1 Introduction
Given any set X with at least two elements,
$b\in [\omega ]^\omega $
and
$A\subseteq \,^\omega X$
, let
$\Gamma _X(A,b)$
be the infinite game of two players in which both players choose the consecutive elements of a sequence
$x\in \, ^\omega X$
. The choice of
$x(n)$
is done by the second player iff
$n\in b$
. The first player wins iff
$x\in A$
. By using such games, Mycielski [Reference Mycielski7] introduced ideals on the space
$^\omega X$
as follows: Given a family
$\mathfrak {B} =\langle b_s :s\in \,^{<\omega }2\rangle $
of infinite subsets of
$\omega $
such that
$b_s=b_{s^{\,\smallfrown \,} 0}\,\dot \cup \,b_{s^{\,\smallfrown \,}1}$
(disjoint union), the Mycielski ideal
$\mathfrak {M}_{X,\mathfrak {B}}$
is defined as the collection of all
$A\subseteq \,^\omega X$
such that for all
$s\in \,^{<\omega }2$
the second player has a winning strategy in the game
$\Gamma _{X,b_s}$
. In [Reference Mycielski7], Mycielski proved, among other things, that if
$X=2$
or
$X=\omega $
, then
$\mathfrak {M}_{X,\mathfrak {B}}$
is a translation invariant
$\sigma $
-ideal that is orthogonal to
$\mathcal {M}\cap \mathcal {N}$
, where
$\mathcal {M}$
and
$\mathcal {N}$
are the ideals of meager and null sets, respectively (see [Reference Rosłanowski8, Theorem 2.2] for a list of Mycielski’s results). Rosłanowski [Reference Rosłanowski8] introduced more ideals, denoted
$\mathfrak {C}_X$
and
$\mathfrak {P}_X$
, that are closely related to Mycielski’s
$\mathfrak {M}_{X,\mathfrak {B}}$
and are also called Mycielski ideals.
Definition 1.1. Let X be a set with at least two elements. Let
$\mathfrak {C}_X$
be the set of all
$A\subseteq \,^\omega X$
for which player II has a winning strategy in the game
$\Gamma _X(A,b)$
for every
$b\in [\omega ]^\omega $
, thus
$$ \begin{align*}\mathfrak{C}_X=\bigcap _{\mathfrak{B}}\mathfrak{M}_{X,\mathfrak{B}}.\end{align*} $$
Moreover, let
$$ \begin{align*}\mathfrak{P}_X=\lbrace A\subseteq \,^\omega X: \forall b\in \,[\omega ]^\omega\quad A\upharpoonright b\ne \, ^bX\rbrace.\end{align*} $$
Here
$A\upharpoonright b=\lbrace x\upharpoonright b: x\in A\rbrace $
. Alternatively,
$\mathfrak {P}_X$
is the set of those
$A\subseteq \, ^\omega X$
for which, for every
$b\in [\omega ]^\omega $
, player II has a winning strategy in the game
$\Gamma _X(A,b)$
that does not depend on the moves of player I. Clearly we have
$\mathfrak {P}_X\subseteq \mathfrak {C}_X$
.
The Mycielski ideals
$\mathfrak {C}_X$
and
$\mathfrak {P}_X$
have been the object of intensive research over the past decades. The main focus has been on their cardinal characteristics. Recall the following cardinals associated with any ideal
$\mathfrak {I}$
on a set Y:
$$ \begin{align*} &\mathrm{add}(\mathfrak{I})\;=\;\min\{ |\mathcal{F}|: \mathcal{F}\subseteq \mathfrak{I}\;\wedge\; \bigcup\mathcal{F}\not\in \mathfrak{I}\}, \notag \\ &\mathrm{cov}(\mathfrak{I})\;=\;\min\{ |\mathcal{F}|: \mathcal{F}\subseteq \mathfrak{I}\;\wedge\; \bigcup\mathcal{F}= Y\}, \notag \\ &\mathrm{non}(\mathfrak{I})\;=\;\min\{ |Z|: Z\subseteq Y\;\wedge\;Z\not\in \mathfrak{I}\},\phantom{\bigcup} \notag \\ &\mathrm{cof}(\mathfrak{I})\;=\;\min\{ |\mathcal{F}|: \mathcal{F}\subseteq \mathfrak{I} \;\wedge\; \forall Z\in \mathfrak{I} \;\: \exists Z'\in \mathcal{F}\;\:Z\subseteq Z'\}. \phantom{\bigcup} \notag \end{align*} $$
Trivially, if
$\mathfrak {I}, \mathfrak {J}$
are ideals such that
$\mathfrak {I}\subseteq \mathfrak {J}$
, then cov
$(\mathfrak {J})\leq \,\mathrm {cov}(\mathfrak {I})$
.
There exists a close relation between
$\mathfrak {P}_k$
and
$\mathfrak {C}_k$
, where
$2\leq k<\omega $
, on the one hand, and the tree ideals
$v^0_k$
,
$u^0_k$
associated with k-dimensional Silver forcing
$\mathbb {SI}_k$
and forcing with k-dimensional uniform Sacks trees
$\mathbb {U}_k$
, respectively, on the other hand. See Definition 2.1 below for their definitions. This relation comes from the fact that a winning strategy for player II in the game
$\Gamma _k(A,b)$
, where b is infinite and coinfinite, can be considered as a perfect tree
$u^b \subseteq \,^{<\omega }k$
with the property that for every
$t\in u^b$
, t is a splitnode of
$u^b$
iff
$\vert t\vert \in \omega \setminus b$
and every splitnode splits into k successor nodes. Such trees are the k-dimensional uniform Sacks trees. If
$u^b$
does not depend on the moves of player II, then
$u^b$
is nothing else than a k-dimensional Silver function with domain b.
The inclusions
$\mathfrak {P}_k\subseteq v^0_k$
and
$\mathfrak {C}_k\subseteq u^0_k$
are pretty obvious (see [Reference Kuiper and Spinas5, Lemma 4.1]); however, no equality is provable here. Moreover, no inclusion between
$v^0_k$
and
$u^0_k$
is provable (see [Reference Kuiper and Spinas5, Section 4] for these results). By general knowledge, if
$j^0$
is the tree ideal associated with some proper tree forcing P, then in the model obtained by a countable support iteration of length
$\omega _2$
of P over a model of CH, which will be called the P-model for short below, cov
$(j^0) =\aleph _2$
holds true. E.g., see [Reference Judah, Miller and Shelah4, Theorem 1.2] where this is proved for the two-dimensional Sacks forcing.
We conclude from the above that in the
$\mathbb {SI}_k$
-model cov(
$\mathfrak {P}_k) =\aleph _2$
holds and in the
$\mathbb {U}_k$
-model cov(
$\mathfrak {C}_k )=\,\mathrm {cov}(\mathfrak {P}_k )=\aleph _2$
. A main result of [Reference Shelah and Steprāns10] is that the equality cov
$(\mathfrak {P}_k)=\,\mathrm {cov}(\mathfrak {P}_{k+1})$
holds in ZFC, for every
$2\leq k<\omega $
. In [Reference Kuiper and Spinas5, Theorem 7.18] it has been shown that in the
$\mathbb {U}_k$
-model cov
$( \mathfrak {C}_{k+1})=\aleph _1$
. By the results just mentioned and as by [Reference Kuiper and Spinas5, Proposition 6.8] cov
$(\mathfrak {C}_{k+1})\leq \, \mathrm {cov}(\mathfrak {C}_{k} )$
is always true, we conclude that in the
$\mathbb {U}_2$
-model we have cov
$ (\mathfrak {C}_{k}) <\,\mathrm {cov}(\mathfrak {P}_{k})$
for every
$3\leq k<\omega $
, but cov
$(\mathfrak {C}_{2})=\,\mathrm {cov}(\mathfrak {P}_{2})=\aleph _2$
. Hence the consistency of cov
$(\mathfrak {C}_{2})<\,\mathrm {cov}(\mathfrak {P}_{2})$
was left open.
Our main result here is that cov
$(\mathfrak {C}_{2})<\,\mathrm {cov}(\mathfrak {P}_{2})$
is true in the
$\mathbb {SI}_2$
-model. For this we have to define
$\aleph _1$
-many
$\mathfrak {C}_{2}$
-sets that cover
$^\omega 2$
in the Silver model. A
$\mathfrak {C}_{2}$
-set can be coded by a family
$C=\langle u^b: b\in \Omega \rangle $
, where
$\Omega \subseteq [\omega ]^\omega $
is
$\subseteq $
-dense and every
$u^b$
is a strategy for player II in the game
$\Gamma _2(\cdot , b)$
, thus a uniform Sacks tree with
$\omega \setminus b$
as its set of split-levels, as described above. Such C will be called coding system below. Then
$$ \begin{align*}A(C)= \,^\omega 2\setminus \bigcup\lbrace [u^b]: b\in\Omega\rbrace\end{align*} $$
is the
$\mathfrak {C}_{2}$
-set coded by C. Clearly, sets of this type form a base of
$\mathfrak {C}_{2}$
. Clearly, no dense
$\Omega $
in the ground model will remain so in a forcing extension adding reals. Indeed, given any new real x, say
$x\in \, ^\omega 2$
, the set
$\lbrace x\upharpoonright n: n<\omega \rbrace \subset \, ^{<\omega } 2$
does not contain any infinite subset from the ground model. Therefore, one of our tasks will be to extend
$\aleph _1$
-many partial coding systems cofinally often during the forcing iteration.
There are two main ingredients for this construction to work. The first one consists in a careful reading of a given
$P_{\omega _2}$
-name
$\dot x$
for a new real, where
$P_{\omega _2}$
is the CS-iteration of
$\mathbb {SI}_2$
. This process will produce (even for a name for a member of
$^\omega \omega $
) a fusion sequence
$\bar S=\langle p_n:n<\omega \rangle $
with limit
$p_\omega $
in
$P_{\omega _2}$
together with a 2-splitting tree T, i.e., every node of T has at most two successor nodes, which is the tree of possibilities for
$\dot x$
below
$p_\omega $
, such that the family of refining finite maximal antichains below
$p_\omega $
associated with
$\bar S$
corresponds to the split-levels of T. For this we apply what we call the Grigorieff dichotomy, i.e., an idea that appears in seminal form in Grigorieff’s paper [Reference Grigorieff3] where it is shown that
$\mathbb {SI}_2$
(as well as many more forcings of a similar type) adds a minimal real.
As a small digression let me mention that in [Reference Rosłanowski and Steprāns9] the property of some forcing that in its extension every new real is a branch through some k-ary tree in the ground model is called k-localization property. In [Reference Rosłanowski and Steprāns9] it is shown that the CS-iteration of
$\mathbb {SI}_k$
has the k-localization property. For this, Shelah-style preservation theorems are used and it is said that“maybe some old wisdom got lost”, but that it seemed impossible to prove this by classical methods. I think that this old wisdom are Grigorieff’s ideas, as building on them the
$2$
-localization property for
$P_{\omega _2}$
can be shown by using a classical fusion construction. I conjecture that this can be generalized to every finite dimension. But then certainly some extra complexity is added, as
$\mathbb {SI}_k$
is no longer minimal if
$k\geq 3$
.Footnote
1
The second ingredient for our main result is a new idea by which it is possible to define, for any
$P_{\omega _2} $
-name
$\dot x$
for a real with fusion
$\bar S$
and limit
$p_\omega $
as above, a coding system
$C=\langle u^b:b\in \Omega \rangle $
in the ground model such that
$$ \begin{align*}p_\omega\Vdash_{P_{\omega_2}}\; \forall b\in\Omega \quad\dot x\not\in [u^b].\end{align*} $$
Moreover, the definition of C depends only on the isomorphism type of
$\dot x$
(see Definition 4.2 below). As by CH in the ground model there are only
$\aleph _1$
-many such types, this will be a good start for the construction of the
$\aleph _1 \mathfrak {C}_2$
-sets we need to have in the final model. This new idea is inspired by the well-known proof that
$\mathbb {SI}_2$
adds a real which splits every infinite subset of
$\omega $
in the ground model.
