# An Iteration Theorem for $\omega_{1}$-preserving Forcings

Andreas Lietz111Institut für Mathematische Logik und Grundlagenforschung,
Universität Münster, Einsteinstrasse 62, 48149 Münster, FRG.  222Current
address: Institut für Diskrete Mathematik und Geometrie, TU Wien, Wiedner
Hauptstrasse 8-10/104, 1040 Wien, AT
This paper is part of the authors PhD thesis.

###### Abstract

We prove an iteration theorem which guarantees for a wide class of nice
iterations of $\omega_{1}$-preserving forcings that $\omega_{1}$ is not
collapse, at the price of needing large cardinals to burn as fuel. More
precisely, we show that a nice iteration of $\omega_{1}$-preserving forcings
which force $\mathrm{SRP}$ at successor steps and preserves old stationary
sets does not collapse $\omega_{1}$.

##  1 Introduction

The method of iterated forcing is a powerful yet flexible tool in establishing
independence results. Say, the goal is to produce a forcing extension of the
universe with a specific property. Frequently, it is the case that it is much
easier to find a forcing $\mathbb{P}$, which solves this problem for “a single
instance” or ”all instances in $V$”, but may add new “unresolved instances” at
the same time. One can then hope to iterate $\mathbb{P}$ up to some closure
point, usually a sufficiently large regular cardinal $\kappa$ so that the
whole iteration is $\kappa$-c.c., so that in end all instances have been dealt
with and the full desired property holds. This can only work if the iteration
in question preserves the progress of earlier stages up until the end.
Theorems which guarantee such a preservation are often called iteration
theorems. If the property in question is one about $H_{\omega_{2}}$ then at
the very least it is required that $\omega_{1}$ is preserved or maybe the
somewhat stronger property that stationary sets are preserved. We give some
examples.

###  1.1 Iterations of c.c.c. Forcings

The earliest iteration theorem is due to Solovay-Tennenbaum.

######  Theorem 1.1 (Solovay-Tennenbaum, [ST71]).

Suppose
$\langle\mathbb{P}_{\alpha},\dot{\mathbb{Q}}_{\beta}\mid\alpha\leq\gamma,\beta<\gamma\rangle$
is a finite support iteration of c.c.c. forcings. Then $\mathbb{P}_{\gamma}$
is c.c.c..

A Suslin tree is a tree of height $\omega_{1}$ with no uncountable chains and
antichains. Forcing with a Suslin tree $T$ (with the reverse order) is a
c.c.c. forcing and in the forcing extension $T$ is no longer Suslin. Not being
a Suslin tree is a $\Sigma_{1}(\omega_{1})$-property and so is upwards
absolute to forcing extensions preserving $\omega_{1}$. As c.c.c. forcings
preserve $\omega_{1}$, iterating forcing with Suslin trees produces models in
which there are no Suslin tree and hence in which Suslin’s hypothesis holds.

######  Theorem 1.2 (Solovay-Tennenbaum, [ST71]).

There is a c.c.c. forcing $\mathbb{P}$ so that
$V^{\mathbb{P}}\models``\text{Suslin's hypothesis}"$.

###  1.2 Iterations of Proper Forcings

The class of forcings with the countable chain condition is rather small, so
not suitable in all cases. Shelah discovered the beautiful notion of proper
forcing which is large enough to include both c.c.c. and $\sigma$-closed
forcing, but nonetheless all such forcings preserve $\omega_{1}$.

######  Definition 1.3 (Shelah,[She98]).

A forcing $\mathbb{P}$ is proper if for any large enough regular $\theta$ and
any countable $X\prec H_{\theta}$ with $\mathbb{P}\in X$, whenever
$p\in\mathbb{P}\cap X$ then there is some $q\leq p$ with

$q\Vdash\check{X}\cap\mathrm{Ord}=\check{X}[\dot{G}]\cap\mathrm{Ord}.$

Shelah proved a famous iteration theorem for proper forcings. Though, as
finite support iterations of non-c.c.c. forcings usually collapse
$\omega_{1}$, the preferred support in this instance is countable support.

######  Theorem 1.4 (Shelah,[She98]).

Suppose
$\langle\mathbb{P}_{\alpha},\dot{\mathbb{Q}}_{\beta}\mid\alpha\leq\gamma,\beta<\gamma\rangle$
is a countable support iteration of proper forcings. Then
$\mathbb{P}_{\gamma}$ is proper.

An Aronszajn tree $T$ is a tree of height $\omega_{1}$ with all countable
levels and no cofinal branch. For a tree $T$ of height $\omega_{1}$ and
$A\subseteq\omega_{1}$, let $T\upharpoonright A$ denote the tree with nodes of
a level $\alpha$ of $T$ with $\alpha\in A$ and the tree order inherited from
$T$. Two trees $S,T$ of height $\omega_{1}$ are club-isomorphic iff there is a
club $C\subseteq\omega_{1}$ so that $S\upharpoonright C\cong T\upharpoonright
C$ as partial orders. Given two Aronszajn trees $S,T$, Abraham-Shelah
discovered a proper forcing $\mathbb{P}(T,S)$ which forces $S$ and $T$ to be
club-isomorphic. Note that the property “$S,T$ are club-isomorphic” is
$\Sigma_{1}(S,T,\omega_{1})$ and thus upwards-absolute to any
$\omega_{1}$-preserving forcing extension.

######  Theorem 1.5 (Abraham-Shelah, [AS85]).

There is a proper forcing $\mathbb{P}$ so that

$V^{\mathbb{P}}\models``\text{Any two Aronszajn trees }S,T\text{ are club-
isomorphic}".$

We remark that Suslin’s hypothesis is an immediate consequence of “any two
Aronszajn trees are club-isomorphic”. There provably is an Aronszajn tree
which is special, i.e. the union of countably many antichains. Such a tree
cannot be club-isomorphic to a Suslin tree.

###  1.3 Iterations of Semiproper Forcings

Later, Shelah proved another iteration theorem for the even larger class of
semiproper forcings.

######  Definition 1.6 (Shelah, [She98]).

A forcing $\mathbb{P}$ is semiproper if for any large enough regular $\theta$
and any countable $X\prec H_{\theta}$ with $\mathbb{P}\in X$, whenever
$p\in\mathbb{P}\cap X$ then there is some $q\leq p$ with

$q\Vdash\check{X}\cap\omega_{1}=\check{X}[\dot{G}]\cap\omega_{1}.$

From now on, we will denote $X\subseteq Y\wedge
X\cap\omega_{1}=Y\cap\omega_{1}$ by $X\sqsubseteq Y$. So for example above we
have $q\Vdash\check{X}\sqsubseteq\check{X}[\dot{G}]$.

######  Theorem 1.7 (Shelah).

Suppose
$\langle\mathbb{P}_{\alpha},\dot{\mathbb{Q}}_{\beta}\mid\alpha\leq\gamma,\beta<\gamma\rangle$
is a $\mathrm{RCS}$-iteration of semiproper forcings. Then
$\mathbb{P}_{\gamma}$ is semiproper.

Once again, the notion of support had to be changed. In the argument of
Theorem 1.7 it is crucial that if $\alpha<\gamma$ and $G_{\alpha}$ is
$\mathbb{P}_{\alpha}$-generic over $V$, then the tail iteration
$\langle\mathbb{P}_{\alpha,\xi},\dot{\mathbb{Q}}_{\beta}\mid\xi\leq\gamma,\beta<\gamma\rangle$
is still a $\mathrm{RCS}$-iteration. This can fail for countable support
iterations as, unlike proper forcings, semiproper forcings can turn regular
cardinals into cardinals of countable cofinality. In fact, Theorem 1.7 fails
if $\mathrm{RCS}$-support is replaced with countable support.

Suppose $\mathcal{I}$ is an ideal on $\omega_{1}$. An $\mathcal{I}$-antichain
is a set $\mathcal{A}\subseteq\mathcal{P}(\omega_{1})-\mathcal{I}$ so that
$S\cap T\in\mathcal{I}$ for any $S\neq T\in\mathcal{A}$. The ideal
$\mathcal{I}$ is saturated if for all $\mathcal{I}$-antichains $\mathcal{A}$
we have $|\mathcal{A}|{\leq}\omega_{1}$.

######  Theorem 1.8 (Shelah, see [Sch11] for a proof).

Assume there is a Woodin cardinal. Then there is a semiproper forcing
$\mathbb{P}$ so that

$V^{\mathbb{P}}\models``\mathrm{NS}_{\omega_{1}}\text{ is saturated}".$

If $\mathcal{A}$ is a maximal $\mathrm{NS}_{\omega_{1}}$-antichain then the
sealing forcing $\mathcal{S}_{\mathcal{A}}$ is a natural stationary-set-
preserving forcing which turns $\mathcal{A}$ into a maximal
$\mathrm{NS}_{\omega_{1}}$-antichain of size ${\leq}\omega_{1}$ and the
statement “$\mathcal{A}$ is a maximal antichain of size ${\leq}\omega_{1}$”
turns out to be $\Sigma_{1}(\mathcal{A},\omega_{1})$. Now, an instance of the
sealing forcing is not semiproper in general, but Shelah shows that when
iterating up to a Woodin cardinal and using a sealing forcing only when it is
semiproper, it can be arranged that often enough sealing forcings are
semiproper that in the end, $\mathrm{NS}_{\omega_{1}}$ is saturated.

###  1.4 Iterations of Stationary-Set-Preserving Forcing

So what are the limits of iteration theorems? We have

$\text{c.c.c}\Rightarrow\text{proper}\Rightarrow\text{semiproper}\Rightarrow\text{stationary
set preserving}.$

and none of the implications can be reversed. However, while there are always
non-c.c.c. proper forcings and non-proper semiproper forcings, consistently
the class of semiproper forcings can agree with the class of stationary set
preserving forcing, so these two notions are quite close. Nonetheless, there
is no analogue of Theorem 1.7 for stationary set preserving forcings.
Consistently, a counterexample can be given along the lines of the discussion
of Theorem 1.8. In the argument, the Woodin cardinal is used solely to verify
that instances of sealing forcing are semiproper often enough, an inaccessible
cardinal would suffice otherwise. But a Woodin cardinal is indeed required for
the conclusion.

######  Theorem 1.9 (Steel, Jensen-Steel [JS13]).

Suppose that there is a normal saturated ideal on $\omega_{1}$. Then there is
an inner model with a Woodin cardinal.

So suppose we work in an model without an inner model with a Woodin cardinal,
say $V=L$, and there is an inaccessible cardinal. One could then try to
iterate instances of the Sealing forcing along a suitable bookkeeping up to
$\kappa$. In light of Theorem 1.9, this cannot result in a forcing extension
in which $\mathrm{NS}_{\omega_{1}}$ is saturated. It follows that the
iteration collapses $\omega_{1}$ at some point, yet instances of the sealing
forcing are always stationary set preserving.

A much more serious example is due to Shelah.

######  Theorem 1.10 (Shelah [She98]).

There is a full support iteration

$\coloneqq\langle\mathbb{P}_{n},\dot{\mathbb{Q}}_{m}\mid
n\leq\omega,m<\omega\rangle$

of stationary set preserving forcings so that $\mathbb{P}_{\omega}$ collapses
$\omega_{1}$.

In fact, in the above example it does not matter at all which kind of limit is
taken, though we want to mention that countable support, $\mathrm{RCS}$ and
full support iterations agree on length $\omega$ iterations. The first forcing
in Shelah’s example is semiproper, but all subsequent forcings are not
semiproper in the relevant extension. Semiproper forcing is the correct
regularity property for stationary set preserving forcings in terms of
iterations in the sense that

  1. 1.

all semiproper forcings are stationary set preserving,

  2. 2.

consistently, all stationary set preserving forcings are semiproper and

  3. 3.

semiproper forcings can be iterated.

We will define the class of respectful forcing which, in a slightly weaker
sense, is a regularity property corresponding to the wider class of
$\omega_{1}$-preserving forcings.

###  1.5 Iterations of $\omega_{1}$-Preserving Forcings

When iterating $\omega_{1}$-preserving forcing which kill stationary sets
there is another threat to preserving $\omega_{1}$ in the limit as illustrated
in the following folklore example: For $S\subseteq\omega_{1}$ stationary, the
club shooting forcing $\mathrm{CS}(S)$ is the canonical forcing that shoots a
club trough $S$. Conditions are closed countable sets $c\subseteq S$ ordered
by $d\leq_{\mathrm{CS}(S)}c$ iff $d\cap(\max(c)+1)=c$. If $G$ is generic for
$\mathrm{CS}(S)$ then $\bigcup G$ is a club contained in $S$, so
$\omega_{1}-S$ is nonstationary in $V[G]$, but $\omega_{1}$ is not collapsed,
that is $\omega_{1}^{V[G]}=\omega_{1}^{V}$. Now suppose $\langle S_{n}\mid
n<\omega\rangle$ be a partition of $\omega_{1}$ into stationary sets. Let
$\mathbb{P}$ be a length $\omega$ iteration of the forcings
$\mathrm{CS}(\omega_{1}-S_{n})$ (it does not matter which limit we take at
$\omega$). Then $\mathbb{P}$ must collapse $\omega_{1}$ as in
$V^{\mathbb{P}}$, $\omega_{1}^{V}=\bigcup_{n<\omega}S_{n}$ is a countable
union of nonstationary sets and hence must be nonstationary itself. Clearly,
this is only possible if $\omega_{1}^{V}<\omega_{1}^{V^{\mathbb{P}}}$.

