2}$ \- though given the results of this paper this seems difficult to
prove by methods currently available - or else new techniques for obtaining
$2^{\aleph_{0}}\geq\aleph_{3}$ are needed, which is well known to be in
general an open and difficult area on the frontiers of set theory.

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