In [Reference Kuiper and Spinas6], the ideas of this work have been used to show that in the model obtained by a CS-iteration of Sacks forcing the inequality
$\mathrm {cov}(\mathfrak {C}_2)<\mathrm {cov}(s^0)$
holds true. The value of
$\mathrm {cov}(\mathfrak {P}_2)$
in the Sacks model is not known.
2 Basic definitions
Definition 2.1. Let
$2\leq k<\omega $
.
(1) Let
$\mathbb {SI}_k$
denote k-dimensional Silver forcing, i.e., the set of all partial functions f from
$\omega $
to k such that
$\omega \setminus \,\mathrm {dom}(f)$
is infinite, ordered by extension. We shall denote
$\omega \setminus \,\mathrm {dom}(f)$
by
$\mathrm {com}(f)$
. So Silver forcing
$\mathbb {SI}$
is
$\mathbb {SI}_2$
. By
$e_f$
we shall denote the increasing enumeration of
$\mathrm {com}(f)$
.
Given such k-dimensional Silver function f and
$s\in \, ^{<\omega }k$
, by
$f^s$
we denote the Silver function extending f such that we have
$\mathrm {dom}(f^s)=\,\mathrm {dom}(f)\cup \lbrace e_f(i):i<\vert s\vert \rbrace $
and
$f^s(e_f(i))=s(i)$
for every
$i<\vert s\vert $
.
Given
$f,f'\in \mathbb {SI}_k$
and
$n<\omega $
, we define
$f'\leq ^n f$
iff
$f'\leq f$
and
$\mathrm {com}(f)$
and
$\mathrm {com}(f')$
agree on their first
$n+1$
elements. It is well-known that, equipped with these orderings,
$\mathbb {SI}_k$
satisfies Axiom A (see [Reference Baumgartner and Mathias1] for this notion).
Given
$f\in \mathbb {SI}_k$
, by
$[f]$
we denote its body, which is defined as
$$ \begin{align*}[f] =\lbrace x\in\, ^{\omega} k: f\subseteq x\rbrace.\end{align*} $$
The k-dimensional Silver ideal
$v^0_k$
is defined as the set of all
$X\subseteq \, ^{\omega }k$
such that
$$ \begin{align*}\forall f\in\mathbb{SI}_k\,\exists f'\in\mathbb{SI}_k\quad (f'\leq f\,\wedge \, [f']\cap X=\emptyset).\end{align*} $$
(2) By
$\mathbb {U}_k$
we denote k-dimensional uniform (Sacks) tree forcing, i.e., the set of all perfect trees
$u\subseteq \, ^{<\omega }k$
such that there exists an infinite and coinfinite set
$a^{u}\subset \omega $
with the property that for every node
$t\in u$
, t is a splitnode of u iff
$\vert t\vert \in a^{u}$
and every splitnode of u has full splitting, i.e., it has k immediate successors. The order on
$\mathbb {U}_k$
is inclusion. We shall denote
$\mathbb {U}_2$
by
$\mathbb {U}$
.
If
$u\in \mathbb {U}_k $
, by
$[u]$
we denote the body of u, i.e., the set of all its branches. The tree ideal associated with
$\mathbb {U}_k$
, denoted by
$u^0_k$
, is defined analogously as
$v^0_k$
: It contains all
$X\subseteq \,^\omega k$
such that for every
$u\in \mathbb {U}_k$
there is
$u'\in \mathbb {U}_k$
with
$u'\leq u$
and
$[u']\cap X=\emptyset $
.
(3) Given any tree T, by
$\mathrm {split}(T)$
we denote the set of all splitnodes of T, i.e., those
$t\in T$
which have at least two immediate successors. By
$\mathrm {split}_n(T)$
we denote the nth splitlevel of T, i.e., the set of those
$t\in \,\mathrm {split}(T)$
which have precisely n proper initial segments which are splitnodes of T.
Definition 2.2. Let
$P_\alpha $
be the CS-iteration of
$\mathbb {S I} $
of length
$\alpha $
, in particular, its elements are functions p with countable
$\mathrm {supp}(p):=\mathrm {dom}(p)\subseteq \alpha $
such that for every
$\beta \in \mathrm {supp}(p)$
we have that
$p(\beta )$
is a
$P_\beta $
-name and the maximal condition in
$P_\beta $
forces
$ p(\beta )\in \mathbb {SI} ^{\mathbf {V}^{P_\beta }}.$
As explained in detail in [Reference Baumgartner and Mathias1, Section 5], given
$\beta <\alpha $
,
$P_\alpha $
is forcing equivalent to an iteration
$P_\beta \ast P_\alpha /\dot {G}_\beta $
, where
$\dot {G}_\beta $
is the canonical
$P_\beta $
-name for the
$P_\beta $
-generic filter, such that
$P_\alpha /\dot {G}_\beta $
is again (the
$P_\beta $
-name of) a CS-support iteration of
$\mathbb {SI}$
. Using this identification, given
$p\in P_\alpha $
and a
$P_\beta $
-generic filter
$G_\beta $
, we can partially evaluate
$p[ G_\beta ] $
as a pair of sequences
$(\underline {f},\overline {f})$
such that
$\underline {f}=\langle f_\gamma :\gamma \in \mathrm {supp}(p)\cap \beta \rangle $
is a sequence of Silver functions
$f_\gamma =p(\gamma )[G_\beta ]$
and
$\overline {f}=\langle \dot {f}_\gamma : \gamma \in \mathrm {supp}(p)\setminus \beta \rangle $
is a condition of the CS-iteration of
$\mathbb {SI}$
along
$\alpha \setminus \beta $
in
$\mathbf {V}[G_\beta ]$
.
As is well-known from proper forcing, we may assume that all
$p\in P_\alpha $
are hereditarily countable (see [Reference Shelah11, Chapter III, Definition 4.1] for this notion and proof of this fact), hence in particular, for every
$\beta \in \,\mathrm {supp}(p) $
, the
$P_\beta $
-name
$p(\beta )$
can be evaluated as a Silver function by
$\langle g_\gamma : \gamma \in \, \mathrm {supp}(p)\cap \beta \rangle $
, where
$g_\gamma $
is the Silver real added by the
$\gamma $
th iterand.
Wlog we may assume
$0,1\in \,\mathrm {supp}(p) $
for every p.
(1) For
$p,q\in P_\alpha $
,
$F\in \left [ \mathrm {{supp}}(p)\right ] ^{<\aleph _0}$
and
$\eta : F\rightarrow \omega $
we call the triple
$(p,F,\eta )$
a fusion condition and we write
$q\leq _{F,\eta }p$
for
$$ \begin{align*}q\leq p\quad \wedge\quad \forall\beta\in F\quad q\upharpoonright \beta \Vdash_{P_\beta} \;\; q(\beta )\leq ^{\eta (\beta)} p(\beta).\end{align*} $$
(2) A fusion sequence in
$P_\alpha $
is a sequence
$\bar S=\langle (p_n,F_n,\eta _n): n<\omega \rangle $
of fusion conditions such that
-
(a)
$F_0=F_1=\left \lbrace 0\right \rbrace $
,
$F_2=\lbrace 0,1\rbrace $
(as long as
$\alpha>1$
of course),
$F_n\subseteq F_{n+1}$
and
$\vert F_{n+1}\setminus F_n\vert \leq 1$
; -
(b)
$\eta _0 (0)=0$
,
$\eta _1(0)=1$
,
$\forall \beta \in F_n\quad \eta _n(\beta )\leq \eta _{n+1}(\beta )\leq \eta _n(\beta )+1$
and there is precisely one
$\beta \in F_{n+1}$
, which we call the active coordinate of
$F_{n+1}$
, such that
$\eta _{n+1}(\beta )=\eta _n(\beta )+1$
in case
$\beta \in F_n$
, and
$\eta _{n+1}(\beta )=0$
in case
$\beta \not \in F_n$
; we also call
$0\in F_0$
active; -
(c)
$\forall \beta \in F_n\,\exists m \quad \eta _m(\beta )\geq n$
; -
(d)
$\bigcup \left \lbrace F_n:n<\omega \right \rbrace =\bigcup \left \lbrace \mathrm {{supp}}(p_n):n<\omega \right \rbrace $
; -
(e)
$p_{n+1}\leq _{F_n, \eta _n} p_n.$
(3) Every fusion sequence
$\langle (p_n,F_n,\eta _n): n<\omega \rangle $
determines its fusion limit
$$ \begin{align*}p_\omega= \inf \left\lbrace p_n:n<\omega\right\rbrace\end{align*} $$
in
$P_\alpha $
. Clearly,
$\mathrm {{supp}}(p_\omega ) =\bigcup \left \lbrace F_n:n<\omega \right \rbrace $
.
(4) For every
$f\in \mathbb {SI}$
and
$s\in 2^{<\omega }$
we have defined
$f^{s}\in \mathbb {SI}$
(see Definition 2.1(1)). If
$(p,F,\eta ) $
is a fusion condition, for
$\sigma \in \prod \limits _{\beta \in F} \, ^{\eta (\beta )}2$
we define
$p\ast \sigma \in P_\alpha $
as follows: For every
$\beta \in F$
let
$(p\ast \sigma )(\beta )$
a
$P_\beta $
-name for a condition in
$\mathbb {SI}$
such that
$$ \begin{align*}p\upharpoonright \beta\, \Vdash_{P_\beta}\quad (p\ast \sigma)(\beta)=p(\beta)^{\sigma(\beta)}.\end{align*} $$
For every
$\beta \in \alpha \setminus F$
let
$(p\ast \sigma )(\beta )= p(\beta )$
.
(5) For
$\beta \in \,\mathrm {supp}(p_\omega )$
let
$Z(\beta )=Z(\bar S, \beta )$
denote the set
$$ \begin{align*}\left\lbrace n<\omega: \beta\mathrm{\; is\, the\; active\; coordinate\; of\;} F_n\right\rbrace \!.\end{align*} $$
Clearly
$\langle Z(\beta ):\beta \in \, \mathrm {supp}(p_\omega )\rangle $
is a partition of
$\omega $
into infinite sets.
(6) For
$\beta \in \,\mathrm {supp}(p_\omega )$
, the increasing bijection between
$Z(\beta ) $
and
$\mathrm {com}(p_\omega (\beta ))$
will be denoted by the
$P_\beta $
-name
$\dot {c}(p_\omega , \beta )$
.
Remark 2.1. (1) The demand
$F_0=F_1= \left \lbrace 0\right \rbrace $
and
$F_2=\lbrace 0,1\rbrace $
in Definition 2.2(2)(a) is only to avoid having to consider several cases in some proofs below.
(2) If
$(p,F,\eta )$
is a fusion condition then
$$ \begin{align*}\langle p\ast \sigma: \sigma\in \prod\limits_{\beta\in F} \, ^{\eta(\beta)}2\rangle\end{align*} $$
is a (finite) maximal antichain below p.