The issue here does not stem from a lack of regularity of the forcings we
used. In fact, for a stationary set $S\subseteq\omega_{1}$, the club shooting
$\mathrm{CS}(S)$ is $S$-proper. The problem is much more that at each step of
the iteration, we come back to $V$ to kill an “old” stationary set. If we
avoid the two presented issues of

  1. 1.

using too many forcings lacking regularity properties and

  2. 2.

killing old stationary sets

then we can prove an iteration theorem for $\omega_{1}$-preserving forcings.
Without defining respectful forcings, a special case of our main result can be
stated as follows.

######  Theorem 1.11.

Suppose
$\langle\mathbb{P}_{\alpha},\dot{\mathbb{Q}}_{\beta}\mid\alpha\leq\gamma,\beta<\gamma\rangle$
is a nice iteration of $\omega_{1}$-preserving forcings so that

  1. $(i)$

if $\alpha+2<\gamma$ then $\Vdash_{\mathbb{P}_{\alpha+2}}``\text{Strong
Reflection Principle}"$ and

  2. $(ii)$

if $\alpha<\gamma$ then $\dot{\mathbb{Q}}_{\alpha}$ is forced to preserve old
stationary sets, i.e.

$\forall\beta<\alpha\
\Vdash_{\mathbb{P}_{\alpha+1}}\mathrm{NS}_{\omega_{1}}\cap
V[\dot{G}_{\beta}]=\mathrm{NS}_{\omega_{1}}^{V[\dot{G}_{\alpha}]}\cap
V[\dot{G}_{\beta}].$

Then $\mathbb{P}_{\gamma}$ preserves $\omega_{1}$. Moreover, we have for all
$\alpha+1\leq\gamma$

$\forall\beta\leq\alpha\
\Vdash_{\mathbb{P}_{\gamma}}\mathrm{NS}_{\omega_{1}}\cap
V[\dot{G}_{\beta}]=\mathrm{NS}_{\omega_{1}}^{V[\dot{G}_{\beta+1}]}\cap
V[\dot{G}_{\beta}].$

In fact we will prove something more general which allows, e.g. the
preservation of a Suslin tree on the side.

Here, the Strong Reflection Principle is the reflection principle isolated by
Todorčević.

######  Definition 1.12 (Todorčević, [Tod87]).

  1. $(i)$

For $\theta$ an uncountable cardinal and
$\mathcal{S}\subseteq[H_{\theta}]^{\omega}$ we define

$\mathcal{S}^{\perp}=\\{X\in[H_{\theta}]^{\omega}\mid\forall
Y\in[H_{\theta}]^{\omega}(X\sqsubseteq Y\rightarrow Y\notin S)\\}.$

  2. $(ii)$

The Strong Reflection Principle ($\mathrm{SRP}$) holds if: Whenever
$\theta\geq\omega_{2}$ is regular, $a\in H_{\theta}$ and
$S\subseteq[H_{\theta}]^{\omega}$ then $\mathcal{S}\cup\mathcal{S}^{\perp}$
contains a continuous increasing $\omega_{1}$-chain of countable elementary
substructures of $H_{\theta}$ containing $a$, i.e. there is $\langle
X_{\alpha}\mid\alpha<\omega_{1}\rangle$ so that for all $\alpha<\omega_{1}$

    1. $(\vec{X}.i)$

$X_{\alpha}\prec H_{\theta}$ is countable,

    2. $(\vec{X}.ii)$

$X_{\alpha}\in\mathcal{S}\cup\mathcal{S}^{\perp}$,

    3. $(\vec{X}.iii)$

$a\in X_{0}$,

    4. $(\vec{X}.iv)$

$X_{\alpha}\in X_{\alpha+1}$ and

    5. $(\vec{X}.v)$

if $\alpha\in\mathrm{Lim}$ then $X_{\alpha}=\bigcup_{\beta<\alpha}X_{\beta}$.

We note that $\mathrm{SRP}$ can always be forced assuming large cardinals.

######  Theorem 1.13 (Shelah).

Suppose there is a supercompact cardinal. Then there is a semiproper forcing
$\mathbb{P}$ so that $V^{\mathbb{P}}\models\mathrm{SRP}$.

As a consequence of this, assuming large cardinals, Theorem 1.11 can be
understood as a strategic iteration theorem. Consider the following two player
game $\mathrm{IG}_{\gamma}$ of length $\gamma$.

Player $I$ | $\dot{\mathbb{Q}}_{0}$ |  | $\dot{\mathbb{Q}}_{2}$ |  | … | $\dot{\mathbb{Q}}_{\omega}$ |  | …
---|---|---|---|---|---|---|---|---
Player $II$ |  | $\dot{\mathbb{Q}}_{1}$ |  | $\dot{\mathbb{Q}}_{3}$ | … |  | $\dot{\mathbb{Q}}_{\omega+1}$ | …

Player $I$ plays at all even stages, including limit steps. Player $I$ and
$II$ cooperate in this way to produce an $\mathrm{RCS}$-iteration
$\langle\mathbb{P}_{\alpha},\dot{\mathbb{Q}}_{\beta}\mid\alpha\leq\gamma,\beta<\gamma\rangle$
of forcings which do not kill old stationary sets. Player $II$ wins iff
$\mathbb{P}_{\gamma}$ preserves $\omega_{1}$.

######  Corollary 1.14.

Suppose there is a proper class of supercompact cardinals. Then for any
$\gamma$, Player $II$ has a winning strategy for the game
$\mathrm{IG}_{\gamma}$.

### Acknowledgements

The author thanks his PhD advisor Ralf Schindler for many fruitful discussions
as well as his guidance and support during the completion of this project.
Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) under Germany’s Excellence Strategy EXC 2044 390685587,
Mathematics Münster: Dynamics - Geometry - Structure.

##  2 Notation

First, we fix some notation. We will extensively deal with countable
elementary substructures $X\prec H_{\theta}$ for large regular $\theta$. We
will make frequent use of the following notation:

######  Definition 2.1.

Suppose $X$ is any extensional set.

  1. $(i)$

$M_{X}$ denotes the transitive isomorph of $X$.

  2. $(ii)$

$\pi_{X}\colon M_{X}\rightarrow X$ denotes the inverse collapse.

  3. $(iii)$

$\delta^{X}\coloneqq\omega_{1}\cap X$.

In almost all cases, we will apply this definition to a countable elementary
substructure $X\prec H_{\theta}$ for some uncountable cardinal $\theta$. In
some cases, the $X$ we care about lives in a generic extension of $V$, even
though it is a substructure of $H_{\theta}^{V}$. In that case, $\delta^{X}$
will always mean $X\cap\omega_{1}^{V}$.

We will also sometimes make use of the following convention in order to
“unclutter” arguments.

######  Convention 2.2.

If $X\prec H_{\theta}$ is an elementary substructure and some object $a$ has
been defined before and $a\in X$ then we denote $\pi_{X}^{-1}(a)$ by
$\bar{a}$.

We will make use of this notation only if it is unambiguous.

######  Definition 2.3.

If $X,Y$ are sets then $X\sqsubseteq Y$ holds just in case

  1. $(i)$

$X\subseteq Y$ and

  2. $(ii)$

$\delta^{X}=\delta^{Y}$.

We use the following notions of clubs and stationarity on
$[H_{\theta}]^{\omega}$:

######  Definition 2.4.

Suppose $A$ is an uncountable set.

  1. $(i)$

$[A]^{\omega}$ is the set of countable subsets of $A$.

  2. $(ii)$

$\mathcal{C}\subseteq[A]^{\omega}$ is a club in $[A]^{\omega}$ if

    1. $a)$

for any $X\in[A]^{\omega}$ there is a $Y\in\mathcal{C}$ with $X\subseteq Y$
and

    2. $b)$

if $\langle Y_{n}\mid n<\omega\rangle$ is a $\subseteq$-increasing sequence of
sets in $\mathcal{C}$ then $\bigcup_{n<\omega}Y_{n}\in\mathcal{C}$.

  3. $(iii)$

$\mathcal{S}\subseteq[A]^{\omega}$ is stationary in $[A]^{\omega}$ if
$\mathcal{S}\cap\mathcal{C}\neq\emptyset$ for any club $\mathcal{C}$ in
$[A]^{\omega}$.

Next, we explain our notation for forcing iterations.

######  Definition 2.5.

Suppose
$\mathbb{P}=\langle\mathbb{P}_{\alpha},\dot{\mathbb{Q}}_{\beta}\mid\alpha\leq\gamma,\beta<\gamma\rangle$
is an iteration and $\beta\leq\gamma$. We consider elements of $\mathbb{P}$ as
functions of domain (or length) $\gamma$.

  1. $(i)$

If $p\in\mathbb{P}_{\beta}$ then $\mathrm{lh}(p)=\beta$.

  2. $(ii)$

If $G$ is $\mathbb{P}$-generic then $G_{\beta}$ denotes the restriction of $G$
to $\mathbb{P}_{\beta}$, i.e.

$G_{\beta}=\\{p\upharpoonright\beta\mid p\in G\\}.$

Moreover, $\dot{G}_{\beta}$ is the canonical $\mathbb{P}$-name for
$G_{\beta}$.

  3. $(iii)$

If $G_{\beta}$ is $\mathbb{P}_{\beta}$-generic then
$\mathbb{P}_{\beta,\gamma}$ denotes (by slight abuse of notation) the
remainder of the iteration, that is

$\mathbb{P}_{\beta,\gamma}=\\{p\in\mathbb{P}_{\gamma}\mid
p\upharpoonright\beta\in G_{\beta}\\}.$

$\dot{\mathbb{P}}_{\beta,\gamma}$ denotes a name for
$\mathbb{P}_{\beta,\gamma}$ in $V$.

  4. $(iv)$

If $G$ is $\mathbb{P}$-generic and $\alpha<\beta$ then $G_{\alpha,\beta}$
denotes the projection of $G$ onto $\mathbb{P}_{\alpha,\beta}$.

There will be a number of instances were we need a structure to satsify a
sufficiently large fragment of $\mathrm{ZFC}$. For completeness, we make this
precise.

######  Definition 2.6.

Sufficiently much of $\mathrm{ZFC}$ is the fragment
$\mathrm{ZFC}^{-}+``\omega_{1}\text{ exists}"$. Here, $\mathrm{ZFC}^{-}$ is
$\mathrm{ZFC}$ without the powerset axiom and with the collection scheme
instead of the replacement scheme.

##  3 $\diamondsuit(\mathbb{B})$ and $\diamondsuit^{+}(\mathbb{B})$

We will introduce the combinatorial principle which will parameterize the main
iteration theorem. These are generalizations of the principles
$\diamondsuit(\omega_{1}^{{<}\omega})$ and
$\diamondsuit^{+}(\omega_{1}^{{<}\omega})$ isolated by Woodin [Woo10] in his
study of $\mathbb{Q}_{\mathrm{max}}$ [Woo10, Section 6.2]. Most results in
this Section are essentially due to Woodin and proven in [Woo10, Section 6.2].

######  Definition 3.1.

Suppose $\mathbb{B}\subseteq\omega_{1}$ is a forcing.

  1. $(i)$

We say that $f$ guesses $\mathbb{B}$-filters if $f$ is a function

$f\colon\omega_{1}\rightarrow H_{\omega_{1}}$

and for all $\alpha<\omega_{1}$, $f(\alpha)$ is a
$\mathbb{B}\cap\alpha$-filter333We consider the empty set to be a filter..

  2. $(ii)$

Suppose $\theta\geq\omega_{2}$ is regular and $X\prec H_{\theta}$ is an
elementary substructure. We say $X$ is $f\text{{-slim}}$444We use the
adjective “slim” for the following reason: An $f\text{-slim}$ $X\prec
H_{\theta}$ cannot be too fat compared to its height below $\omega_{1}$, i.e.
$\delta^{X}$. If $X\sqsubseteq Y\prec H_{\theta}$ and $Y$ is $f\text{-slim}$
then $X$ is $f\text{-slim}$ as well, but the converse can fail. if

    1. $(X.i)$

$X$ is countable,

    2. $(X.ii)$

$f,\mathbb{B}\in X$ and

    3. $(X.iii)$

$f(\delta^{X})$ is $\mathbb{B}\cap\delta^{X}$-generic over $M_{X}$.