(3) If
$\bar {S} =\langle (p_n,F_n,\eta _n): n<\omega \rangle $
is a fusion sequence with limit
$p_\omega $
, then
$$ \begin{align*}\langle p_{n+1}\ast \sigma: \sigma\in \prod\limits_{\beta\in F_n} \, ^{\eta_n(\beta)+1}2\rangle\end{align*} $$
is a maximal antichain below
$p_{n+1}$
of size
$2^{n+1}$
and it induces a maximal antichain
$A_{n+1}(\bar {S})$
below
$p_\omega $
of the same size, as
$$ \begin{align*}(p_{n+1}\ast \sigma)\wedge p_\omega =p_\omega \ast \sigma,\end{align*} $$
such that every member of
$A_{n}(\bar {S})$
gets refined by two members of
$A_{n+1}(\bar {S})$
, where we let
$A_0(\bar S)= \left \lbrace p_\omega \right \rbrace $
. Hence we naturally enumerate
$A_n(\bar {S})$
by
$\langle a^s: s\in \,^{n}2\rangle $
such that
$a^{s^{\smallfrown }0}$
,
$a^{s^{\smallfrown }1}$
are the two members of
$A_{n+1}(\bar {S})$
below
$a^s$
and, moreover, if
$\gamma $
is the active coordinate of
$F_n$
, thus
$n\in Z(\gamma )$
, then
$a^{s^{\smallfrown }i}=p_\omega \ast \sigma $
for some
$\sigma \in \prod \limits _{\beta \in F_n} \, ^{\eta _n(\beta )+1}2$
with the last digit of
$\sigma (\gamma )$
being i.
Definition 2.3. Let
$\dot y$
be a
$P_\alpha $
-name for an element of
$\omega $
and
$(p,F,\eta )$
a fusion condition. We say that
$(p,F,\eta )$
weakly decides
$\dot y$
if for every
$\sigma \in \prod \limits _{\beta \in F} \, ^{\eta (\beta )+1}2$
,
$p\ast \sigma $
decides
$\dot y$
.
The following lemma is well-known (e.g., see [Reference Kuiper and Spinas5, Lemma 7.6] where it is proved for
$\mathbb {U}_k)$
.
Lemma 2.1. Suppose
$\Vdash _{P_\alpha } \, \dot {x}\in \omega ^\omega $
and
$p\in P_\alpha $
. There exists a fusion sequence
$\langle (p_n,F_n,\eta _n): n<\omega \rangle $
in
$ P_\alpha $
such that the following hold:
-
(1)
$p_0= p$
; -
(2)
$(p_{n}, F_{n},\eta _{n})$
weakly decides
$\dot {x}\upharpoonright n$
, and hence, for every
$n<\omega $
,
$(p_\omega , F_{n},\eta _{n})$
weakly decides
$\dot {x}\upharpoonright n$
.
3 The Grigorieff dichotomy
Definition 3.1. Let
$\alpha \leq \omega _2$
and suppose that
$P_\alpha $
is a CS-iteration of
$\mathbb {SI}$
,
$\dot {x}$
is a
$P_\alpha $
-name such that
$\Vdash _{P_\alpha }\, \dot x\in 2^\omega \setminus \mathbf {V}$
and
$p\in P_\alpha $
. By
$\dot x\left [ p\right ] $
we denote the longest
$t\in \, ^{< \omega }2$
such that
$$ \begin{align*}p\Vdash _{P_\alpha}\; t\subseteq \dot x.\end{align*} $$
Following Grigorieff [Reference Grigorieff3] we say that
$k<\omega $
is indifferent to
$p, \dot {x}$
, if
$k\in \,\mathrm {com}(p(0))$
and there is no
$q\leq p$
such that
$k\in \,\mathrm {com}(q(0))$
and
$$ \begin{align*}\dot{x}\left[ q^{(k,0)} \right] \perp \dot{x}\left[ q^{(k,1)}\right] ,\end{align*} $$
where
$$ \begin{align*}q^{(k,i)}= (q(0) \cup \left\lbrace (k,i)\right\rbrace )^{\smallfrown}q\upharpoonright \left[ 1,\alpha\right) \!.\end{align*} $$
We have the following Grigorieff dichotomy: Either (G1) or (G2) holds, where
-
(G1)
$\forall p\exists q\leq p\forall r\leq q \forall k \quad k$
is not indifferent to
$r, \dot {x}$
; -
(G2)
$\exists p\forall q\leq p\exists r\leq q\exists k\quad k$
is indifferent to
$r,\dot {x} $
.
The following lemma, whose prototype is [Reference Grigorieff3, Lemma 4.7], shows that if
$\dot x$
is a name for a new real, then (G1) must hold.
Lemma 3.1. Suppose
$\Vdash _{P_\alpha } \,\dot {x}\in \omega ^\omega $
and
$p\in P_\alpha $
witnesses that (G2) of Grigorieff’s dichotomy holds. Then
$p \Vdash _{P_\alpha } \,\dot {x}\in \mathbf {V}.$
Proof. We construct a fusion sequence
$\langle (p_n, F_n, \eta _n): n<\omega \rangle $
in
$P_\alpha $
and families
$\langle n_k:k<\omega \rangle $
,
$\langle j_k:k<\omega \rangle $
and
$\langle \xi _n: n<\omega \rangle $
such that
$p_0\leq p$
and for every
$n<\omega $
the following hold:
-
(1)
$\langle n_k:k<\omega \rangle $
increasingly enumerates
$Z(0)$
(hence
$n_0=0$
and
$n_1=1$
); -
(2) if
$Z(0)\cap n= \lbrace n_0,...,n_{k-1}\rbrace $
, then
$\lbrace j_0<...<j_{k-1}\rbrace $
is an initial segment of com
$(p_n(0))$
and
$j_\ell $
is indifferent to
$p_{n}, \dot {x}$
for every
$\ell < k$
; -
(3) if
$n=n_k$
then
$\lbrace j_0<...<j_{k}\rbrace $
is an initial segment of com
$(p_n(0))$
and
$j_\ell $
is indifferent to
$p_{n}, \dot {x}$
for every
$\ell \leq k$
; -
(4)
$\xi _n\in \, ^{n}2$
and
$p_{n} \Vdash _{P_\alpha } \, \dot {x}\upharpoonright n=\xi _{n}$
.
We present the first three steps of the recursive construction in detail to make clear the crucial arguments in a simple situation. After that we shall give the general step.
We apply (G2) and obtain
$p_0\leq p$
and
$j_0\in \,\mathrm {com}(p_0(0))$
such that
$j_0$
is indifferent to
$p_0, \dot {x}$
. Wlog we may assume
$j_0=\min ( \mathrm {com}(p_0(0)))$
. Letting
$\xi _0=\emptyset $
, (2), (3), and (4) hold for
$n=n_0=0$
.
By Lemma 2.1, we can choose
$q_1\leq _{F_0,\eta _0}p_0$
(hence
$j_0\in \,\mathrm {com}(q_1(0))$
) such that
$(q_1, F_0, \eta _0)$
weakly decides
$\dot {x}\upharpoonright 1$
. Note that then
$q_1$
even decides
$\dot x\upharpoonright 1$
, say as
$\xi _1$
, as otherwise we had
$$ \begin{align*}\dot x \left[ q_1\ast \sigma_0\right] \perp \dot x \left[ q_1\ast \sigma_1\right]\! ,\end{align*} $$
where
$\sigma _i=\langle \langle i\rangle \rangle $
, which contradicts the indifference of
$j_0$
(note that
$q_1^{(j_0,i)} = q_1\ast \sigma _i$
). By (G2) we can find
$p_1\leq _{F_0,\eta _0} q_1$
and
$j_1$
such that
$j_1$
is indifferent to
$p_1\ast \sigma _0, \dot {x}$
. Hence
$j_1>j_0$
and wlog we may assume that
$j_1$
is the second member of com
$(p_1(0))$
.
We claim that
$j_1$
is indifferent even to
$p_1,\dot {x}$
, and hence (2), (3), and (4) hold for
$n=n_1=1$
. Indeed, otherwise we could find
$q\leq _{F_1,\eta _1} p_1$
such that, letting
$\sigma _{ij}= \langle \langle i,j\rangle \rangle $
(recall that by Definition 2.2(2)
$F_1=\left \lbrace 0\right \rbrace $
and
$\eta _1(0)=1$
),
$$ \begin{align*}\dot x \left[ q\ast \sigma_{10}\right] \perp \dot x \left[ q\ast \sigma_{11}\right] ,\end{align*} $$
say
$$ \begin{align*}\dot x \left[ q\ast \sigma_{10}\right](m)\ne \dot x \left[ q\ast \sigma_{11}\right](m)\end{align*} $$
for some m. Wlog we may assume that
$(q,F_1,\eta _1)$
weakly decides
$\dot {x}\upharpoonright m+1$
. By the indifference of
$j_1$
to
$q\ast \sigma _0, \dot {x}$
, as above we conclude that
$q\ast \sigma _0$
decides
$\dot {x}(m)$
. But now we can choose j such that
$$ \begin{align*}\dot x \left[ q\ast \sigma_{0j}\right]\perp \dot x \left[ q\ast \sigma_{1j}\right]\!,\end{align*} $$
contradicting that
$j_0$
is indifferent to
$p_0, \dot {x}$
.
In order to find
$p_2$
as desired we first choose
$q_2 \leq _{F_1,\eta _1} p_1$
such that
$(q_2, F_2, \eta _2)$
weakly decides
$\dot {x}(1) $
. Again we claim that
$q_2$
even decides
$\dot {x}(1) $
. Otherwise, as each
$ \sigma \in \prod \limits _{\beta \in F_2} \, ^{\eta _2(\beta )+1}2$
is of the form
$\sigma _{ijk}=\langle \langle i,j\rangle , \langle k\rangle \rangle $
, we find distinct
$\sigma =\sigma _{ijk}$
and
$\sigma '=\sigma _{i'j'k'}$
such that
$q_2\ast \sigma $
and
$q_2\ast \sigma '$
force different values to
$\dot {x}(1)$
, say x and
$x'$
. The most interesting case is
$\langle i,j\rangle =\langle i',j'\rangle $
and hence
$k\ne k'$
. By indifference of
$j_1$
we have that, letting
$\sigma " = \langle \langle i,1-j\rangle , k\rangle $
,
$q_2\ast \sigma "$
decides
$\dot {x}(1)$
as x. Now we can define
$r\leq _{F_0,\eta _0} q_2 \ast \langle i\rangle $
such that
$r\ast \langle 1-j\rangle $
decides
$\dot {x}(1)$
as x while
$r\ast \langle j\rangle $
decides
$\dot {x}(1)$
as
$x'$
, which contradicts the indifference of
$j_1$
. Indeed, let r such that
$r(0)= (q_2\ast \langle i\rangle )(0)$
(which is the same thing as
$q_2(0)^{\langle i\rangle } $
) and
$$\begin{align*}r \upharpoonright \left[ 1,\alpha\right) = \left\{\begin{array}{cl} {q_2(1)^{\langle k\rangle} }^{\smallfrown} q_2\upharpoonright \left[ 2,\alpha\right)\!, & {\mathrm{if}}\; r(0)^{\langle i, 1-j\rangle} \;\mathrm{(belongs \; to \;the \; generic\;} G(0))\\ {q_2(1)^{\langle k'\rangle} }^{\smallfrown} q_2\upharpoonright \left[ 2,\alpha\right)\!, & {\mathrm{otherwise.}} \end{array} \right.\end{align*}$$
The other cases, i.e.,
$\langle i,j\rangle \ne \langle i',j'\rangle $
, are similar.