######  Definition 3.2.

Let $\mathbb{B}\subseteq\omega_{1}$ be a forcing. $\diamondsuit(\mathbb{B})$
states that there is a function $f$ so that

  1. $(i)$

$f$ guesses $\mathbb{B}$-filters and

  2. $(ii)$

for any $b\in\mathbb{B}$ and regular $\theta\geq\omega_{2}$

$\displaystyle\\{X\prec H_{\theta}\mid X\text{ is }f\text{-slim}\wedge$
$\displaystyle b\in f(\delta^{X})\\}$

is stationary in $[H_{\theta}]^{\omega}$.

$\diamondsuit^{+}(\mathbb{B})$ is the strengthening of
$\diamondsuit(\mathbb{B})$ where $(ii)$ is replaced by:

  1. $(ii)^{+}$

For any regular $\theta\geq\omega_{2}$

$\\{X\prec H_{\theta}\mid X\text{ is }f\text{-slim}\\}$

contains a club of $[H_{\theta}]^{\omega}$. Moreover, for any $b\in\mathbb{B}$

$\\{\alpha<\omega_{1}\mid b\in f(\alpha)\\}$

is stationary.

We say that $f$ witnesses $\diamondsuit(\mathbb{B})$,
$\diamondsuit^{+}(\mathbb{B})$ respectively.

######  Remark 3.3.

Observe that if $f$ witnesses $\diamondsuit(\mathbb{B})$ and $\mathbb{B}$ is
separative then $\mathbb{B}$ can be “read off” from $f$: We have
$\mathbb{B}=\bigcup_{\alpha<\omega_{1}}f(\alpha)$ and for $b,c\in\mathbb{B}$,
$b\leq_{\mathbb{B}}c$ iff whenever $b\in f(\alpha)$ then $c\in f(\alpha)$ as
well. Thus, it is usually not necessary to mention $\mathbb{B}$.

We introduce some convenient shorthand notation.

######  Definition 3.4.

If $\mathbb{B}\subseteq\omega_{1}$ is a forcing, $f$ guesses
$\mathbb{B}$-filters and $b\in\mathbb{B}$ then

$S^{f}_{b}\coloneqq\\{\alpha<\omega_{1}\mid b\in f(\alpha)\\}.$

If $f$ is clear from context we will sometimes omit the superscript $f$.

Note that if $f$ witnesses $\diamondsuit(\mathbb{B})$, then $S^{f}_{b}$ is
stationary for all $b\in\mathbb{B}$. This is made explicit for
$\diamondsuit^{+}(\mathbb{B})$. This is exactly the technical strengthening
over Woodin’s definition of
$\diamondsuit(\omega_{1}^{{<}\omega}),\diamondsuit^{+}(\omega_{1}^{{<}\omega})$.
Lemma 3.11 shows that this strengthening is natural. Moreover, this implies

$\diamondsuit(\mathbb{B}\oplus\mathbb{C})\Rightarrow\diamondsuit(\mathbb{B})\wedge\diamondsuit(\mathbb{C})$

whenever $\mathbb{B},\mathbb{C}\subseteq\omega_{1}$ are forcings and
$\mathbb{B}\oplus\mathbb{C}$ is the disjoint union of $\mathbb{B}$ and
$\mathbb{C}$ coded into a subset of $\omega_{1}$. This becomes relevant in
Subsection LABEL:splitwitnessessubsection. Nonetheless, the basic theory of
these principles is not changed by a lot.

######  Definition 3.5.

If $f$ witnesses $\diamondsuit(\mathbb{B})$ and $\mathbb{P}$ is a forcing, we
say that $\mathbb{P}$ preserves $f$ if whenever $G$ is $\mathbb{P}$-generic
then $f$ witnesses $\diamondsuit(\mathbb{B})$ in $V[G]$.

We remark that if $f$ witnesses $\diamondsuit^{+}(\mathbb{B})$ then
“$\mathbb{P}$ preserves $f$” still only means that $f$ witnesses
$\diamondsuit(\mathbb{B})$ in $V^{\mathbb{P}}$.

Next, we define a variant of stationary sets related to a witness of
$\diamondsuit(\mathbb{B})$. Suppose $\theta\geq\omega_{2}$ is regular. Then
$S\subseteq\omega_{1}$ is stationary iff for any club
$\mathcal{C}\subseteq[H_{\theta}]^{\omega}$, there is some $X\in\mathcal{C}$
with $\delta^{X}\in S$. $f$-stationarity results from restricting to $f$-slim
$X\prec H_{\theta}$ only.

######  Definition 3.6.

Suppose $f$ guesses $\mathbb{B}$-filters.

  1. $(i)$

A subset $S\subseteq\omega_{1}$ is $f$-stationary iff whenever
$\theta\geq\omega_{2}$ is regular and
$\mathcal{C}\subseteq[H_{\theta}]^{\omega}$ is club then there is some
$f\text{-slim}$ $X\in\mathcal{C}$ with $\delta^{X}\in S$.

  2. $(ii)$

A forcing $\mathbb{P}$ preserves $f$-stationary sets iff any $f$-stationary
set is still $f$-stationary in $V^{\mathbb{P}}$.

We make use of $f$-stationarity only when $f$ witnesses
$\diamondsuit(\mathbb{B})$. However, with the above definition it makes sense
to talk about $f$-stationarity in a forcing extension before we know that $f$
has been preserved. Note that all $f$-stationary sets are stationary, but the
converse might fail, see Proposition LABEL:noplusprop. We will later see that
$f$-stationary sets are the correct replacement of stationary set in our
context. Most prominently this notion will be used in the definition of the
$\mathrm{MM}^{++}$-variant $\mathrm{MM}^{++}(f)$ we introduce in Subsection
LABEL:fmppandrelatedforcingaxiomssection. It will be useful to have an
equivalent formulation of $f$-stationarity at hand.

######  Proposition 3.7.

Suppose $f$ guesses $\mathbb{B}$-filters. The following are equivalent for any
set $S\subseteq\omega_{1}$:

  1. $(i)$

$S$ is $f$-stationary.

  2. $(ii)$

Whenever $\langle D_{\alpha}\mid\alpha<\omega_{1}\rangle$ is a sequence of
dense subsets of $\mathbb{B}$, the set

$\\{\alpha\in S\mid\forall\beta<\alpha\ f(\alpha)\cap
D_{\beta}\neq\emptyset\\}$

is stationary.

######  Proposition 3.8.

Suppose $f$ guesses $\mathbb{B}$-filters. The following are equivalent:

  1. $(i)$

$f$ witnesses $\diamondsuit(\mathbb{B})$.

  2. $(ii)$

$S^{f}_{b}$ is $f$-stationary for all $b\in\mathbb{B}$.

  3. $(iii)$

For any $b\in\mathbb{B}$ and sequence $\langle
D_{\alpha}\mid\alpha<\omega_{1}\rangle$ of dense subsets of $\mathbb{B}$,

$\\{\alpha\in S^{f}_{b}\mid\forall\beta<\alpha\ f(\alpha)\cap
D_{\beta}\neq\emptyset\\}$

is stationary.

We mention a handy corollary.

######  Corollary 3.9.

Suppose $f$ witnesses $\diamondsuit(\mathbb{B})$. Any forcing preserving
$f$-stationary sets preserves $f$.

######  Proposition 3.10.

Suppose $f$ guesses $\mathbb{B}$-filters. The following are equivalent:

  1. $(i)$

$f$ witnesses $\diamondsuit^{+}(\mathbb{B})$.

  2. $(ii)$

For any $b\in\mathbb{B}$, $S^{f}_{b}$ is stationary and all stationary sets
are $f$-stationary.

  3. $(iii)$

If $D$ is dense in $\mathbb{B}$ then

$\\{\alpha<\omega_{1}\mid\ f(\alpha)\cap D\neq\emptyset\\}$

contains a club and for all $b\in\mathbb{B}$, $S^{f}_{b}$ is stationary.

  4. $(iv)$

All countable $X\prec H_{\theta}$ with $f\in X$ and $\theta\geq\omega_{2}$
regular are $f\text{-slim}$ and moreover for all $b\in\mathbb{B}$, $S^{f}_{b}$
is stationary.

We will now give a natural equivalent formulation of
$\diamondsuit^{+}(\mathbb{B})$.

Witnesses of $\diamondsuit^{+}(\mathbb{B})$ are simply codes for regular
embeddings555Regular embeddings, also known as complete embeddings, are
embeddings between partial orders which preserve maximal antichains. of
$\mathbb{B}$ into $\mathrm{NS}_{\omega_{1}}^{+}$.

######  Lemma 3.11.

The following are equivalent:

  1. $(i)$

$\diamondsuit^{+}(\mathbb{B})$.

  2. $(ii)$

There is a regular embedding
$\eta\colon\mathbb{B}\rightarrow(\mathcal{P}(\omega_{1})/\mathrm{NS}_{\omega_{1}})^{+}$.

The argument above suggests the following definition.

######  Definition 3.12.

Suppose $f$ witnesses $\diamondsuit(\mathbb{B})$. We define

$\eta_{f}\colon\mathbb{B}\rightarrow(\mathcal{P}(\omega_{1})/\mathrm{NS}_{\omega_{1}})^{+}$

by $b\mapsto[S^{f}_{b}]_{\mathrm{NS}_{\omega_{1}}}$ and call $\eta_{f}$ the
embedding associated to $f$.

We will now show that $\diamondsuit(\mathbb{B})$ is consistent for any forcing
$\mathbb{B}\subseteq\omega_{1}$, even simultaneously so for all such
$\mathbb{B}$. We will deal with the consistency of
$\diamondsuit^{+}(\mathbb{B})$ in the next section.

######  Proposition 3.13.

Assume $\diamondsuit$. Then $\diamondsuit(\mathbb{B})$ holds for any poset
$\mathbb{B}\subseteq\omega_{1}$.

######  Corollary 3.14.

Suppose $\mathbb{B}\subseteq\omega_{1}$ is a forcing. Then
$\diamondsuit(\mathbb{B})$ holds in $V^{\mathrm{Add}(\omega_{1},1)}$.

In a number of arguments, we will deal with $f\text{-slim}$ $X\prec
H_{\theta}$ that become thicker over time, i.e. at a later stage there will be
some $f\text{-slim}$ $X\sqsubseteq Y\prec H_{\theta}$.

######  Definition 3.15.

In the above case of $X\sqsubseteq Y$, we denote the canonical elementary
embedding from $M_{X}$ to $M_{Y}$ by

$\mu_{X,Y}\colon M_{X}\rightarrow M_{Y}.$

$\mu_{X,Y}$ is given by $\pi_{Y}^{-1}\circ\pi_{X}$.

Usually, both $X$ and $Y$ will be $f\text{-slim}$. It is then possible to lift
$\mu_{X,Y}$.

######  Proposition 3.16.

Suppose $f$ guesses $\mathbb{B}$-filters and $X,Y\prec H_{\theta}$ are both
$f\text{-slim}$ with $X\sqsubseteq Y$. Then the lift of $\mu_{X,Y}$ to

$\mu_{X,Y}^{+}\colon M_{X}[f(\delta^{X})]\rightarrow M_{Y}[f(\delta^{X})]$

exists.

###### Proof.

As $\delta^{X}=\delta^{Y}$, the critical point of $\mu_{X,Y}$ is
${>}\delta^{X}$ (if it exists). As $\pi_{X}^{-1}(\mathbb{B})$ is a forcing of
size ${\leq}\omega_{1}^{M_{X}}=\delta^{X}$ and $f(\delta^{X})$ is generic over
both $M_{X}$ and $M_{Y}$, the lift exists. ∎

We consider the above proposition simultaneously as a definition: From now on
$\mu_{X,Y}^{+}$ will refer to this lift if it exists.

######  Definition 3.17.

Suppose $f$ witnesses $\diamondsuit(\mathbb{B})$. $\mathrm{NS}_{f}$ is the
ideal of $f$-nonstationary sets, that is

$\mathrm{NS}_{f}=\\{N\subseteq\omega_{1}\mid N\text{ is not
}f\text{-stationary}\\}.$

######  Lemma 3.18.

Suppose $f$ witnesses $\diamondsuit(\mathbb{B})$. $\mathrm{NS}_{f}$ is a
normal uniform ideal.

###  3.1 Miyamoto’s theory of nice iterations

For all our intents and purposes, it does not matter in applications how the
limit our iterations look like as long as we can prove a preservation theorem
about it.