Now we can let
$p_2=q_2$
and
$\xi _2$
is what
$q_2$
decides about
$\dot {x}\upharpoonright 2$
. Then clearly (2) and (4) hold for
$n=2$
and (3) does not apply.
Now we consider the general recursive step, where we have to apply essentially the same arguments we have seen above. Suppose that
$(p_0, F_0, \eta _0), \dots , (p_n,F_n,\eta _n)$
,
$\xi _0,\dots ,\xi _n$
for
$n\geq 2$
have been constructed for some
$n<\omega $
together with some initial segment
$\langle j_\ell :\ell \leq k\rangle $
of com
$(p_n(0))$
, where k is such that
$Z(0)\cap (n+1) =\lbrace n_0<...<n_k\rbrace $
such that
$$ \begin{align*}p_n\Vdash_{P_\alpha}\, \dot{x}\upharpoonright n=\xi_n\end{align*} $$
and every
$j_\ell $
(
$\ell \leq k$
) is indifferent to
$p_n, \dot {x}$
.
We have to distinguish two cases:
Case 1:
$n+1\in Z(0)$
, thus
$n+1 = n_{k+1}$
and clearly
$n>k$
,
$\eta _n(0)= k$
and
$\eta _{n+1}(0)=k+1$
.
Fix some
$ \tau \in \prod \limits _{\beta \in F_n} \, ^{\eta _n(\beta )+1}2$
. By (G2) and Lemma 2.1 we can find
$p_{n+1}\leq _{F_n,\eta _n} p_n$
and
$j_{k+1}$
such that
$j_{k+1}$
is indifferent to
$p_{n+1}\ast \tau , \dot {x}$
and
$p_{n+1}$
weakly decides
$\dot {x}(n)$
. Then clearly
$j_{k+1}>j_k$
. Wlog we may assume that
$\lbrace j_0,...,j_{k+1}\rbrace $
is an initial segment of com
$(p_{n+1}(0))$
.
At first we show that
$p_{n+1}$
even decides
$\dot x(n)$
(and hence
$\dot {x}\upharpoonright n+1$
). Then a similar argument together with the one above showing indifference of
$j_1$
to
$p_1,\dot {x}$
will prove that
$j_{k+1}$
is even indifferent to
$p_{n+1},\dot {x}$
.
For
$ \sigma \in \prod \limits _{\beta \in F_n} \, ^{\eta _n(\beta )+1}2$
let
$x_\sigma $
such that
$$ \begin{align*}p_{n+1}\ast \sigma \Vdash _{P_\alpha}\, \dot x (n)=x_\sigma.\end{align*} $$
For
$\sigma , \sigma ' \in \prod \limits _{\beta \in F_n} \, ^{\eta _n(\beta )+1}2$
let
$\mathit {s}=\mathit {s}(\sigma , \sigma ')$
be the size of the set
$$ \begin{align*}\left\lbrace \ell<\eta_n(0)+1: \sigma(0)(\ell)\ne \sigma'(0)(\ell)\right\rbrace \!.\end{align*} $$
We shall prove
$x_\sigma =x_{\sigma '}$
by induction on s. Suppose first that
$\mathit {s}(\sigma , \sigma ')=0$
(thus
$\sigma (0)=\sigma '(0)$
) and
$\sigma \ne \sigma '$
. Let
$ \sigma ^{\prime }_1$
be defined as follows: If
$(\sigma (0))(0)=i$
,
$\sigma _1'$
equals
$\sigma '$
except for
$(\sigma _1'(0))(0)=1-i$
.
As
$j_0$
is indifferent to
$p_{n+1}$
we must have
$x_{\sigma '}=x_{\sigma ^{\prime }_1}$
. Hence if we had
$x_\sigma \ne x_{\sigma '}$
, then also
$x_\sigma \ne x_{\sigma ^{\prime }_1}$
. But this leads to a contradiction to the indifference of
$j_0$
, as follows: Define
$r\in P_\alpha $
such that
$$ \begin{align*}r(0)= p_{n+1}(0) ^{\sigma (0)\upharpoonright \eta _n(0)+1 \setminus \lbrace 0\rbrace}\end{align*} $$
(hence
$j_0=\min (\mathrm {com}(r(0)))$
and
$r(0)$
does not change if
$\sigma $
is replaced by
$\sigma '$
) and, for
$j<2$
,
$$ \begin{align*}r(0)^{ (j_0,j)}\Vdash _{\mathbb{SI}} \quad r\upharpoonright \left[ 1,\alpha\right) =p_{n+1}\ast \tau_ j,\end{align*} $$
where
$\tau _i =\sigma \upharpoonright F_n\setminus \left \lbrace 0\right \rbrace $
and
$\tau _{1-i}= \sigma ' \upharpoonright F_n\setminus \left \lbrace 0\right \rbrace $
. Clearly
$r\leq p_n$
, but
$$ \begin{align*}\dot x\left[ r^{(j_0,i)}\right](n) =x_\sigma \ne x_{\sigma'}= \dot x\left[ r^{(j_0,1-i)}\right](n),\end{align*} $$
a contradiction.
Now suppose
$\mathit {s}(\sigma , \sigma ')=m+1$
and for
$\mathit {s}\leq m$
the claim is true. Let
$\ell ^*\leq \eta _n(0)$
(hence
$\ell ^*\leq k$
) be maximal such that
$(\sigma (0))(\ell ^*)\ne (\sigma '(0))(\ell ^*)$
. Now define
$ \sigma ^{\prime }_1$
as follows: If
$(\sigma (0))(\ell ^*)=i$
let
$\sigma _1'$
be equal to
$\sigma '$
except for
$(\sigma _1'(0))(\ell ^*)=i$
.
By the indifference of
$j_{\ell ^*}$
we must have
$x_{\sigma '}=x_{\sigma ^{\prime }_1}$
. By the inductive hypothesis we have
$x_\sigma = x_{\sigma ^{\prime }_1}$
. Hence we conclude
$x_\sigma = x_{\sigma ^{\prime }_{1}} =x_{\sigma '}$
and thus
$p_{n+1}$
decides
$\dot {x}\upharpoonright n+1$
as claimed. Denote this decision by
$\xi _{n+1}$
. Then clearly (2), (3), and (4) hold for
$n+1$
.
Now let us sketch why
$j_{k+1}$
is indifferent to
$p_{n+1},\dot {x}$
. Otherwise we could find
$q\leq _{F_{n+1}, \eta _{n+1}}$
,
$\tau '\in \prod _{\beta \in F_n}\,^{\eta _n(\beta )+1}2$
and
$m<\omega $
such that
$\dot {x}\left [ q\ast \tau '\,^{\smallfrown }0 \right ] (m)\ne \dot {x}\left [ q\ast \tau '\,^{\smallfrown }1\right ] (m)$
and q weakly decides
$\dot {x}\upharpoonright m+1$
. Similarly as we showed above that
$p_{n+1}$
decides
$\dot {x}(n)$
, applying indifference of
$j_{k+1}$
to
$p_{n+1}\ast \tau $
, we can show that q even decides
$\dot {x}(m)$
, which is a contradiction.
Case 2:
$n+1\not \in Z(0)$
.
Choose
$p_{n+1}\leq _{F_n,\eta _n} p_n$
such that
$p_{n+1}$
weakly decides
$\dot {x}(n)$
. Very similarly as in Case 1 we can show that
$p_{n+1}$
even decides
$\dot {x}(n)$
, hence also
$\dot {x}\upharpoonright n+1$
, and we denote this decision by
$\xi _{n+1}$
. Then clearly (2) and (4) hold for
$n+1$
and (3) does not apply.
By Lemma 3.1 we conclude that if
$\dot x$
is a
$P_\alpha $
-name for a new real, then (G1) of the Grigorieff dichotomy must hold. As Theorem 3.1 below will show, this enables us to find a very precise form of continuous reading of
$\dot x$
. The prototype of this result is [Reference Grigorieff3, Lemma 4.6].
The following definition was introduced in [Reference Kuiper and Spinas5] (see Definition 7.8).
Definition 3.2. Let
$\dot {x}$
be a
$P_\alpha $
-name for an element of
$\omega ^\omega $
. We say that a fusion condition
$(p,F,\eta )$
splits
$\dot x$
at
$\gamma \in F$
if for every
$$ \begin{align*}\sigma\in \prod\limits_{\beta\in F\setminus \left\lbrace \gamma \right\rbrace } \, ^{\eta(\beta)+1}2\quad \mathrm{and }\quad s_0,s_1\in \, ^{(\eta(\gamma)+1)}2\end{align*} $$
with
$s_0\upharpoonright \eta (\gamma )=s_1\upharpoonright \eta (\gamma )$
and
$s_0(\eta (\gamma ))<s_1(\eta (\gamma ))$
,
$$ \begin{align*}\dot{x}\left[ p\ast (\sigma \cup \left\lbrace (\gamma, s_0)\right\rbrace )\right] \perp \dot{x}\left[ p\ast (\sigma \cup \left\lbrace (\gamma, s_1)\right\rbrace )\right].\end{align*} $$
Theorem 3.1. Suppose
$p\in P_\alpha $
and
$$ \begin{align*}p\Vdash_{P_\alpha} \,\dot{x}\in \omega^\omega \setminus \bigcup\limits_{\beta<\alpha}\mathbf{V}^{P_\beta}.\end{align*} $$
There exists a fusion sequence
$\bar {S}=\langle (p_n,F_n,\eta _n):n<\omega \rangle $
below p such that for every n, if
$\gamma $
is the active coordinate of
$F_{n}$
then
$(p_{n}, F_{n}, \eta _{n})$
splits
$\dot x$
at
$\gamma $
.
Moreover, letting
$A_n(\bar {S})$
and
$\langle a^s: s\in \, ^{n}2\rangle $
be defined as in Remark 2.1 and letting
$t_s$
the longest common initial segment of
$\dot {x}\left [ a^{s^{\smallfrown } 0}\right ] $
and
$\dot {x}\left [ a^{s^{\smallfrown } 1}\right ] $
, the following hold:
-
(0)
$\dot {x}\left [ a^{s^{\smallfrown } 0}\right ] (\vert t_s\vert )\perp \dot {x}\left [ a^{s^{\smallfrown } 1}\right ](\vert t_s\vert )$
; -
(1) if
$\vert s\vert <\vert s'\vert $
then
$\vert t_s\vert < \vert t_{s'}\vert $
; -
(2) if
$\gamma $
is the active coordinate of
$F_n$
and
$s, s'\in \, ^{n}2 $
are such that we have
$$ \begin{align*}s\upharpoonright n\setminus \bigcup \left\lbrace Z(\beta):\beta\geq \gamma\right\rbrace \ne s'\upharpoonright n\setminus \bigcup \left\lbrace Z(\beta):\beta\geq \gamma\right\rbrace\end{align*} $$
$\vert t_s\vert \ne \vert t_{s'}\vert $
. (For the definition of
$Z(\beta )$
see Definition 2.2(5).)
Proof. Our recursive construction will guarantee that the following demand holds for every n:
-
(*) n If
$\gamma $
is the active coordinate of
$F_n$
then
$$ \begin{align*}p_n\upharpoonright \gamma \Vdash_{P_\gamma}\, \forall r\leq p_n \upharpoonright \left[ \gamma , \alpha\right)\forall k\quad k\mathrm{\; is \;not \; indifferent\; to\; } r,\dot{x}.\end{align*} $$
For
$(\ast )_n$
to make sense we use the well-known fact that the quotient forcing
$P_\alpha /\dot G_\gamma $
is again a CS-support iteration of
$\mathbb {SI}$
(see [Reference Baumgartner and Mathias1, Section 5]).
We have to find
$p_0$
such that
$(p_0,F_0,\eta _0)$
splits
$\dot {x}$
at
$0$
. For this we first apply Lemma 3.1 and the assumption and conclude that (G2) of Grigorieff’s dichotomy fails below p, hence (G1) holds below p and we obtain
$ q$
as in (G1). Let
$k=\min (\mathrm {com}(q(0)))$
. As k is not indifferent to
$q,\dot {x}$
we find
$p_0\leq q$
as desired. By (G1) we also know that at every later stage
$n\in Z(0)$
we will have
$(\ast )_n$
.
Now suppose we have gotten
$(p_0,F_0,\eta _0),...,(p_n,F_n,\eta _n)$
as desired such that
$(\ast )_m$
holds for every
$m\leq n$
.
Let
$\gamma $
be the active coordinate of
$F_{n+1}$
. We perform a recursion along the lexicographic order of the set
$$ \begin{align*}\Sigma=\left\lbrace s\upharpoonright n+1\setminus \bigcup \left\lbrace Z(\beta):\beta\geq \gamma\right\rbrace :s\in \, ^{n+1}2\right\rbrace \!.\end{align*} $$
Note that its members correspond to functions
$$ \begin{align}\sigma \in \prod\limits_{\beta\in F_{n+1}\cap \gamma } \, ^{\eta_{n+1}(\beta)+1}2\end{align} $$
which are then ordered accordingly, say by
$\prec $
.
We have two subcases according to whether
$n+1=\min (Z(\gamma ))$
or not. In the second case we shall apply
$(\ast )_{m}$
for some
$m<n+1$
in
$Z(\gamma )$
and perform the same recursion as will be done in the first case after some preliminary step. Hence we treat only the first case.
In the first case, as for the construction of
$p_0$
, (working in
$\mathbf {V}[ \dot {G}_\gamma ] $
) we first have to apply our assumption together with (G1) to extend
$p_n\upharpoonright [ \gamma ,\alpha )$
to some
$p^1\in P_\alpha /\dot {G}_\gamma $
such that
$p^1\leq _{F_{n+1}\setminus \gamma +1, \eta _{n+1}\upharpoonright F_{n+1}\setminus \gamma +1}p_n\upharpoonright [ \gamma ,\alpha )$
and
$$ \begin{align}p_n\upharpoonright \gamma \Vdash_{P_\gamma}\, \forall r\leq p^1\,\forall k\quad k\mathrm{\; is \;not \; indifferent\; to\; } r,\dot{x}.\end{align} $$
Such
$p^1$
is obtained as the last element of a
$\leq _{F_{n+1}\setminus \gamma +1, \eta _{n+1}\upharpoonright F_{n+1}\setminus \gamma +1 }$
-decreasing chain considering each
$$ \begin{align}\tau \in \prod\limits_{\beta\in F_{n+1}\setminus \gamma +1 } \, ^{\eta_{n+1}(\beta)+1}2.\end{align} $$
As pedantically,
$p^1$
is a only name for a condition in the quotient forcing (hence
$(p_n\upharpoonright \gamma , p^1)$
is not a member of
$P_\alpha $
), by properness we obtain
$p^0\leq _{F_n\cap \gamma , \eta _n\upharpoonright \gamma } p_n\upharpoonright \gamma $
which decides supp
$(p^1)$
, i.e., turns
$p^1$
into a countable sequence of names so that
$(p^0,p^1)$
will be a member of
$P_\alpha $
. (See [Reference Baumgartner and Mathias1, Section 5] for more details about this.)
Now we start our recursion below
$(p^0,p^1)$
. Let
$\sigma $
be the first sequence as in (3). We have to construct
$q\leq _{F_n,\eta _n} (p^0,p^1) $
such that for every pair of
$$ \begin{align*}u_0,u_1\in\,^{\eta_{n+1}(\gamma)+1}2\end{align*} $$
and every
$\tau $
as in (5), if
$u_0\upharpoonright \eta _{n+1}(\gamma ) =u_1\upharpoonright \eta _{n+1}(\gamma )$
and
$u_0( \eta _{n+1}(\gamma ))<u_1 (\eta _{n+1}(\gamma ))$
then
$$ \begin{align*}\dot{x}\left[ q\ast (\sigma \cup \left\lbrace (\gamma, u_0)\right\rbrace \cup\tau)\right] \perp \dot{x}\left[ q\ast (\sigma \cup \left\lbrace (\gamma, u_1)\right\rbrace\cup\tau )\right],\end{align*} $$
and, letting
$$ \begin{align*}t(q,\sigma, (u_0,u_1), \tau)\end{align*} $$
denote the longest common initial segment of these two incompatible nodes, we have
$$ \begin{align}\vert t_s\vert < \vert t(q,\sigma, (u_0,u_1),\tau)\vert\end{align} $$
for every
$s\in \,^{n}2$
. Note that
$t(q,\sigma , (u_0,u_1),\tau )=t_s$
for some
$s\in \,^{n+1}2$
.
This is easy to achieve. Simply build a finite
$\leq _{F_n,\eta _n}$
-descending chain of conditions
$r\leq (p^0,p^1)$
taking care of every
$(u_0,u_1)$
and
$\tau $
as above. More precisely, if we have obtained r and have to consider
$(u_0,u_1)$
and
$\tau $
, by (4) we know
$$ \begin{align*}r\upharpoonright \gamma \Vdash_{P_\gamma}\, \forall r'\leq r\upharpoonright [ \gamma,\alpha) \,\forall k\quad k\mathrm{\; is \;not \; indifferent\; to\; } r',\dot{x}.\end{align*} $$
Choose
$G_\gamma $
a
$P_\gamma $
-generic filter with
$r\upharpoonright \gamma \ast \sigma \in G_\gamma $
. Work in
$\mathbf {V}\left [ G_\gamma \right ] $
. Let
$u:= u_0\upharpoonright \eta _{n+1}(\gamma ) = u_1\upharpoonright \eta _{n+1}(\gamma ) $
. Choose
$r'\leq (r(\gamma )^{u}, r\upharpoonright [ \gamma +1,\alpha )\ast \tau )$
deciding an initial segment of
$\dot x$
, say
$\xi $
, that is longer than all the
$t_s$
for
$s\in \,^{n}2$
, and let
$k=\min (\mathrm {com}(r'(\gamma )))$
. By (4) we can find
$r"\leq r' $
such that
$k\in \mathrm {com}(r"(\gamma ))$
and
$$ \begin{align*}\dot x[ (r" (\gamma) \cup \left\lbrace (k,0)\right\rbrace , r"\upharpoonright [\gamma+1,\alpha) ] \perp \dot x[ (r" (\gamma) \cup \left\lbrace (k,1)\right\rbrace , r"\upharpoonright [\gamma+1,\alpha) ] .\end{align*} $$
Then let
$r_1$
be such that
$r_1(\gamma )$
is
$r(\gamma )$
except that
$r_1(\gamma )^{u} =r"(\gamma )$
and
$$ \begin{align*}r_1(\gamma)^{u}\Vdash_{P_\alpha/\dot G_{\gamma +1}} r_1\upharpoonright [ \gamma+1,\alpha)\ast \tau=r"\upharpoonright [ \gamma+1,\alpha).\end{align*} $$
Finally choose
$r_0\leq _{F_n\cap \gamma , \eta _n} r\upharpoonright \gamma $
such that
$r_0\ast \sigma \in G_\gamma $
forces all this, decides
$\xi $
and also decides supp
$(r_1)$
(see the above remark how to get
$(p^0,p^1)$
). Then
$r=(r_0,r_1) $
is the next condition in our finite chain. Let q be its last element. We denote
$q=(q\upharpoonright \gamma , q\upharpoonright \left [ \gamma ,\alpha \right )) $
by
$ (q_\sigma ,q^\sigma )$
.
Now suppose we have already dealt with an initial segment of
$\sigma $
’s as in (3), built a
$\leq _{F_n,\eta _n}$
-descending chain of conditions below
$(p^0,p^1)$
with last element
$q $
, and
$\sigma '$
is the next sequence we have to consider. We essentially repeat the above recursion below
$q $
, but this time deciding long enough initial segments of
$\dot x$
so that
$$ \begin{align*}\vert t((q_\sigma, q^\sigma),\sigma, (u_0,u_1),\tau)\vert<\vert t((q_{\sigma'},q^{\sigma'}), \sigma ', (u_0',u_1'),\tau')\vert\end{align*} $$
will hold for every
$\sigma \prec \sigma '$
and all
$(u_0,u_1), (u_0',u_1'),\tau ,\tau '$
as above. Suppose now we have done this for every
$\sigma $
. We define
$p_{n+1}$
as the last condition we have obtained. Then by construction we have
$$ \begin{align*}\dot x\left[ p_{n+1}\ast (\sigma\cup \left\lbrace (\gamma, u_0)\right\rbrace )\cup\tau\right] \perp \dot x\left[ p_{n+1}\ast (\sigma\cup \left\lbrace (\gamma, u_1)\right\rbrace\cup \tau )\right]\end{align*} $$
for every
$\sigma $
,
$(u_0,u_1)$
and
$\tau $
, and if
$\sigma \prec \sigma '$
then
$$ \begin{align*}\vert t(p_{n+1}, \sigma, (u_0,u_1),\tau)\vert < \vert t(p_{n+1}, \sigma', (u_0',u_1'),\tau')\vert\end{align*} $$
for any
$(u_0,u_1),(u_0',u_1'),\tau ,\tau '$
. Hence the theorem is proved.
Remark 3.1. Suppose that
$\dot x$
, the fusion sequence
$\bar S=\langle (p_n,F_n,\eta _n): n<\omega \rangle $
and
$\langle t_s:s\in \, ^{<\omega }2\rangle $
are as in Theorem 3.1 and let
$p_\omega $
be the fusion limit of
$\bar S$
. Moreover, associated with these we have the refining finite antichains
$A_n(\bar {S})=\langle a^s:s\in \,^n2\rangle $
as explained in Remark 2.1(3).
(1) Let
$T=T(\dot x, \bar S)$
be the tree generated by
$\bar {t}=\langle t_s:s\in \, ^n2\rangle $
. By construction,
$T\subseteq \, ^{<\omega }\omega $
is a
$2$
-ary tree such that
$$ \begin{align*}p_\omega \Vdash_{P_\alpha}\, \dot{x}\in \left[ T\right]\end{align*} $$
and
${\mathrm {split}}(T)= \left \lbrace t_s: s\in \, ^{<\omega }2\right \rbrace $
. Moreover
$$ \begin{align*}T=\left\lbrace t\in T: \lnot\,\, p_\omega\Vdash _{P_\alpha}\, \lnot \,t\subseteq \dot{x}\right\rbrace\end{align*} $$
is the tree of possibilities for
$\dot {x}$
below
$p_\omega $
.