We give a brief introduction to Miyamoto’s theory of nice iterations. These
iterations are an alternative to $\mathrm{RCS}$-itertaions when dealing with
the problem described above. In the proof of the iteration theorem for
($f$-)proper forcings, one constructs a generic condition $q$ by induction as
the limit of a sequence $\langle q_{n}\mid n<\omega\rangle$. In case of
($f$-)semiproper forcings, the length of the iteration may have uncountable
cofinality in $V$ but become $\omega$-cofinal along the way. In this case, a
sequence $\langle q_{n}\mid n<\omega\rangle$ with the desired properties
cannot be in $V$. The key insight to avoid this issue is that one should give
up linearity of this sequence and instead build a tree of conditions in the
argument. Nice supports follow the philosophy of form follows function, i.e.
its definitions takes the shape of the kind of arguments it is intended to be
involved in. The conditions allowed in a nice limit are represented by
essentially the kind of trees that this inductive nonlinear constructions we
hinted at above produces.

Miyamoto works with a general notion of iteration. For our purposes, we will
simply define nice iterations by induction on the length. Successor steps are
defined as usual, that is if
$\mathbb{P}_{\gamma}=\langle\mathbb{P}_{\alpha},\dot{\mathbb{Q}}_{\beta}\mid\alpha\leq\gamma,\beta<\gamma\rangle$
is a nice iteration of length $\gamma$ and $\dot{\mathbb{Q}}_{\gamma}$ is a
$\mathbb{P}_{\gamma}$-name for a forcing then
$\langle\mathbb{P}_{\alpha},\dot{\mathbb{Q}}_{\beta}\mid\alpha\leq\gamma+1,\beta\leq\gamma\rangle$
is a nice iteration of length $\gamma+1$ where
$\mathbb{P}_{\gamma+1}\cong\mathbb{P}_{\gamma}\ast\dot{\mathbb{Q}}_{\gamma}$.

######  Definition 3.19 (Miyamoto, [Miy02]).

Let
$\vec{\mathbb{P}}=\langle\mathbb{P}_{\alpha},\dot{\mathbb{Q}}_{\alpha}\mid\alpha<\gamma\rangle$
be a potential nice iteration, that is

  1. $(\vec{\mathbb{P}}.i)$

$\mathbb{P}_{\alpha}$ is a nice iteration of length $\alpha$ for all
$\alpha<\gamma$,

  2. $(\vec{\mathbb{P}}.ii)$

$\mathbb{P}_{\alpha+1}\cong\mathbb{P}_{\alpha}\ast\dot{\mathbb{Q}}_{\alpha}$
for all $\alpha+1<\gamma$ and

  3. $(\vec{\mathbb{P}}.iii)$

$\mathbb{P}_{\beta}\upharpoonright\alpha=\mathbb{P}_{\alpha}$ for all
$\alpha\leq\beta<\gamma$.

A nested antichain in $\vec{\mathbb{P}}$ is of the form

$(T,\langle T_{n}\mid n<\omega\rangle,\langle\mathrm{suc}^{n}_{T}\mid
n<\omega\rangle)$

so that for all $n<\omega$ the following hold666Usually, we identify the
nested antichain with $T$, its first component and write $\mathrm{suc}(a)$
instead of $\mathrm{suc}^{n}_{T}(a)$ if $n,T$ are clear from context.:

  1. $(i)$

$T=\bigcup_{n<\omega}T_{n}$.

  2. $(ii)$

$T_{0}=\\{a_{0}\\}$ for some
$a_{0}\in\bigcup_{\alpha<\gamma}\mathbb{P}_{\alpha}$.

  3. $(iii)$

$T_{n}\subseteq\bigcup_{\alpha<\gamma}\mathbb{P}_{\alpha}$ and
$\mathrm{suc}^{n}_{T}\colon T_{n}\rightarrow\mathcal{P}(T_{n+1})$.

  4. $(iv)$

For $a\in T_{n}$ and $b\in\mathrm{suc}^{n}_{T}(a)$,
$\mathrm{lh}(a)\leq\mathrm{lh}(b)$ and $b\upharpoonright\mathrm{lh}(a)\leq a$.

  5. $(v)$

For $a\in T_{n}$ and distinct $b,b^{\prime}\in\mathrm{suc}^{n}_{T}(a)$,
$b\upharpoonright\mathrm{lh}(a)\perp b^{\prime}\upharpoonright\mathrm{lh}(a)$.

  6. $(vi)$

For $a\in T_{n}$, $\\{b\upharpoonright\mathrm{lh}(a)\mid
b\in\mathrm{suc}^{n}_{T}(a)\\}$ is a maximal antichain below $a$ in
$\mathbb{P}_{\mathrm{lh}(a)}$.

  7. $(vii)$

$T_{n+1}=\bigcup\\{\mathrm{suc}^{n}_{T}(a)\mid a\in T_{n}\\}$.

Abusing notation, we will usually identify $T$ with

$(T,\langle T_{n}\mid n<\omega\rangle,\langle\mathrm{suc}^{n}_{T}\mid
n<\omega\rangle).$

If $b\in\mathrm{suc}^{n}_{T}(a)$ then we also write
$a=\mathrm{pred}^{n}_{T}(b)$. If $\beta<\gamma$ then $p\in\mathbb{P}_{\beta}$
is a mixture of $T$ up to $\beta$ iff for all $\alpha<\beta,\
p\upharpoonright\alpha$ forces

  1. $(p.i)$

$p(\alpha)=a_{0}(\alpha)$ if $\alpha<\mathrm{lh}(a_{0})$ and
$a_{0}\upharpoonright\alpha\in G_{\alpha}$,

  2. $(p.ii)$

$p(\alpha)=b(\alpha)$ if there are $a,b\in T$, $n<\omega$ with
$b\in\mathrm{suc}^{n}_{T}(a)$, $\mathrm{lh}(a)\leq\alpha<\mathrm{lh}(b)$ and
$b\upharpoonright\alpha\in G_{\alpha}$,

  3. $(p.iii)$

$p(\alpha)=\mathbbm{1}_{\dot{\mathbb{Q}}_{\alpha}}$ if there is a sequence
$\langle a_{n}\mid n<\omega\rangle$ with
$a_{n+1}\in\mathrm{suc}^{n}_{T}(a_{n})$, $\mathrm{lh}(a_{n})\leq\alpha$ and
$a_{n}\in G_{\mathrm{lh}(a_{n})}$ for all $n<\omega$.

If $\xi\leq\gamma$ is a limit, and $q$ is a sequence of length $\xi$ (may or
may not be in $\mathbb{P}_{\xi}$), $q$ is $(T,\xi)$-nice if for all
$\beta<\xi$, $q\upharpoonright\beta\in\mathbb{P}_{\beta}$ is a mixture of $T$
up to $\beta$.

We refer to [Miy02] for basic results on nested antichains and mixtures. We go
on and define nice limits.

######  Definition 3.20 (Miyamoto, [Miy02]).

Suppose
$\vec{\mathbb{P}}=\langle\mathbb{P}_{\alpha},\dot{\mathbb{Q}}_{\alpha}\mid\alpha<\gamma\rangle$
is a potential nice iteration of limit length $\gamma$. Let $\bar{\mathbb{P}}$
denote the inverse limit along $\vec{\mathbb{P}}$. The nice limit of
$\vec{\mathbb{P}}$ is defined as

$\mathrm{nicelim}(\vec{\mathbb{P}})=\\{p\in\bar{\mathbb{P}}\mid\exists T\text{
a nested antichain of }\vec{\mathbb{P}}\text{ and }p\text{ is
}(T,\gamma)\text{-nice}\\}.$

$\mathrm{nicelim}(\vec{\mathbb{P}})$ inherits the order from
$\bar{\mathbb{P}}$.

Finally, if
$\vec{\mathbb{P}}=\langle\mathbb{P}_{\alpha},\dot{\mathbb{Q}}_{\alpha}\mid\alpha<\gamma\rangle$
is a potential nice iteration then

$\langle\mathbb{P}_{\alpha},\dot{\mathbb{Q}}_{\beta}\mid\alpha\leq\gamma,\beta<\gamma\rangle$

is a nice iteration of length $\gamma$ where
$\mathbb{P}_{\gamma}=\mathrm{nicelim}(\vec{\mathbb{P}})$.

The fundamental property of nice iterations is:

######  Fact 3.21 (Miyamoto,[Miy02]).

Suppose
$\mathbb{P}=\langle\mathbb{P}_{\alpha},\dot{\mathbb{Q}}_{\beta}\mid\alpha\leq\gamma,\beta<\gamma\rangle$
is a nice iteration and $T$ is a nested antichain in $\mathbb{P}$. Then there
is a mixture of $T$.

######  Definition 3.22 (Miyamoto,[Miy02]).

Let
$\mathbb{P}=\langle\mathbb{P}_{\alpha},\dot{\mathbb{Q}}_{\beta}\mid\alpha\leq\gamma,\beta<\gamma\rangle$
be a nice iteration. If $S,T$ are nested antichains in $\mathbb{P}$ then
$S\angle\ T$ iff for any $n<\omega$ and $a\in S_{n}$ there is $b\in T_{n+1}$
with

$\mathrm{lh}(b)\leq\mathrm{lh}(a)\text{ and
}a\upharpoonright\mathrm{lh}(b)\leq b.$

######  Fact 3.23 (Miyamoto, [Miy02]).

Let
$\mathbb{P}=\langle\mathbb{P}_{\alpha},\dot{\mathbb{Q}}_{\beta}\mid\alpha\leq\gamma,\beta<\gamma\rangle$
be a nice iteration of limit length $\gamma$. Suppose that

  1. $(i)$

$T$ is a nested antichain in $\mathbb{P}$,

  2. $(ii)$

$p$ is a mixture of $T$ and $s\in\mathbb{P}$,

  3. $(iii)$

$r\in T_{1}$,

  4. $(iv)$

$s\leq r^{\frown}p\upharpoonright[\mathrm{lh}(r),\gamma)$ and

  5. $(v)$

$A\subseteq\gamma$ is cofinal.

Then there is a nested antichain $S$ in $\mathbb{P}$ with

  1. $(a)$

$s$ is a mixture of $S$,

  2. $(b)$

If $S_{0}=\\{c\\}$ then $\mathrm{lh}(r)\leq\mathrm{lh}(c)\in A$ and
$c\upharpoonright\mathrm{lh}(r)\leq r$ and

  3. $(c)$

$S\angle\ T$.

The following describes the tool we use to construct conditions.

######  Definition 3.24 (Miyamoto, [Miy02]).

Let
$\mathbb{P}=\langle\mathbb{P}_{\alpha},\dot{\mathbb{Q}}_{\beta}\mid\alpha\leq\gamma,\beta<\gamma\rangle$
be a nice iteration of limit length $\gamma$. A fusion structure in
$\mathbb{P}$ is

$T,\langle p^{(a,n)},\ T^{(a,n)}\mid n<\omega,\ a\in T_{n}\rangle$

where

  1. $(i)$

$T$ is a nested antichain in $\mathbb{P}$

and for all $n<\omega$ and $a\in T_{n}$

  1. $(ii)$

$T^{(a,n)}$ is a nested antichain in $\mathbb{P}$,

  2. $(iii)$

$p^{(a,n)}\in\mathbb{P}$ is a mixture of $T^{(a,n)}$,

  3. $(iv)$

$a\leq p^{(a,n)}\upharpoonright\mathrm{lh}(a)$ and if $T_{0}^{(a,n)}=\\{c\\}$
then $\mathrm{lh}(a)=\mathrm{lh}(c)$ and

  4. $(v)$

for any $b\in\mathrm{suc}_{T}^{n}(a)$, $T^{(b,n+1)}\angle\ T^{(a,n)}$, thus
$p^{(b,n+1)}\leq p^{(a,n)}$.

If $q\in\mathbb{P}$ is a mixture of $T$ then $q$ is called a fusion of the
fusion structure.

######  Fact 3.25 (Miyamoto, [Miy02]).

Let
$\mathbb{P}=\langle\mathbb{P}_{\alpha},\dot{\mathbb{Q}}_{\beta}\mid\alpha\leq\gamma,\beta<\gamma\rangle$
be a nice iteration of limit length $\gamma$. If $q\in\mathbb{P}$ is a fusion
of a fusion structure

$T,\langle p^{(a,n)},\ T^{(a,n)}\mid n<\omega,\ a\in T_{n}\rangle$

and $G$ is $\mathbb{P}$-generic with $q\in G$ then the following holds in
$V[G]$: There is a sequence $\langle a_{n}\mid n<\omega\rangle$ so that for
all $n<\omega$

  1. $(i)$

$a_{0}\in T_{0}$,

  2. $(ii)$

$a_{n}\in G_{\mathrm{lh}(a_{n})}$,

  3. $(iii)$

$a_{n+1}\in\mathrm{suc}_{T}^{n}(a_{n})$ and

  4. $(iv)$

$p^{(a_{n},n)}\in G$.

We mention one more convenient fact:

######  Fact 3.26 (Miyamoto, [Miy03]).

Suppose $\kappa$ is an inaccessible cardinal,
$\mathbb{P}=\langle\mathbb{P}_{\alpha},\dot{\mathbb{Q}}_{\beta}\mid\alpha\leq\kappa,\beta<\kappa\rangle$
is a nice iteration so that

  1. $(i)$

$|\mathbb{P}_{\alpha}|<\kappa$ for all $\alpha<\kappa$ and

  2. $(ii)$

$\mathbb{P}$ preserves $\omega_{1}$.

Then $\mathbb{P}$ is $\kappa$-c.c..

Miyamoto proves this for so called simple iterations of semiproper forcings.
The proof works just as well for nice iterations of semiproper forcings and
finally the proof can be made to work with assuming only $\mathbb{P}$
preserves $\omega_{1}$ instead of $\mathbb{P}$ being a semiproper iteration.

##  4 The Iteration Theorem

The full main theorem we are going to prove is the following.

######  Theorem 4.1.

Suppose $f$ witnesses $\diamondsuit(\mathbb{B})$ and
$\mathbb{P}=\langle\mathbb{P}_{\alpha},\dot{\mathbb{Q}}_{\beta}\mid\alpha\leq\gamma,\beta<\gamma\rangle$
is a nice iteration of $f$-preserving forcings. Suppose that

  1. $(\mathbb{P}.i)$

$\Vdash_{\mathbb{P}_{\alpha+2}}\mathrm{SRP}$ for all $\alpha+2\leq\gamma$ and

  2. $(\mathbb{P}.ii)$

$\Vdash_{\mathbb{P}_{\alpha}}``\dot{\mathbb{Q}}_{\alpha}\text{ preserves
}f\text{-stationary sets from }\bigcup_{\beta<\alpha}V[\dot{G}_{\beta}]$.

Then $\mathbb{P}$ preserves $f$.

Note that if $\mathbb{B}$ is the trivial forcing $\\{\mathbbm{1}\\}$ and we
take $f$ to be the witness of $\diamondsuit^{+}(\mathbb{B})$ with
$f(\alpha)=\\{\mathbbm{1}\\}$ for all $0<\alpha<\omega_{1}$, then we recover
the special case mentioned in the introduction.

So what is the basic idea? For the moment, let us assume that $f$ is the
trivial witness of $\diamondsuit(\\{\mathbbm{1}\\})$ above for simplicity. As
always, we want to imitate the argument of the mother of all iteration
theorems, the iteration theorem for proper forcings. Suppose we have a full
support iteration

$\mathbb{P}=\langle\mathbb{P}_{n},\dot{\mathbb{Q}}_{m}\mid
n\leq\omega,m<\omega\rangle$

and for the moment assume only that

$\Vdash_{\mathbb{P}_{n}}``\dot{\mathbb{Q}}_{n}\text{ preserves }\omega_{1}".$

We try to motivate some additional reasonable constraints imply $\mathbb{P}$
to be $\omega_{1}$-preserving. For the moment, we try to consider Shelah’s
argument as a game: In the beginning there some countable $X\prec H_{\theta}$
as well as $p_{0}\in X\cap\mathbb{P}$. The argument proceeds as follows: In
round $n$, we have already constructed a $(X,\mathbb{P}_{n})$-semigeneric
condition $q_{n}\in\mathbb{P}\upharpoonright n$ and have

$q_{n}\Vdash\dot{p}_{n}\upharpoonright
n\in\dot{G}_{n}\cap\check{X}[\dot{G}_{n}].$

Next, our adversary hits us with a dense subset $D\subseteq\mathbb{P}$ in $X$
and we must find $\dot{p}_{n+1}\in V^{\mathbb{P}_{n}}$ and some
$(X,\mathbb{P}_{n+1})$-semigeneric $q_{n+1}$ with $q_{n+1}\upharpoonright
n=q_{n}$ and777Here, we consider $\dot{p}_{n}$ also as a
$\mathbb{P}_{n+1}$-name.

$q_{n+1}\Vdash\dot{p}_{n+1}\in\check{D}\wedge p_{n+1}\upharpoonright
n+1\in\dot{G}_{n+1}\cap\check{X}[\dot{G}_{n+1}].$

Our job is to survive this game for $\omega$-many steps. If we have a winning
strategy then we can find a $(X,\mathbb{P})$-semigeneric condition, so in
particular $\mathbb{P}$ preserves $\omega_{1}$.

Destroying stationarity makes it significantly more difficult to survive the
above game: Suppose for example that

$p_{0}(0)\Vdash\check{S}\in\mathrm{NS}_{\omega_{1}}$

for some $S\in X$ with $\delta^{X}\in S$. Then there is no hope of finding a
$(X,\mathbb{P}_{1})$-semigeneric condition $q$ with $q\leq
p_{0}\upharpoonright 1$. Hence, we must already be careful with what $X$ we
start the game. This leads us to the following definitions.

######  Definition 4.2.

Suppose $\theta$ is sufficiently large and regular, $X\prec H_{\theta}$ is
countable. If $I$ is an ideal on $\omega_{1}$, we say that $X$ respects $I$ if
for all $A\in I\cap X$ we have $\delta^{X}\notin A$.

Note that all countable $X\prec H_{\theta}$ respect $\mathrm{NS}_{\omega_{1}}$
and countable $Y\prec H_{\theta}$ with $f\in Y$ respects $\mathrm{NS}_{f}$ if
and only if $Y$ is $f\text{-slim}$.

######  Definition 4.3.

Suppose $\mathbb{P}$ is a forcing and $\dot{I}\in V^{\mathbb{P}}$ is a name
for an ideal on $\omega_{1}$. For $p$ in $\mathbb{P}$, we denote the partial
evaluation of $\dot{I}$ by $p$ by

$\dot{I}^{p}\coloneqq\\{S\subseteq\omega_{1}\mid
p\Vdash\check{S}\in\dot{I}\\}.$

Back to the discussion, we need to start with an $X$ so that $X$ respects
$\dot{I}^{p_{0}\upharpoonright 1}$ where $\dot{I}$ is a name for the
nonstationary ideal. This gives us a shot at getting past the first round.
Luckily, there are enough of these $X$.

######  Definition 4.4.

Let $A$ be an uncountable set with $\omega_{1}\subseteq A$ and $I$ a normal
uniform ideal on $\omega_{1}$. Then $\mathcal{S}\subseteq[A]^{\omega}$ is
projective $I$-positive if for any $S\in I^{+}$ the set

$\\{X\in\mathcal{S}\mid\delta^{X}\in S\\}$

is stationary in $[A]^{\omega}$.

######  Proposition 4.5.

Suppose $\theta$ is sufficiently large and regular. Let $I$ be a normal
uniform ideal on $\omega_{1}$. Then

$\mathcal{S}=\\{X\in[H_{\theta}]^{\omega}\mid X\prec H_{\theta}\text{ respects
}I\\}$

is projective $I$-positive.

###### Proof.

Let $\mathcal{C}$ be a club in $[H_{\theta}]^{\omega}$ and assume that all
elements of $\mathcal{C}$ are elementary substructures of $H_{\theta}$ and
contain $I$ as an element. Let

$\vec{X}\coloneqq\langle X_{\alpha}\mid\alpha<\omega_{1}\rangle$

be a continuous increasing chain of elements in $\mathcal{C}$. Let
$X\coloneqq\bigcup_{\alpha<\omega_{1}}X_{\alpha}$ and let

$\vec{A}\coloneqq\langle A_{\alpha}\mid\alpha<\omega_{1}\rangle$

be an enumeration of $X\cap I$. Let $C\subseteq\omega_{1}$ be the set of
$\alpha$ so that

  1. $(C.i)$

$\delta^{X_{\alpha}}=\alpha$ and

  2. $(C.ii)$

$\vec{A}\upharpoonright\alpha$ is an enumeration of $X_{\alpha}\cap I$

and note that $C$ is club. Let
$A=\bigtriangledown_{\alpha<\omega_{1}}I_{\alpha}$. As $I$ is normal, $A\in
I$. Then $C-A$ is a complement of a set in $I$ and for any $\alpha\in C-A$ we
have

$\delta^{X_{\alpha}}=\alpha\notin I_{\beta}$

for all $\beta<\alpha$. Hence $X_{\alpha}\in\mathcal{S}\cap\mathcal{C}$. ∎

Of course, the problem continues. What if we have found a suitable $q_{1}$ and
now we work in $V[G_{1}]$ with $q_{1}\in G_{1}$. At the very least, we need
that $X[G_{1}]$ respects $\dot{I}^{p_{0}\upharpoonright[1,2)}$, where
$\dot{I}$ is now a $\mathbb{P}_{1,2}$-name for the nonstationary ideal.
Ensuring this is a matter of being able to pick the right $q_{1}$ to begin
with. This motivates the following class of forcings.

######  Definition 4.6.

We say that a forcing $\mathbb{P}$ is respectful if $\mathbb{P}$ preserves
$\omega_{1}$ and the following is true: Whenever

  * •

$\theta$ is sufficiently large and regular,

  * •

$X\prec H_{\theta}$ is countable with $\mathbb{P}\in X$,

  * •

$\dot{I}\in X$ is a $\mathbb{P}$-name for a normal uniform ideal and

  * •

$p\in\mathbb{P}\cap X$

then exactly one of the following holds:

  1. $(\mathrm{Res}.i)$

Either there is some $(X,\mathbb{P})$-semigeneric $q\leq p$ so that

$q\Vdash``\check{X}[\dot{G}]\text{ respects }\dot{I}"$

or

  2. $(\mathrm{Res}.ii)$

$X$ does not respect $\dot{I}^{p}$.

Roughly, this condition states that we can find a $\mathbb{P}$-generic filter
$G$ with $p\in G$ so that $X\sqsubseteq X[G]$ respects $\dot{I}^{G}$ as long
as there is no obvious obstruction to it.

######  Remark 4.7.

If $\mathbb{P}$ is respectful and preserves stationary sets then $\mathbb{P}$
is semiproper. However, the converse is not true in general. Similarly, a
respectful $f$-stationary set preserving forcing is $f$-semiproper, which
follows from plugging in a name for $\mathrm{NS}_{f}$ as $\dot{I}$ in the
definition of respectfulness.

We require888This excludes the first counterexample due to Shelah, but not yet
all the counterexamples of the second kind. now that

$\Vdash_{\mathbb{P}_{n}}``\dot{\mathbb{Q}}_{n}\text{ is respectful}"$

for all $n<\omega$. We then aim to make sure (assuming $\dot{p}_{n+1}$ is
already defined) to find $q_{n+1}$ in round $n$ so that in addition to the
prior constraints,

$q_{n+1}\Vdash``\check{X}[\dot{G}_{n+1}]\text{ respects }\dot{I}"$

where $\dot{I}$ is a $\mathbb{P}_{n+1}$ name for the ideal of sets forced to
be nonstationary by $\dot{p}_{n+1}(n+1)$. Consider $\dot{I}$ as a
$\mathbb{P}_{n}$-name $\ddot{I}$ for a $\dot{\mathbb{Q}}_{n}$-name. By
respectfulness, this reduces to avoiding an instance of the “bad case”
$(\mathrm{Res}.ii)$, namely we should make sure that whenever $G_{n}$ is
$\mathbb{P}_{n}$-generic with $q_{n}\in G_{n}$ then

$X[G_{n}]\text{ respects }\left(\ddot{I}^{G_{n}}\right)^{p_{n+1}(n+1)}$

where $p_{n+1}=\dot{p}_{n+1}^{G_{n+1}}$. he next key insight is that this
reduces to

$``X[G_{n}]\text{ respects }J\coloneqq\\{S\subseteq\omega_{1}\mid
p_{n+1}(n)\Vdash\check{S}\in\mathrm{NS}_{\omega_{1}}\\}"$

which we have (almost)999We made sure of this if $p_{n+1}$ is replaced by
$p_{n}$ in the definition of $J$, we ignore this small issue for now. already
justified inductively, assuming $\dot{\mathbb{Q}}_{n+1}$ only kills new
stationary sets: Our final requirement101010It is readily seen that this
eliminates the counterexamples of the second kind. is that

$\Vdash_{\mathbb{P}_{n+1}}``\dot{\mathbb{Q}}_{n+1}\text{ preserves stationary
sets which are in }V[\dot{G}_{n}]"$

for all $n<\omega$. The point is that trivially
$\left(\ddot{I}^{G_{n}}\right)^{p_{n+1}(n)}$ only contains sets in $V[G_{n}]$,
so all such sets will be preserved by $\dot{\mathbb{Q}}_{n+1}$. The sets that
are killed are then already killed in the extension by
$\dot{\mathbb{Q}}_{n}^{G_{n}}$.
Modulo some details we have shown the following.