(2) If we step into
$\mathbf {V}[G_\gamma ]$
for some
$\gamma <\omega _2$
, where
$G_\gamma $
is
$P_\gamma $
-generic containing
$p_\omega \upharpoonright \gamma $
, then
$G_\gamma $
evaluates
$\dot {x}$
partially. More precisely, the generic reals
$\langle g_\beta :\beta \in \mathrm {supp}( {p_\omega })\cap \gamma \rangle $
determine a possibly partial function
$H_\gamma =H_\gamma (\bar {S}):\omega \rightarrow 2$
with
$$ \begin{align*}\mathrm{dom}(H_\gamma)=\bigcup \lbrace Z( \beta) : \beta \in \mathrm{supp}( {p_\omega})\cap\gamma\rbrace\end{align*} $$
such that for every
$n<\omega $
, only those
$a^s(\bar {S})$
in
$A_n(\bar {S})$
are compatible with
$p_\omega \left [ G_\gamma \right ] $
for which s is compatible with
$H_\gamma $
, and hence
$$ \begin{align*}A^\gamma_n(\bar{S})\left[ G_\gamma\right] :=\lbrace a^s(\bar{S})\left[ G_\gamma\right] : s\in \, {^{n}2}\,\wedge \,\forall i\in\mathrm{dom}(s)\cap\mathrm{dom}(H_\gamma)\,\, s(i)=H_\gamma(i)\rbrace\end{align*} $$
is a maximal antichain in the quotient forcing
$P_{\omega _2}/G_\gamma $
below
$p_\omega \left [ G_\gamma \right ] $
. More explicitly, given
$\beta \in \mathrm {supp}(p_\omega )\cap \gamma $
and
$i\in Z(\beta )$
, we have
$$ \begin{align*}H_\gamma(i) = g_\beta\circ \dot{c}(p_\omega,\beta)[G_\beta](i).\end{align*} $$
(See Definition 2.2(6).)
Note that for any
$a^s(\bar {S})\left [ G_\gamma \right ] , a^{s'}(\bar {S})\left [ G_\gamma \right ] \in A^\gamma _n(\bar {S})\left [ G_\gamma \right ]$
we have
$$ \begin{align*}a^s(\bar{S})\left[ G_\gamma\right] \upharpoonright \gamma = a^{s'}(\bar{S})\left[ G_\gamma\right]\upharpoonright \gamma .\end{align*} $$
Clearly,
$H_\gamma $
is a total function iff
$\mathrm {supp}( {p_\omega })\subseteq \gamma $
, otherwise
$H_\gamma $
is a Silver function, and if
$\beta <\gamma $
,
$H_\gamma \upharpoonright Z(\beta )$
is completely determined by
$g_\beta $
, hence belongs to
$\mathbf {V}^{P_{\beta +1 }}$
.
Corollary 3.1 [Reference Rosłanowski and Steprāns9, Corollary 2.6].
The CS-iteration of
$\mathbb {SI}$
has the 2-localization property.
As outlined in the introduction, [Reference Rosłanowski and Steprāns9, Corollary 2.6] proves the k-localization property for the CS-iteration of
$\mathbb {SI}_k$
, as well as for
$\mathbb {U}_k$
and k-dimensional Sacks forcing
$\mathbb {S}_k$
, for every
$2\leq k<\omega $
.
4
$\mathfrak {C}_2$
-sets covering new reals
In this section we prove that, given any
$P_{\omega _2}$
-name
$\dot x$
for a new real, where
$P_{\omega _2}$
is the CS-iteration of
$\mathbb {SI}$
, it is possible to define a coding system
$\langle u^b:b\in \Omega \rangle $
in the ground model such that
$\Vdash _{P_{\omega _2}}\; \forall b\in \Omega \quad \dot x\not \in [ u^b ] .$
Let me first explain the core idea in the simple case where we replace
${P_{\omega _2}}$
by
$\mathbb {SI}$
and
$\dot x$
by the
$\mathbb {SI}$
-name
$\dot g$
for the Silver real.
From now on let
$\Omega $
denote the set of all infinite and coinfinite subsets of
$\omega $
. For
$b\in \Omega $
we define
$u^b\in \mathbb {U}$
with
$a^{u^b}=\omega \setminus b$
(see Definition 2.1(2)) by recursion on levels as follows: Suppose we have
$n\in b$
and
$t\in u^b\cap \,^n2$
. For any finite partial function s from
$\omega $
to
$2$
we let
$$ \begin{align*}i_s=\vert \left\lbrace j\in\,\mathrm{dom}(s): s(j)=1\right\rbrace \vert \;\mathrm{mod}\; 2.\end{align*} $$
We stipulate that
$t\,^\smallfrown \, i_t$
is the (only) successor node of t in
$u^b$
. Hence
$u^b$
is defined and thus also the coding system
$C=\langle u^b: b\in \Omega \rangle $
. Now we can prove:
$$ \begin{align}\Vdash_{\mathbb{SI}}\; \dot g \in A(C).\end{align} $$
Recall
$$ \begin{align*}A(C)= \,^\omega 2\setminus \bigcup\lbrace [u^b]: b\in\Omega\rbrace.\end{align*} $$
For this, let
$f\in \mathbb {SI}$
and
$b\in \Omega $
. Let
$m=\min (\mathrm {com}(f))$
and
$n=\min (b\setminus m+1)$
. Wlog we may assume that
$n+1\setminus \left \lbrace m\right \rbrace \subseteq \;\mathrm {dom}(f)$
. Define
$$\begin{align*}k = \left\{\begin{array}{cl} 0, & {\mathrm{if}}\; i_{f\upharpoonright n\setminus\left\lbrace m\right\rbrace }=1-f(n)\\ 1, & {\mathrm{otherwise}} \end{array} \right.\end{align*}$$
and
$f'=f\cup \left \lbrace (m,k)\right \rbrace $
. Then clearly
$f'\Vdash _{\mathbb {SI}}\;\dot g\not \in \left [ u^b\right ] $
, as
$f'\upharpoonright n+1\not \in u^b$
. This proves
$(\ast )$
. In the general case, the correct definition of C is more complex. It is given in the following definition. Then Lemma 4.1 will generalize the above argument.
Definition 4.1. (1) Let Y be a nonempty set of ordinals and
$\bar {Z} =\langle Z(\beta ): \beta \in Y \rangle $
a partition of
$\omega $
into infinite sets. Let
$$ \begin{align*}F_n =\left\lbrace \beta \in Y: Z(\beta)\cap n+1 \ne \emptyset\right\rbrace \!.\end{align*} $$
Let
$T\subseteq \,^{<\omega }2$
be a perfect tree such that
$\bar {t}=\langle t_s:s\in \, ^{<\omega }2\rangle $
enumerates canonically
$\mathrm {split}(T)$
, hence
$\left \lbrace t_s: s\in \, ^n2\right \rbrace =\mathrm {split}_n(T)$
, such that the following are satisfied:
-
(i)
$\vert t_s\vert <\vert t_{s'}\vert $
whenever
$\vert s\vert <\vert s'\vert $
; -
(ii) if
$n\in Z(\gamma )$
and
$s, s'\in \, ^{n}2 $
are such that we have
$$ \begin{align*}s\upharpoonright n\setminus \bigcup \left\lbrace Z(\beta):\beta\geq \gamma\right\rbrace\ne s'\upharpoonright n\setminus \bigcup \left\lbrace Z(\beta):\beta\geq \gamma\right\rbrace\!,\end{align*} $$
$\vert t_s\vert \ne \vert t_{s'}\vert $
.
Such a tree
$T $
will be called a coding tree and we write
$T= T(\bar Z)$
to indicate with respect to which
$\bar Z$
property (ii) is satisfied.
Given such T we define a coding system
$C=C(T)=\langle u^b:b\in \Omega (T)\rangle $
as follows: For
$\beta \in Y$
let
$$ \begin{align*}L(\beta)= \left\lbrace \vert t_s\vert : \vert s\vert \in Z(\beta) \right\rbrace \!.\end{align*} $$
Clearly, by (i) the
$L(\beta )$
are infinite and pairwise disjoint. We define
$\Omega (T )$
as follows:
$\Omega (T)= \Omega \cap \lbrace b : \left [ \exists \beta \in Y\,\, b\subseteq L(\beta ) \right ] \,\vee \, [ b\subseteq \bigcup \limits _{\beta \in Y} L(\beta ) \;\wedge $
$\qquad \qquad \qquad \qquad \qquad \forall \beta \in Y\,\, \vert b\cap L(\beta ))\vert \leq 1 ] \,\vee \, [ b\cap \bigcup \limits _{\beta \in Y} L(\beta ) =\emptyset ] \rbrace .$
Clearly
$\Omega (T)$
is dense in
$\left [ \omega \right ] ^{\omega }$
and every
$b\in \Omega (T)$
is coinfinite.
We define
$C= \langle u^b:b\in \Omega (T)\rangle $
such that for
$b\in \Omega (T)$
we have
$u^b\in \mathbb {U}$
with
$a^{u^b}=\omega \setminus b$
defined according to the three types of members of
$\Omega (T)$
as follows:
-
(I) Suppose
$b\subseteq L(\beta )$
for
$\beta \in Y$
. We determine whether
$t\in \, ^{n+1}2$
belongs to
$u^b$
by recursion on
$n\in b$
. Suppose
$t\in u^b\cap \, ^n2$
. Now we require the following:-
• if
$t\not \in T$
let
$t\,^\smallfrown \, 0\in u^b$
; -
• if
$t \in T\setminus \mathrm {split}(T)$
let
$t\,^\smallfrown \, i\in u^b$
such that
$t\,^\smallfrown \, i\not \in T$
; -
• if
$t\in \mathrm {split}(T)$
, thus
$t=t_s$
for some
$s\in \,^{<\omega }2$
, letting we stipulate that
$$ \begin{align*}i(t,\beta):= \vert \left\lbrace j\in \vert s\vert \cap Z(\beta) :s(j)=1\right\rbrace \vert \,\mathrm{mod}\,2,\end{align*} $$
$$ \begin{align*}t\,^\smallfrown\, t_{s\,^\smallfrown\, i(t,\beta)}(\vert t\vert )\in u^b.\end{align*} $$
-
-
(II) Suppose
$\forall \beta \in Y\; \vert b\cap L(\beta )\vert \leq 1$
and
$b\subseteq \bigcup \limits _{\beta \in Y} L(\beta ) $
. We require the following:-
• if
$t\not \in T$
or
$t\in \mathrm {split}(T)$
let
$t\,^\smallfrown \, 0\in u^b$
; -
• if
$t \in T\setminus \mathrm {split}(T)$
let
$t\,^\smallfrown \, i\in u^b$
such that
$t\,^\smallfrown \, i\not \in T$
;
-
-
(III) Suppose
$ b\cap \bigcup \limits _{\beta \in Y} L(\beta ) =\emptyset $
. Hence for no
$t\in \,\mathrm {split}(T)$
do we have
$\vert t\vert \in b$
. We define
$u^b$
as in case II.