######  Theorem 4.8.

Suppose $\mathbb{P}=\langle\mathbb{P}_{n},\dot{\mathbb{Q}}_{m}\mid
n\leq\omega,m<\omega\rangle$ is a full support iteration so that

  1. $(\mathbb{P}.i)$

$\Vdash_{\mathbb{P}_{n}}\dot{\mathbb{Q}}_{n}\text{ is respectful}$ and

  2. $(\mathbb{P}.ii)$

$\Vdash_{\mathbb{P}_{n+1}}\dot{\mathbb{Q}}_{n+1}\text{ preserves stationary
sets which are in }V[\dot{G}_{n}]"$

for all $n<\omega$. Then $\mathbb{P}$ does not collapse $\omega_{1}$.

Two issues arise when generalizing this to longer iterations. The first issue
is the old problem that new relevant indices may appear along the iteration in
the argument, which we deal with by using nice supports. The second problem is
that it seemingly no longer suffices that each iterand individually is
respectful. For longer iterations, say of length $\gamma$, the argument then
requires that

$\Vdash_{\alpha}``\dot{\mathbb{P}}_{\alpha,\beta}\text{ is respectful}"$

for sufficiently many $\alpha<\beta<\gamma$. This is problematic as we will
not prove an iteration theorem of any kind for respectful forcings111111Indeed
it seems that no useful iteration theorem for respectful forcings is provable
in $\mathrm{ZFC}$, see Subsection LABEL:disrespectfulforcingsubsection.. This
is where we take out the sledgehammer.

######  Definition 4.9.

$(\ddagger)$ holds if and only if all $\omega_{1}$-preserving forcings are
respectful.

######  Lemma 4.10.

$\mathrm{SRP}$ implies $(\ddagger)$.

###### Proof.

Let $\mathbb{P}$, $\theta$, $\dot{I}$, $p$ be as Definition 4.6. It is easy to
see that $(\mathrm{Res}.i)$ and $(\mathrm{Res}.ii)$ cannot hold
simultaneously. It is thus enough to prove that one of them holds. Let
$\lambda$ be regular, $2^{|\mathbb{P}|}<\lambda<\theta$ and $\lambda\in X$ and
consider the set

$\displaystyle\mathcal{S}=\\{Y\in[H_{\lambda}]^{\omega}\mid$ $\displaystyle
Y\prec H_{\lambda}\wedge\neg(\exists q\leq p\ q\text{ is }$
$\displaystyle(Y,\mathbb{P})\text{-semigeneric}\text{ and
}q\Vdash``\check{Y}[\dot{G}]\text{ respects }\dot{I}")\\}.$

By $\mathrm{SRP}$, there is a continuous increasing elementary chain

$\vec{Y}=\langle Y_{\alpha}\mid\alpha<\omega_{1}\rangle$

so that

  1. $(\vec{Y}.i)$

$\mathbb{P},p,\dot{I}\in Y_{0}$ and

  2. $(\vec{Y}.ii)$

for all $\alpha<\omega_{1}$, either $Y_{\alpha}\in\mathcal{S}$ or there is no
$Y_{\alpha}\sqsubseteq Z\prec H_{\theta}$ with $Z\in\mathcal{S}$.

Let $S=\\{\alpha<\omega_{1}\mid Y_{\alpha}\in\mathcal{S}\\}$.

######  Claim 4.11.

$p\Vdash\check{S}\in\dot{I}$.

###### Proof.

Let $G$ be generic with $p\in G$ and let $I=\dot{I}^{G}$. Assume toward a
contradiction that $S$ is $I$-positive. Note that $\langle
Y_{\alpha}[G]\mid\alpha<\omega_{1}\rangle$ is a continuous increasing sequence
of elementary substructure of $H_{\theta}^{V[G]}$. Hence there is a club $C$
of $\alpha$ so that for $\alpha\in C$

$\delta^{Y_{\alpha}}=\delta^{Y_{\alpha}[G]}=\alpha$

and thus there is a $(Y_{\alpha},\mathbb{P})$-semigeneric condition $q\leq p$,
$q\in G$. Hence by definition of $S$, for any $\alpha\in S\cap C$, we may find
some $N_{\alpha}\in I\cap Y_{\alpha}[G]$ so that $\delta^{Y_{\alpha}}\in
N_{\alpha}$. By normality of $I$, there is some $I$-positive $T\subseteq S\cap
C$ and some $N$ so that $N=N_{\alpha}$ for all $\alpha\in T$. But then for
$\alpha\in T$, we have

$\alpha=\delta^{Y}\in N$

so that $T\subseteq N$. But $N\in I$, contradiction. ∎

Thus if $\delta^{X}\in S$, then $S$ witnesses $(\mathrm{Res}.ii)$ to hold.
Otherwise, $\delta^{X}\notin S$. Note that
$\delta^{Y_{\delta^{X}}}=\delta^{X}$ as $\vec{Y}\in X$. We find that
$Y_{\delta^{X}}\sqsubseteq X\cap H_{\lambda}\prec H_{\lambda}$. Thus, $X\cap
H_{\lambda}\notin\mathcal{S}$, so that there must be some $q\leq p$ that is
$(X\cap H_{\lambda},\mathbb{P})$-semigeneric and

$q\Vdash``(\widecheck{X\cap H_{\lambda}})[\dot{G}]\text{ respects }\dot{I}".$

This $q$ witnesses that $(\mathrm{Res}.i)$ holds. ∎

We will get around this second issue by forcing $\mathrm{SRP}$ often along the
iteration. Remember that what we really care about is preserving a witness $f$
of $\diamondsuit(\mathbb{B})$ along an iteration of $f$-preserving forcings,
so fix such an $f$ now. It will be quite convenient to introduce some short
hand notation.

######  Definition 4.12.

Suppose $\mathbb{P}$ is a forcing and $p\in\mathbb{P}$. Then we let
$I^{\mathbb{P}}_{p}$ denote $\dot{I}^{p}$ where $\dot{I}$ is a
$\mathbb{P}$-name for $\mathrm{NS}_{f}$. That is

$I^{\mathbb{P}}_{p}\coloneqq\\{S\subseteq\omega_{1}\mid
p\Vdash\check{S}\in\mathrm{NS}_{f}\\}.$

######  Definition 4.13.

Suppose $f$ witnesses $\diamondsuit(\mathbb{B})$. An $f$-ideal is an ideal $I$
on $\omega_{1}$ so that

  1. $(i)$

whenever $S\in I^{+}$ and $\langle D_{i}\mid i<\omega_{1}\rangle$ is a
sequence of dense subsets of $\mathbb{B}$, then

$\\{\alpha\in S\mid\forall\beta<\alpha\ f(\alpha)\cap
D_{\beta}\neq\emptyset\\}\in I^{+}$

  2. $(ii)$

and $S^{f}_{b}\in I^{+}$ for all $b\in\mathbb{B}$.

Recall that $\mathrm{NS}_{f}$ is clearly an $f$-ideal and it is normal and
uniform by Lemma 3.18.

######  Proposition 4.14.

Suppose $\mathbb{P}$ is a forcing that preserves $f$ and $p\in\mathbb{P}$.
Then $I^{\mathbb{P}}_{p}$ is a normal uniform $f$-ideal.

We leave the proof to the reader. The next Lemma gives us a criterion that
guarantees the relevant witness $f$ of $\diamondsuit(\mathbb{B})$ to be
preserved. We first introduce the notion of a $f$-semigeneric condition.

######  Definition 4.15.

Suppose $f$ witnesses $\diamondsuit(\mathbb{B})$, $\mathbb{P}$ is a forcing,
$\theta$ is sufficiently large and $X\prec H_{\theta}$ is a $f\text{-slim}$
elementary substructure of $H_{\theta}$ with $\mathbb{P}\in X$. A condition
$p\in\mathbb{P}$ is called $(X,\mathbb{P},f)$-semigeneric if $p$ is
$(X,\mathbb{P})$-semigeneric and

$p\Vdash\check{X}[\dot{G}]\text{ is }\check{f}\text{-slim}.$

######  Lemma 4.16.

Suppose $f$ witnesses $\diamondsuit(\mathbb{B})$ and $\mathbb{P}$ is a forcing
with the following property: For any sufficiently large regular $\theta$ and
$p\in\mathbb{P}$ there is a normal uniform $f$-ideal $I$ so that

$\\{X\in[H_{\theta}]^{\omega}\mid X\prec H_{\theta}\wedge\mathbb{P},p\in
X\wedge\exists q\leq p\ q\text{ is }(X,\mathbb{P},f)\text{-semigeneric}\\}$

is projective $I$-positive. Then $\mathbb{P}$ preserves $f$.

###### Proof.

Assume $p\in\mathbb{P}$, $\theta$ is sufficiently large and regular. Let
$b\in\mathbb{B}$,

$\vec{\dot{D}}=\langle\dot{D}_{\alpha}\mid\alpha<\omega_{1}\rangle$

be a sequence of $\mathbb{P}$-names for dense subsets of $\mathbb{B}$ and
$\dot{C}$ a $\mathbb{P}$-name for a club in $\omega_{1}$. We will find $q\leq
p$ so that

$q\Vdash\exists\alpha\in S^{\check{f}}_{b}\cap\dot{C}\forall\beta<\alpha\
\check{f}(\alpha)\cap\dot{D}_{\beta}\neq\emptyset.$

By our assumption, there is some normal uniform $f$-ideal $I$ so that

$\\{X\in[H_{\theta}]^{\omega}\mid X\prec H_{\theta}\wedge\mathbb{P},p\in
X\wedge\exists q\leq p\ q\text{ is }(X,\mathbb{P},f)\text{-semigeneric}\\}$

is projective $I$-positive. It follows that we can find some countable $X\prec
H_{\theta}$ so that

  1. $(X.i)$

$\mathbb{P},p,\vec{\dot{D}},\dot{C}\in X$ as well as

  2. $(X.ii)$

$b\in f(\delta^{X})$

and some $q\leq p$ that is $(X,\mathbb{P},f)$-semigeneric. If $G$ is then any
$\mathbb{P}$-generic with $q\in G$, we have

$X\sqsubseteq X[G]\text{ is }f\text{-slim}$

and hence $\delta^{X}\in\dot{C}^{G}$ as well as

$\forall\beta<\delta^{X}\ f(\delta^{X})\cap\dot{D}^{G}_{\beta}\neq\emptyset.$

∎

We also need to resolve a small issue that we glossed over in the sketch of a
proof of Theorem 4.8.

######  Lemma 4.17.

Suppose $f$ witnesses $\diamondsuit(\mathbb{B})$. Further assume that

  * •

$\mathbb{P}$ is a respectful, $f$-preserving forcing and $p\in\mathbb{P}$,

  * •

$\theta$ is sufficiently large and regular,

  * •

$X\prec H_{\theta}$ is countable, respects $I^{\mathbb{P}}_{p}$ and
$\mathbb{P},p\in X$ and

  * •

$M_{X}[f(\delta^{X})]\models``D\text{ is dense below }\pi_{X}^{-1}(p)\text{ in
}\pi_{X}^{-1}(\mathbb{P})"$.

Then there are $Y,q$ with

  1. $(i)$

$X\sqsubseteq Y\prec H_{\theta}$ is countable,

  2. $(ii)$

$q\leq p$,

  3. $(iii)$

$Y$ respects $I^{\mathbb{P}}_{q}$, in particular $Y$ is $f\text{-slim}$ and

  4. $(iv)$

$q\in\pi_{Y}[\mu_{X,Y}^{+}(D)]$.

###### Proof.

We may assume that $X$ is an elementary substructure of

$\mathcal{H}\coloneqq(H_{\theta};\in,\unlhd)$

where $\unlhd$ is a wellorder of $H_{\theta}$. As $\mathbb{P}$ is respectful
and $X$ respects $I^{\mathbb{P}}_{p}$, there is a $(X,\mathbb{P})$-semigeneric
condition $r\leq p$ so that

$r\Vdash``\check{X}[\dot{G}]\text{ respects }\mathrm{NS}_{\check{f}}"$

i.e. $r$ is $(X,\mathbb{P},f)$-semigeneric. Let $G$ be $\mathbb{P}$-generic
with $r\in G$. Then $X[G]$ is $f\text{-slim}$. Let $Z=X[G]\cap V$, note that
$\mu_{X,Z}^{+}$ exists by Proposition 3.16. Now there is thus some $q\leq p$,
$q\in G$ with

$q\in\pi_{Z}[\mu_{X,Z}^{+}(D)].$

Finally, note that $q$ and
$Y\coloneqq\mathrm{Hull}^{\mathcal{H}}(X\cup\\{q\\})$ have the desired
properties. ∎

###### Proof of Theorem 4.1.

Let
$\mathbb{P}=\langle\mathbb{P}_{\alpha},\dot{\mathbb{Q}}_{\beta}\mid\alpha\leq\gamma,\beta<\gamma\rangle$
be an iteration of $f$-preserving forcings which preserve old $f$-stationary
sets and forces $\mathrm{SRP}$ at successor steps. We may assume inductively
that $\mathbb{P}_{\alpha}$ preserves $f$ for all $\alpha<\gamma$. The
successor step is trivial, so we may restrict to $\gamma\in\mathrm{Lim}$. Note
that we may further assume that $(\ddagger)$ holds in $V$, otherwise we could
work in $V^{\mathbb{P}_{1}}$. Let $p\in\mathbb{P}$ and let $I\coloneqq
I^{\mathbb{Q}_{0}}_{p(0)}$. $I$ is a normal uniform $f$-ideal by Proposition
4.14. Now let $\theta$ be sufficiently large and regular, $X\prec H_{\theta}$
countable with

  1. $(X.i)$

$\mathbb{P},p,f\in X$ and

  2. $(X.ii)$

$X$ respects $I$.

By Proposition 4.5 and Lemma 4.16, it suffices to find $q\leq p$ that is
$(X,\mathbb{P},f)$-semigeneric. Note that $X$ is $f\text{-slim}$ as $I$ is a
$f$-ideal. Let

$h\colon\omega\rightarrow\omega\times\omega$

be a surjection with $i\leq n$ whenever $h(n)=(i,j)$.
Let $\delta$ denote $\delta^{X}$. We will construct a fusion structure

$T,\langle p^{(a,n)},T^{(a,n)}\mid a\in T_{n},\ n<\omega\rangle$

in $\mathbb{P}$ as well as names

$\left\langle\dot{X}^{(a,n)},\dot{Z}^{(a,n)}\left(\dot{D}_{j}^{(a,n)}\right)_{j<\omega},\dot{I}^{(a,n)}\mid
a\in T_{n},n<\omega\right\rangle$

so that for any $n<\omega$ and $a\in T_{n}$

  1. $(F.i)$

$T_{0}=\\{\mathbbm{1}\\}$, $p^{(\mathbbm{1},0)}=p$,
$\dot{X}^{(\mathbbm{1},0)}=\check{X}$, $\dot{I}^{(\mathbbm{1},0)}=\check{I}$,

  2. $(F.ii)$

$T^{(\mathbbm{1},0)}\in X$ is a nested antichain that $p$ is a mixture of with
$T_{0}^{(\mathbbm{1},0)}=\\{\mathbbm{1}\\}$,

  3. $(F.iii)$

$a\Vdash_{\mathrm{lh}(a)}\dot{Z}^{(a,n)}=\dot{X}^{(a,n)}\cap V$,

  4. $(F.iv)$

$\left(\dot{D}_{j}^{(a,n)}\right)_{j<\omega}$ is forced by $a$ to be an
enumeration of all dense subsets of
$\pi_{\dot{Z}^{(a,n)}}^{-1}(\check{\mathbb{P}})$ in

$M_{\dot{Z}^{(a,n)}}\left[\widecheck{f(\delta)}\right],$

  5. $(F.v)$

$a\leq p^{(a,n)}\upharpoonright\mathrm{lh}(a)$,

  6. $(F.vi)$

$\mathrm{lh}(a)$ is not a limit ordinal,

  7. $(F.vii)$

$a\Vdash_{\mathrm{lh}(a)}\check{p}^{(a,n)},\check{T}^{(a,n)},\dot{G}_{\mathrm{lh}(a)}\in\dot{X}^{(a,n)}$,

  8. $(F.viii)$

$a\Vdash_{\mathrm{lh}(a)}\dot{I}^{(a,n)}=I^{\dot{\mathbb{Q}}_{\mathrm{lh}(a)}}_{\check{p}^{(a,n)}(\mathrm{lh}(a))}$
and

  9. $(F.ix)$

$a\Vdash``\check{X}\sqsubseteq\dot{X}^{(a,n)}\prec
H_{\check{\theta}}^{V[\dot{G}_{\mathrm{lh}(a)}]}\text{ is countable and
respects }\dot{I}^{(a,n)}"$.

Moreover, for any $b\in\mathrm{suc}^{n}_{T}(a)$

  1. $(F.x)$

$b\upharpoonright\mathrm{lh}(a)\Vdash_{\mathrm{lh}(a)}``\check{p}^{(b,n+1)},\check{T}^{(b,n+1)}\in\dot{X}^{(a,n)},\text{
in particular
}\mathrm{lh}(\check{b}),\mathbb{P}_{\mathrm{lh}(\check{a}),\mathrm{lh}(\check{b})}\in\dot{X}^{(a,n)}$”,

  2. $(F.xi)$

$b\Vdash_{\mathrm{lh}(b)}\dot{X}^{(a,n)}[\dot{G}_{\mathrm{lh}(a),\mathrm{lh}(b)}]\sqsubseteq\dot{X}^{(b,n+1)}$
and

  3. $(F.xii)$

if $h(n)=(i,j)$ and $c=\mathrm{pred}_{T}^{i}(b)$ then

$b\Vdash_{\mathrm{lh}(n)}\check{p}^{(b,n+1)}\in\pi_{\dot{X}^{(a,n)}}[\dot{\mu}_{c,a}^{+}(\dot{D}_{j}^{(c,i)})].$

Here, $\mu_{c,a}^{+}$ denotes121212There is some slight abuse of notation here
in an effort to improve readability.

$\mu_{\dot{Z}^{(c,i)},\dot{Z}^{(a,n)}}^{+}\colon
M_{\dot{Z}^{(c,i)}}[\check{f}(\check{\delta})]\rightarrow
M_{\dot{Z}^{(a,n)}}[\check{f}(\check{\delta})].$

We define all objects by induction on $n<\omega$.

$T_{0}=\\{\mathbbm{1}\\},p^{(\mathbbm{1},0)},T^{(\mathbbm{1},0)},\dot{X}^{(\mathbbm{1},0)},\dot{Z}^{(\mathbbm{1},0)}\left(\dot{D}_{j}^{(\mathbbm{1},0)}\right)_{j<\omega},\dot{I}^{(\mathbbm{1},0)}$

are given by $(F.i)$-$(F.iv)$ and $(F.viii)$. Suppose we have already defined

$T_{n},\left\langle
p^{(a,n)},T^{(a,n)},\dot{X}^{(a,n)},\dot{Z}^{(a,n)},\left(\dot{D}_{j}^{(a,n)}\right)_{j<\omega}\mid
a\in T_{n}\right\rangle$

and we will further construct

$T_{n+1},\left\langle
p^{(b,n+1)},T^{(b,n+1)},\dot{X}^{(b,n+1)},\dot{Z}^{(b,n+1)},\left(\dot{D}_{j}^{(b,n+1)}\right)_{j<\omega}\mid
b\in T_{n+1}\right\rangle.$

Fix $a\in T_{n}$. Let $E$ be the set of all $b$ so that

  1. $(E.i)$

$b\in\mathbb{P}_{\mathrm{lh}(b)}$ and $\mathrm{lh}(b)<\gamma$,

  2. $(E.ii)$

$\mathrm{lh}(a)\leq\mathrm{lh}(b)$ and $b\upharpoonright\mathrm{lh}(a)\leq a$,

and there are a nested antichain $S$ in $\mathbb{P}$, $s\in\mathbb{P}$ and
names $\dot{X}$, $\dot{I}$ with

  1. $(E.iii)$

$S\angle\ T^{(a,n)}$,

  2. $(E.iv)$

$s\leq p^{(a,n)}$ is a mixture of $S$,

  3. $(E.v)$

if $h(n)=(i,j)$ and $c=\mathrm{pred}_{T}^{i}(a)$ then

$b\Vdash_{\mathrm{lh}(b)}\check{s}\in\pi_{\dot{Z}^{(a,n)}}[\dot{\mu}^{+}_{c,a}(\dot{D}_{j}^{(c,i)})],$

  4. $(E.vi)$

$\mathrm{lh}(b)$ is not a limit ordinal,

  5. $(E.vii)$

$b\upharpoonright\mathrm{lh}(a)\Vdash_{\mathrm{lh}(a)}\check{s},\check{S}\in\dot{X}$,

  6. $(E.viii)$

$b\Vdash_{\mathrm{lh}(b)}\check{s}\upharpoonright\mathrm{lh}(b)\in\dot{G}_{\mathrm{lh}(b)}$,

  7. $(E.ix)$

$b\Vdash_{\mathrm{lh}(b)}\dot{X}^{(a,n)}\sqsubseteq\dot{X}^{(a,n)}[\dot{G}_{\mathrm{lh}(a),\mathrm{lh}(b)}]\sqsubseteq\dot{X}\prec
H_{\check{\theta}}^{V[\dot{G}_{\mathrm{lh}(\check{b})}]}$,

  8. $(E.x)$

$b\Vdash_{\mathrm{lh}(b)}``\dot{X}\text{ is countable and respects }\dot{I}"$,

  9. $(E.xi)$

$b\Vdash_{\mathrm{lh}(b)}\dot{I}=I^{\dot{\mathbb{Q}}_{\mathrm{lh}(b)}}_{\check{s}(\mathrm{lh}(b))}$
and

  10. $(E.xii)$

if $S_{0}=\\{c_{0}\\}$ then $\mathrm{lh}(b)=\mathrm{lh}(c_{0})$ and $b\leq
c_{0}$.

######  Claim 4.18.

$E\upharpoonright\mathrm{lh}(a)\coloneqq\\{b\upharpoonright\mathrm{lh}(a)\mid
b\in E\\}$ is dense in $\mathbb{P}_{\mathrm{lh}(a)}$.

###### Proof.

Let $a^{\prime}\leq a$ and let $G$ be $\mathbb{P}_{\mathrm{lh}(a)}$-generic
with $a^{\prime}\in G$. By $(F.v)$, $p^{(a,n)}\upharpoonright\mathrm{lh}(a)\in
G$. Work in $V[G]$. Let $h(n)=(i,j)$ and $c=\mathrm{pred}_{T}^{i}(a)$. Let

$X^{(c,i)}=\left(\dot{X}^{(c,i)}\right)^{G_{\mathrm{lh}(c)}}\text{ and
}X^{(a,n)}=\left(\dot{X}^{(a,n)}\right)^{G}$

as well as $Z^{(c,i)}=X^{(c,i)}\cap V$, $Z^{(a,n)}=X^{(a,n)}\cap V$. Find
$r\in T_{1}^{(a,n)}$ with $r\upharpoonright\mathrm{lh}(a)\in G$. As
$p^{(a,n)}$ is a mixture of $T^{(a,n)}$, we have

$r\leq p^{(a,n)}\upharpoonright\mathrm{lh}(r).$

Let $\hat{r}=r^{\frown}p^{(a,n)}\upharpoonright[\mathrm{lh}(r),\gamma)$. Note
that $\hat{r}\in X^{(a,n)}$, as

$p^{(a,n)},T^{(a,n)},G\in X^{(a,n)}$

by $(F.vii)$. Moreover, $\hat{r}\upharpoonright\mathrm{lh}(a)\in G$. Let
$\mathbb{Q}\coloneqq\dot{\mathbb{Q}}_{\mathrm{lh}(a)}^{G}$ and

$D\coloneqq\mu_{c,a}^{+}((\dot{D}^{i}_{j})^{G_{\mathrm{lh}(c)}})\in
M_{Z^{(a,n)}}[f(\delta)]\subseteq M_{X^{(a,n)}}[f(\delta)].$

######  Subclaim 4.19.

There are $s$, $Y$ with

  1. $(i)$

$X^{(a,n)}\sqsubseteq Y\prec H_{\theta}^{V[G]}$,

  2. $(ii)$

$s\leq p^{(a,n)}$,

  3. $(iii)$

$s\upharpoonright\mathrm{lh}(a)\in G$,

  4. $(iv)$

$s\in\pi_{Y}[\mu_{X^{(a,n)},Y}^{+}(D)]$ and

  5. $(v)$

$Y$ respects $I^{\mathbb{Q}}_{s(\mathrm{lh}(a))}$.