(2) Suppose that the objects as in (1) are given and
$Y'$
is another countable set of ordinals that is order isomorphic to Y. Let
$\pi :Y\rightarrow Y'$
be the isomorphism. Let
$\bar Z'=\langle Z'(\beta ):\beta \in Y'\rangle $
be the partition of
$\omega $
induced by
$\pi $
, i.e.,
$Z'(\pi (\beta ))=Z(\beta )$
for every
$\beta \in Y$
. Similarly, we obtain the induced finite sets
$F_n'\subseteq Y'$
such that
$Y'=\bigcup \left \lbrace F_n':n<\omega \right \rbrace \!.$
Then clearly (1)(ii) holds for
$F_n', Z'(\beta )$
as well and
$L(\beta )=L'(\pi (\beta ))$
, where
$$ \begin{align*}L'(\pi(\beta))=\left\lbrace \vert t_s\vert :\vert s\vert \in Z'(\pi (\beta))\right\rbrace \!.\end{align*} $$
Moreover,
$\Omega (T(\bar Z))=\Omega (T(\bar Z'))$
and we obtain the same trees
$u^b$
if we replace
$Z(\beta )$
by
$Z'(\pi (\beta ))$
and
$L(\beta )$
by
$L'(\pi (\beta ))$
.
Lemma 4.1. Suppose that
$P_\alpha $
is a CS-iteration of
$\mathbb {SI}$
,
$p\in P_\alpha $
and
$$ \begin{align*}p\Vdash_{P_\alpha} \,\dot{x}\in \omega^\omega \setminus \bigcup\limits_{\beta<\alpha}\mathbf{V}^{P_\beta}.\end{align*} $$
Suppose also that
$\bar S=\langle (p_n,F_n,\eta _n):n<\omega \rangle $
is a fusion sequence below p with limit
$p_\omega $
for
$\dot x$
as in Theorem 3.1. Hence we have the associated partition
$\bar {Z}=\langle Z(\beta ):\beta \in \,\mathrm {supp}(p_\omega )\rangle $
and perfect tree
$T=T(\dot x, \bar S)$
with splitnodes
$\bar t =\langle t_s: s\in \, ^n2\rangle $
. Clearly
$T=T(\bar Z)$
is a coding tree as in Definition 4.1. If the coding system
$C=C(T )= \langle u^b:b\in \Omega (T)\rangle $
is defined as there we have
$$ \begin{align*}p_\omega\Vdash _{P_\alpha}\, \forall b\in \Omega(T)\cap\mathbf{V}\quad \dot x\not\in [ u^b ] .\end{align*} $$
Proof. Suppose that
$q\leq p_\omega $
and
$b\in \Omega (T)\cap \mathbf {V}$
are arbitrary. We have to find
$q'\leq q$
such that
$$ \begin{align*}q'\Vdash_{P_\alpha}\; \dot{x}\not\in \left[ u^b\right] .\end{align*} $$
Let the antichains
$A_n(\bar {S})=\langle a^s: s\in \,^{n}2\rangle $
be defined as in Remark 2.1(3).
Define
$$ \begin{align*}R^q=\left\lbrace s\in\, ^{<\omega}2: \lnot \,q\perp a^s\right\rbrace\end{align*} $$
and let
$$ \begin{align*}T^q= \left\lbrace t\in T: \lnot\, q\Vdash _{P_\alpha}\, \lnot \,t\subseteq \dot{x}\right\rbrace\end{align*} $$
be the tree of possibilities for
$\dot {x}$
below q. Clearly,
$R^q$
and
$T^q$
are trees and
$T^q$
is generated by
$ \left \lbrace t_s: s \in R^q \right \rbrace \!.$
Also note that if
$t\in T^q$
and
$t=t_s$
for some s, then
$s\in R^q$
, as otherwise let
$s_0\in R^q$
be maximal with
$s_0\subseteq s$
. Hence every
$s'\in R^q$
that is not an initial segment of
$s_0$
either extends
$s_0\,^\smallfrown \, 1-s(\vert s_0\vert )$
or is incompatible with
$s_0$
. As the
$t_{s'}$
for
$s'\in \,^{<\omega }2$
are the splitnodes of T we conclude that
$t\not \in T^q$
, a contradiction.
Note that wlog we may assume that for every
$l\in b$
and for every
$t\in T^q$
of length l there is
$s\in R^q$
such that
$t=t_s$
. Indeed, otherwise we know by the remark we just made that t is not a splitnode of T but there is
$q'\leq q$
forcing
$\dot {x}\upharpoonright l =t$
and hence by definition of
$u^b$
we have
$$ \begin{align*}q' \Vdash _{P_\alpha}\, \dot{x} \upharpoonright l+1 \not\in u^b ,\end{align*} $$
and we are done.
Now the proof proceeds along the three cases of Definition 4.1.
Case I:
We consider
$b\subseteq L(\beta )$
for some
$\beta \in \,\mathrm {supp}(p_\omega )$
.
Choose
$G_\beta $
a
$P_\beta $
-generic filter containing
$q\upharpoonright \beta $
. Hence
$q(\beta )\left [ G_\beta \right ] \leq ^{\mathbb {SI}}p_\omega (\beta )\left [ G_\beta \right ]$
and there exist k and
$v\in \,^{k}2$
such that
$$ \begin{align*}\mathrm{stem}(q(\beta)\left[ G_\beta\right] )=\,\mathrm{stem}(p_\omega(\beta)\left[ G_\beta\right]^v).\end{align*} $$
Let
$m_0=\, \vert \mathrm {stem}(q(\beta )\left [ G_\beta \right ] )\vert $
. By the bookkeeping we used for the fusion by which we obtained
$p_\omega $
we know the step at which coordinate
$\beta $
was active for the kth time, say it was step
$n_0$
. We choose
$l\in b$
larger than
$\max \lbrace \vert t_s\vert : s\in \,^{n_0}2\rbrace $
. By property (i) of a coding tree (see Definition 4.1) and by our observation above (that every
$t\in T^q$
of length l is of the form
$t_s$
for some
$s\in R^q$
), there is a unique
$n_1\in Z(\beta )$
such that for every
$t\in T^q$
of length l there exists
$s\in \,^{n_1}2$
with
$t= t_s $
.
Moreover, in
$\mathbf {V}[G_\beta ]$
the tree of possibilities for
$\dot {x}$
below q (with respect to the forcing
$P_{\alpha }/G_\beta $
) has been further restricted to
$T^q[G_\beta ]$
defined as follows: Letting
$$ \begin{align*}R^q[G_\beta]=\left\lbrace s\in\,^{<\omega}2: a^s(\bar{S})[G_\beta]\upharpoonright \beta\in G_\beta\,\wedge\, \lnot\,\, a^s(\bar{S}) \,\perp_{ P_{\alpha}/G_\beta} \,q \right\rbrace\!,\end{align*} $$
$T^q[G_\beta ]$
is the tree generated by the set of all
$t_s$
with
$s\in R^q[G_\beta ]$
.
Given
$s\in R^q[G_\beta ]\cap \,^{n_1}2$
and letting
$m_1= \dot {c}(p_\omega ,\beta )[G_\beta ](n_1)$
(see Definition 2.2(6)), the value of
$\dot {x}(\vert t_s\vert )$
is determined by
$\dot {g}_\beta $
as
$t_{s\,^ \smallfrown \, \dot {g}_\beta (m_1)}( \vert t_s\vert ),$
more precisely, we have
$$ \begin{align*}q\wedge a^s(\bar{S}) \Vdash_{P_{\omega_2}/G_\beta} \, \dot{x}(\vert t_s\vert)=t_{s\,^ \smallfrown \, \dot{g}_\beta (m_1)}( \vert t_s\vert).\end{align*} $$
In
$\mathbf {V}$
, we can find
$\hat {q}\leq q $
in
$P_{\alpha }$
as follows:
$\hat q\upharpoonright \beta \leq q\upharpoonright \beta $
forces all the facts we have noticed above after we fixed
$G_\beta $
, and
$\hat q\upharpoonright \beta $
also decides
$k,v,m_0,n_0,l, n_1,m_1$
as above. Moreover,
$$ \begin{align*}\hat q\upharpoonright\beta\Vdash_{P_\beta}\, \mathrm{com}(\hat{q}(\beta))\cap m_1+1= \left\lbrace m_0\right\rbrace \!,\end{align*} $$
$\hat q\upharpoonright \beta $
decides
$\hat {q}(\beta )\upharpoonright m_1+1$
, thus
$\dot g_\beta \upharpoonright (m_1+1 \setminus \lbrace m_0\rbrace ) $
, say as
$\langle g(0),...,g(m_0 -1),g(m_0 +1),...,g(m_1)\rangle $
, and we let
$\hat {q}\upharpoonright [\beta +1,\omega _2)=q\upharpoonright [\beta +1,\omega _2)$
.
Now we can find
$s_0,s_1\in R^{\hat {q}}\cap \, ^{n_1} 2$
such that
$s_i(n_0)=i$
and
$s_0(j) =s_1(j)$
for every
$j\in Z(\beta )\cap n_1\setminus \left \lbrace n_0\right \rbrace $
, and hence
$$ \begin{align*}i( t_{s_0\,^\smallfrown\,g(m_1)},\beta)\ne i(t_{s_1\,^\smallfrown\,g(m_1)},\beta).\end{align*} $$
As remarked above, we know that both
$t_{s_0}$
and
$t_{s_1}$
are splitnodes of T of length l. Now choose j such that
$i(t_{s_j\,^\smallfrown \,g(m_1)},\beta ) \ne g(m_1)$
and a common extension
$q'$
of
$\hat {q}$
and
$a^{s_j}$
. We conclude
$$ \begin{align*}q'\Vdash_{P_\alpha}\; t_{s_j\,^\smallfrown\,g(m_1)} \upharpoonright \vert t_{s_j} \vert +1\subset \dot x\end{align*} $$
and hence
$q'\Vdash _{P_\alpha }\; \dot x\not \in \left [ u^b\right ],$
by definition of
$u^b$
.
Case II:
We have
$b\subseteq \bigcup \limits _{\beta \in \mathrm {supp} (p_\omega )} L(\beta ) $
and
$\forall \beta \in \,\mathrm {supp}(p_\omega )\; \vert b\cap L(\beta )\vert \leq 1$
. In this case we shall apply property (ii) of a coding tree (see Definition 4.1). As
$$ \begin{align*}\lbrace n\in Z(0): {c}(p_\omega, 0)(n) \in \mathrm{com} (q(0)) \rbrace\end{align*} $$
is infinite (see Definition 2.2(6)), certainly we can find a large enough
$\beta \in \,\mathrm {supp}(p_\omega )$
such that there are
$l\in b\cap L(\beta )$
,
$n\in Z(\beta )$
and
$s\in \, ^{n}2$
,
$ s_0,s_1\in \, ^{n}2\cap R^q$
such that
$\vert t_s\vert =l$
and
$$ \begin{align*}s_0\upharpoonright n\setminus \bigcup\left\lbrace Z(\gamma):\gamma \geq \beta\right\rbrace \ne s_1\upharpoonright n\setminus \bigcup\left\lbrace Z(\gamma):\gamma \geq \beta\right\rbrace \!.\end{align*} $$
By property (ii) of a coding tree we know that
$\vert t_{s_0}\vert \ne \vert t_{s_1}\vert $
. Hence we can pick
$i<2$
such that
$\vert t_{s_i} \vert \ne l$
. We can find
$q'$
extending both q and
$a_{s_i}$
which decides
$\dot x\upharpoonright l$
, say as t. By property (i) of a coding tree we know that t is not a splitting node of T, and hence
$$ \begin{align*}q'\Vdash_{P_\alpha}\, \dot x\upharpoonright l+1\not\in u^b\end{align*} $$
by definition of
$u^b$
.