###### Proof.

Let

$D_{0}\coloneqq\\{t\in D\mid\pi_{X^{(a,n)}}(t)\leq
p^{(a,n)}\wedge\pi_{X^{(a,n)}}(t)\upharpoonright\mathrm{lh}(a)\in G\\}$

and $D_{1}$ be the projection of $D_{0}$ onto
$\pi_{X^{(a,n)}}^{-1}(\mathbb{Q})$. Observe that

$M_{X^{(a,n)}}[f(\delta)]\models``D_{1}\text{ is dense below
}\pi_{X^{(a,n)}}^{-1}(p^{(a,n)}(\mathrm{lh}(a))\text{ in
}\pi_{X^{(a,n)}}^{-1}(\mathbb{Q})".$

Applying Lemma 4.17 with (making use of the notation there)

  * •

$\mathbb{P}=\dot{\mathbb{Q}}$,

  * •

$p=p^{(a,n)}(\mathrm{lh}(a))$,

  * •

$X=X^{(a,n)}$ and

  * •

$D=D_{0}$,

we find some countable $Y$ and some $s_{0}$ with

  1. $(i)$

$X^{(a,n)}\sqsubseteq Y\prec H_{\theta}^{V[G]}$,

  2. $(ii)$

$s_{0}\leq p^{(a,n)}(\mathrm{lh}(a))$,

  3. $(iii)$

$s_{0}\in\pi_{Y}[\mu_{X^{(a,n)},Y}^{+}(D_{1})]$ and

  4. $(iv)$

$Y$ respects $I^{\mathbb{Q}}_{s_{0}}$.

By definition of $D_{1}$, there is $s\leq p^{(a,n)}$ with

  1. $(s.i)$

$s\upharpoonright\mathrm{lh}(a)\in G$,

  2. $(s.ii)$

$s\in\pi_{Y}[\mu_{X^{(a,n)},Y}^{+}(D)]$ and

  3. $(s.iii)$

$s(\mathrm{lh}(a))=s_{0}$.

$Y,s$ have the desired properties.

∎

We can now apply Fact 3.23 in $Y$ and get a nested antichain $S\in X^{(a,n)}$
with

  1. $(S.i)$

$s$ is a mixture of $S$,

  2. $(S.ii)$

if $S_{0}=\\{d\\}$ then $\mathrm{lh}(r)\leq\mathrm{lh}(d)$,
$d\upharpoonright\mathrm{lh}(r)\leq r$ and $\mathrm{lh}(d)$ is not a limit
ordinal and

  3. $(S.iii)$

$S\angle\ T^{(a,n)}$.

Let $\dot{X}$ be a name for $Y[\dot{G}_{\mathrm{lh}(a),\mathrm{lh}(d)}]$ and
$\dot{I}$ a name for
$I^{\dot{\mathbb{Q}}_{\mathrm{lh}(d)}}_{s(\mathrm{lh}(d))}$.

######  Subclaim 4.20.

In $V[G]$, we have

$\dot{I}^{s\upharpoonright\mathrm{lh}(d)}=I^{\mathbb{P}_{\mathrm{lh}(a),\mathrm{lh}(d)+1}}_{s\upharpoonright\mathrm{lh}(d)+1}=I^{\mathbb{Q}}_{s(\mathrm{lh}(a))}.$

###### Proof.

The first equality is simply by definition of $\dot{I}$. The second equality
follows as we preserve old $f$-stationary sets along the iteration and since
$\mathbb{P}_{\mathrm{lh}(a),\mathrm{lh}(d)+1}$ preserves $f$ by our inductive
hypothesis. ∎

It follows that

$Y\text{ respects }\dot{I}^{s\upharpoonright\mathrm{lh}(d)}.$

As $\mathrm{lh}(a)$ is not a limit ordinal, $(\ddagger)$ holds in $V[G]$, so
that $\mathbb{P}_{\mathrm{lh}(a),\mathrm{lh}(d)}$ is respectful by Lemma 4.10.
Thus there is $b\in\mathbb{P}_{\mathrm{lh}(a),\mathrm{lh}(d)}$, $b\leq
s\upharpoonright\mathrm{lh}(d)$ so that

$b\Vdash_{\mathrm{lh}(b)}``\check{Y}\sqsubseteq\check{Y}[\dot{G}]\text{
respects }\dot{I}".$

Since $b\upharpoonright\mathrm{lh}(a)\in G$, we may assume further that
$b\upharpoonright\mathrm{lh}(a)\leq a^{\prime}$. $s,S,\dot{X},\dot{I}$ witness
$b\in E$. ∎

To define $T_{n+1}$, fix a maximal antichain $A\subseteq
E\upharpoonright\mathrm{lh}(a)$, and for any $e\in A$ choose $b_{e}\in E$ with
$b_{e}\upharpoonright\mathrm{lh}(a)=e$. We set
$\mathrm{suc}_{T}^{n}(a)=\\{b_{e}\mid e\in A\\}$. For any
$b\in\mathrm{suc}_{T}^{n}(a)$, let $S,s,\dot{X},\dot{I}$ witness $b\in E$. We
then let

  * •

$p^{(b,n+1)}=s,\ T^{(b,n+1)}=S,\ \dot{X}^{(b,n+1)}=\dot{X},\
\dot{I}^{(b,n+1)}=\dot{I}$,

  * •

$\dot{Z}^{(b,n+1)}$ be a name for $\dot{X}\cap V$ and

  * •

$\left(\dot{D}_{j}^{(b,n+1)}\right)_{j<\omega}$ be a sequence of names that
are forced by $b$ to enumerate all dense subsets of
$\pi_{\dot{Z}^{(b,n+1)}}^{-1}(\mathbb{P})$ in
$M_{\dot{Z}^{(b,n+1)}}\left[\widecheck{f(\delta)}\right]$.

This finishes the construction.

By Fact 3.23, there is a mixture $q$ of $T$. Let $G$ be $\mathbb{P}$-generic
with $q\in T$. By Fact 3.25, in $V[G]$ there is a sequence $\langle a_{n}\mid
n<\omega\rangle$ so that for all $n<\omega$

  1. $(\vec{a}.i)$

$a_{0}=q_{0}$,

  2. $(\vec{a}.ii)$

$a_{n+1}\in\mathrm{suc}_{T}^{n}(a_{n})$ and

  3. $(\vec{a}.iii)$

$p^{(a_{n},n)}\in G$.

For $n<\omega$, let $\alpha_{n}=\mathrm{lh}(a_{n})<\gamma$. For $n<\omega$ we
let

$X_{n}\coloneqq\left(\dot{X}^{(a_{n},n)}\right)^{G_{\alpha_{n}}}$

and also

$X_{\omega}=\bigcup_{n<\omega}X_{n}[G_{\alpha_{n},\gamma}].$

Further, for $n\leq\omega$ let

$Z_{n}\coloneqq X_{n}\cap V\text{ and }\pi_{n}\coloneqq\pi_{Z_{n}}.$

We remark that

$X_{n}[G_{\alpha_{n},\gamma}]\sqsubseteq X_{m}[G_{\alpha_{m},\gamma}]\prec
H_{\theta}^{V[G]}$

follows inductively from $(F.vii)$ and $(F.ix)$ for $n\leq m<\omega$ so that
$X_{\omega}\prec H_{\theta}^{V[G]}$. We aim to prove that

$X\sqsubseteq X_{\omega}\text{ is }f\text{-slim}.$

In fact, we will show

  1. $(Z_{\omega}.i)$

$X\sqsubseteq Z_{\omega}$,

  2. $(Z_{\omega}.ii)$

$Z_{\omega}\text{ is }f\text{-slim}$ and

  3. $(Z_{\omega}.iii)$

$\pi_{\omega}^{-1}[G]\text{ is generic over }M_{\omega}[f(\delta)]$,

which implies the above.

######  Claim 4.21.

$Z_{\omega}=\bigcup_{n<\omega}Z_{n}$.

###### Proof.

$``\supseteq"$ is trivial, so we show $``\subseteq"$. Let $x\in Z_{\omega}$
and find $i<\omega$ with $x\in X_{i}[G_{\alpha_{i},\gamma}]$. Note that there
is $\dot{x}\in Z_{i}$ a $\mathbb{P}$-name for a set in $V$ with
$x=\dot{x}^{G}$. Let $D\in M_{i}$ be the dense set of conditions in
$\pi_{n}^{-1}(\mathbb{P})$ deciding $\pi_{i}^{-1}(\dot{x})$. There must be
some $j<\omega$ so that

$\left(\dot{D}_{j}^{(a_{i},i)}\right)^{G}=D.$

Now find $n$ with $h(n)=(i,j)$. We then have

$p^{(a_{n+1},n+1)}\in\pi_{n}[\mu_{a_{i},a_{n+1}}^{+}(D)]$

by $(F.xii)$. We have that $p^{(a_{n+1},n+1)}$ decides $\dot{x}$ to be some
$z\in X_{n}$, and as $p^{(a_{n+1},n+1)}\in G$,

$x=\dot{x}^{G}=z\in X_{n}\cap V=Z_{n}.$

∎

As $X\sqsubseteq X_{n}$ is $f\text{-slim}$ by $(F.ix)$ for $n<\omega$,
$(Z_{\omega}.i)$ and $(Z_{\omega}.ii)$ follow at once. It remains to show
$(Z_{\omega}.iii)$.
As $Z_{\omega}$ is $f\text{-slim}$ and by Claim 4.21, we have that

$\langle M_{\omega}[f(\delta)],\mu_{n,\omega}^{+}\mid
n<\omega\rangle=\varinjlim\langle M_{n}[f(\delta)],\mu_{n,m}^{+}\mid n\leq
m<\omega\rangle$

for some $(\mu_{n,\omega}^{+})_{n<\omega}$. Let $E\in M_{\omega}[f(\delta)]$
be dense in $\pi_{\omega}^{-1}(\mathbb{P})$. Then for some $i,j<\omega$,
$E=\mu_{i,\omega}^{+}(D)$ for

$D\coloneqq\left(\dot{D}_{j}^{(a_{i},i)}\right)^{G}.$

Find $n$ with $h(n)=(i,j)$. By $(F.xii)$,

$p^{(a_{n+1},n+1)}\in\pi_{n}[\mu_{i,n}^{+}(D)]\subseteq\pi_{\omega}[\mu_{i,\omega}^{+}(D)]=\pi_{\omega}[E].$

As $p^{(a_{n+1},n+1)}\in G$, we have $E\cap\pi_{\omega}^{-1}[G]\neq\emptyset$,
which is what we had to show. ∎

##  5 $f$-Proper and $f$-Semiproper Forcings

Suppose $f$ witnesses $\diamondsuit(\mathbb{B})$. We already used the term
$(X,\mathbb{P},f)$-semigeneric which suggests there should be a notion of
$f$-semiproperness. Indeed there is and it behaves roughly like
semiproperness. In fact, there are several other classes associated to $f$
which mirror well-known forcing classes.

######  Definition 5.1.

A forcing $\mathbb{P}$ is $f$-complete if for any sufficiently large regular
$\theta$, for any $f\text{-slim}$ $X\prec H_{\theta}$ with $\mathbb{P}\in X$
and any $g\subseteq\bar{\mathbb{P}}$ generic over $M_{X}[f(\delta^{X})]$,
there is a some $p\in\mathbb{P}$ with

$p\Vdash\dot{G}\cap\check{X}=\pi_{X}[\check{g}].$

######  Definition 5.2.

A forcing $\mathbb{P}$ is $f$-proper if for any sufficiently large regular
$\theta$, any $f\text{-slim}$ $X\prec H_{\theta}$ with $\mathbb{P}\in X$ and
any $p\in X\cap\mathbb{P}$, there is a $(X,\mathbb{P},f)$-generic condition
$q\leq p$, that is a condition $q$ with

$q\Vdash\dot{G}\cap\check{X}\text{ is generic over
}\check{X}\wedge\check{X}[\dot{G}]\text{ is }\check{f}\text{-slim}".$

######  Definition 5.3.

A forcing $\mathbb{P}$ is $f$-semiproper if for any sufficiently large regular
$\theta$, any $f\text{-slim}$ $X\prec H_{\theta}$ with $\mathbb{P}\in X$ and
any $p\in X\cap\mathbb{P}$, there is a $(X,\mathbb{P},f)$-semigeneric
condition $q\leq p$.

The following graphic collects all provable relations between the relevant
forcing classes.

Classical$\diamondsuit$-Forcingcomplete($\approx\sigma$-closed)propersemiproperstationary
set
preserving$\omega_{1}$-preserving$f$-complete$f$-proper$f$-semiproper$f$-stationary
set preserving$f$-preserving

We also get the expected iteration theorems.

######  Theorem 5.4.

Any countable support iteration of $f$-complete (resp. $f$-proper) forcings is
$f$-complete (resp. $f$-proper).

######  Theorem 5.5.

Any nice iteration of $f$-semiproper forcings is $f$-semiproper.

The proof is much easier than that of Theorem 4.1, so we omit it.

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