Case III:
We have
$b\cap \bigcup \limits _{\beta \in \mathrm {supp}(p_\omega )} L(\beta ) =\emptyset $
and we know that no
$t\in \,\mathrm {split}(T)$
has
$\vert t\vert \in b$
. Hence by definition of
$u^b$
we conclude
$$ \begin{align*}p_\omega\Vdash_{P_\alpha}\, \dot x \not\in [u^b].\end{align*} $$
Definition 4.2. Suppose we are given two
$P_{\alpha }$
-names for reals,
$\dot x$
and
$\dot x'$
, together with conditions
$p,p'$
, fusion sequences
$\bar S$
,
$\bar S'$
below
$p,p'$
, respectively, fusion limits
$p_\omega $
,
$p_\omega '$
and associated partitions of
$\omega $
,
$\bar Z$
and
$\bar Z'$
, respectively, as in Lemma 4.1, such that
$$ \begin{align*}p\Vdash_{P_\alpha} \,\dot{x}\in \omega^\omega \setminus \bigcup\limits_{\beta<\alpha}\mathbf{V}^{P_\beta} \;\mathrm{and}\; p'\Vdash_{P_{\alpha'}} \,\dot{x}'\in \omega^\omega \setminus \bigcup\limits_{\beta<\alpha'}\mathbf{V}^{P_\beta}.\end{align*} $$
We call
$\dot x$
and
$\dot x'$
isomorphic if there is an order isomorphism
$$ \begin{align*}\pi: \mathrm{supp}(p_\omega)\rightarrow\mathrm{supp}(p_\omega')\end{align*} $$
such that
$Z(\beta )=Z'(\pi (\beta ))$
and
$T(\dot x,\bar S)=T(\dot x',\bar S') $
with the same sequence
$\bar t=\langle t_s: s\in \,^{<\omega }2\rangle $
of splitnodes. Then Definition 4.1(2) applies and we know that
$\Omega (T(\dot x,\bar S))=\Omega (T(\dot x',\bar S'))$
and
$C(T(\dot x, \bar S))=C(T(\dot x',\bar S'))$
, i.e., isomorphic names are associated with the same coding system. Clearly, under CH there are only
$\aleph _1$
-many isomorphism types of names.
5 Extending the coding systems
As our goal is to prove cov
$(\mathfrak {C}_2) =\aleph _1$
in the iterated Silver model, we need to construct a family of
$\aleph _1$
-many coding systems coding
$\mathfrak {C}_2$
-sets covering
$2^\omega $
. For this, Lemma 4.1 is a good start, as any two isomorphic names of reals give rise to the same coding system as we have noticed at the end of Definition 4.2.
However, whenever by some forcing new reals are added then no dense
$\Theta \subseteq \left [ \omega \right ] ^\omega $
in
$\mathbf {V}$
will be dense in the extension (see the argument in the introduction). Hence the coding system
$C(T(\bar Z))=\langle u^b:b\in \Omega (T(\bar Z))\rangle $
defined in Definition 4.1 where
$T(\bar Z)=T(\dot x,\bar S)$
as in Lemma 4.1 has to be extended cofinally often during the iteration so that it will code a
$\mathfrak {C}_2$
-set A in the final model. This extension has to be done in such a way that Lemma 4.1 will remain true for every name belonging to the equivalence class
$\mathcal {K}$
of
$\dot x$
, i.e., the evaluation of every
$\dot x '\in \mathcal {K}$
belongs to A.
As is well-known, new reals appear only in intermediate models
$\mathbf {V}^{P_{\gamma }}$
where
$\gamma>0$
and cf
$(\gamma )$
is countable (i.e., finite and hence 1, or countably infinite). Hence at stages
$\gamma \leq \omega _2$
of uncountable cofinality we can take unions of the coding systems we already constructed. Hence we actively extend the coding systems only at steps
$\gamma>0$
of countable cofinality. In order to make sure that no old real is a branch through some new tree we are adding to the system at that stage, we make use of the
$\gamma $
th Silver real
$\dot {g}_\gamma $
. Hence this extension will be completed only in the model
$\mathbf {V}^{P_{\gamma +1}}$
.
We let
$\Omega ^0(T(\bar Z))=\Omega (T(\bar Z))$
and
$C^{0}(T(\bar Z))=C(T(\bar Z))$
. So both belong to
$\mathbf {V}$
. Now we define
$\Omega ^{\gamma } (T)$
and the coding system
$C^{\gamma } (T(\bar Z))=\langle u^{\dot {b}} :\dot {b}\in \Omega ^{\gamma } (T)\rangle $
for every
$0<\gamma <\omega _2$
such that
$\Omega ^{\gamma } (T)\in \mathbf {V}^{P_{\gamma }}$
always and
$C^{\gamma } (T(\bar Z))\in \mathbf {V}^{P_{\gamma }}$
if cf
$(\gamma )=\omega _1$
and
$C^{\gamma } (T(\bar Z))\in \mathbf {V}^{P_{\gamma +1}}$
if cf
$(\gamma )$
is countable.
Suppose we have already extended
$C(T(\bar Z))$
to a coding system
$C^{\beta } (T(\bar Z))=\langle u^{\dot {b}} :\dot {b}\in \Omega ^{\beta } (T)\rangle $
in
$\mathbf {V}^{P_{\beta +1}}$
for every
$\beta <\gamma $
, for some
$0<\gamma <\omega _2$
. If
$\gamma $
is a limit of uncountable cofinality we let
$$ \begin{align*}\Omega^{\gamma}(T(\bar Z))=\bigcup\lbrace\Omega^\beta (T(\bar Z)):\beta<\gamma \rbrace\end{align*} $$
and
$$ \begin{align*}C^{\gamma } (T(\bar Z))= \bigcup\lbrace C^\beta (T(\bar Z)):\beta<\gamma \rbrace .\end{align*} $$
If
$ \gamma $
has countable cofinality, let
$$ \begin{align*}\Omega^{\gamma }(T(\bar Z))=\lbrace \dot{b}\in \Omega (T(\bar Z)) ^{\mathbf{V}^{P_{\gamma }}}: \lnot\,\,\exists c\in \left[\omega \right] ^\omega \cap \bigcup_{\beta<\gamma}\mathbf{V}^{P_\beta} \quad c\subseteq\dot{b} \rbrace.\end{align*} $$
(For the definition of
$\Omega (T(\bar Z))$
see Definition 4.1(1).) Clearly,
$\Omega ^{\gamma }(T(\bar Z))$
is dense in
$\left [\omega \right ] ^\omega \cap \mathbf {V}^{P_{\gamma }}$
. In this case, in
$\mathbf {V}^{P_{\gamma +1}}$
we shall define a coding system
$\langle u^{\dot {b}}: \dot {b}\in \Omega ^{\gamma }(T(\bar Z))\rangle $
such that for every
$\dot x'\in \mathcal {K}$
(with associated isomorphism
$\pi '$
, fusion sequence
$\bar S'$
and limit
$p^{\prime }_\omega $
), its evaluation in the final model is not a branch through any of the
$u^{\dot {b}}$
.
As in Definition 4.1, the definition of
$u^{\dot {b}}$
will depend on the three types of members of
$\Omega ^{\gamma }(T(\bar Z))$
. But now, the two first ones of these (i.e.,
$\dot {b}\subset L(\beta )$
for some
$\beta \in \mathrm {supp}(p_\omega )$
or
$\dot {b}\subseteq \bigcup _{\beta \in \mathrm {supp}(p_\omega )}L(\beta )$
and
$\forall \beta \in \mathrm {supp}(p_\omega )\,\vert \dot {b}\cap L(\beta )\vert \leq 1$
) split into two subcases taking care of how
$\mathrm {supp} (p_\omega ')$
and
$\pi '(\beta )$
are positioned with respect to
$\gamma $
. For this, disjoint infinite subsets
$\dot {b}_0,\dot {b}_1$
of
$\dot {b}$
are chosen and then, on levels from
$\dot {b}_0$
,
$u^{\dot {b}} $
is defined in
$\mathbf {V}^{P_\gamma }$
, whereas on levels from
$\dot {b}_1$
we need the Silver real
$\dot {g}_\gamma $
to define it.
For our main result we shall need to know that in the Silver model, the ground model reals are a
$\mathfrak {P}_2$
-set. This seems to be proved by [Reference Cichoń, Rosłanowski, Steprāns and Weglorz2, Corollary 4.4] by using the Sacks property of
$\mathbb {SI}$
. Let us give a direct proof here.
Lemma 5.1. Suppose that
$f_0\in \mathbb {SI}$
and
$\dot b$
is a
$\mathbb {SI}$
-name such that
$f_0\Vdash _{\mathbb {SI}} \,\dot {b}\in [\omega ] ^\omega .$
There exist
$\mathbb {SI}$
-names
$\dot c, \dot y$
and a condition
$f\in \mathbb {SI}$
such that
$f\leq f_0$
and
$$ \begin{align*}f\Vdash_{\mathbb{SI}} \,\dot c\in [ \dot b ] ^{\omega} \wedge \dot y\in \,^{\dot c}2\setminus \left\lbrace x\upharpoonright \dot c : x\in \,^\omega 2\cap \mathbf{V}\right\rbrace .\end{align*} $$
Proof. By a simple fusion we can find
$f\in \mathbb {SI}$
and a one-to-one family
$\langle k_s:s\in \, ^{<\omega } 2\rangle $
in
$\omega $
such that
$f\leq f_0$
and
$$ \begin{align*}f^s\Vdash_{\mathbb{SI}} k_s\in \dot b\end{align*} $$
for every
$s\in \, ^{<\omega } 2$
. Letting
$\dot g$
be the canonical name for the Silver real and defining
$$ \begin{align*}\dot c:= \left\lbrace k_s: f^s\subseteq \dot g\right\rbrace ,\end{align*} $$
clearly
$f\Vdash _{\mathbb {SI}}\, \dot c\in [ \dot b ] ^{\omega }$
. Now we define
$\dot y $
on
$\dot c$
such that
$$ \begin{align*}\dot y (k_s)=0 \quad \mathrm{iff} \quad k_{s^{\,\smallfrown \,} i }\in \dot c\;\wedge \; k_{s^{\,\smallfrown \,} i}<k_{s^{\,\smallfrown \,} 1-i}.\end{align*} $$
Given any
$f'\leq f$
and
$x\in \,^\omega 2\cap \mathbf {V}$
, let s be maximal with
$f'\leq f^s$
and let
$j=\min (\mathrm {com}(f'))$
. Hence
$f'\Vdash _{\mathbb {SI}} k_s\in \dot c$
, and
$\dot y(k_s)$
depends on
$\dot g(j)$
, thus has not yet been decided by
$f'$
. Hence we can find
$f"\leq f'$
forcing
$\dot y(k_s)\ne x(k_s)$
. Note that the lemma is nontrivial only if
$\dot {b}$
is forced to be outside
$\mathbf {V}$
.
Corollary 5.1. (1)
$\mathbf {V}^{\mathbb {SI}}\models \; ^\omega 2\cap \mathbf {V} \in \mathfrak {P}_2$
.
(2) If
$P_{\omega _2}$
is the CS iteration of
$\mathbb {SI}$
of length
$\omega _2$
then
$\mathbf {V}^{P_{\omega _2}}\models \; ^\omega 2\cap \mathbf {V} \in \mathfrak {P}_2.$
Proof. (1) In
$\mathbf {V}^{\mathbb {SI}}$
, let
$\Theta \subseteq \,^\omega [\omega ]$
be dense such that for every
$b\in \Theta $
we have
$[b]^\omega \cap \mathbf {V} =\emptyset $
. By Lemma 5.1, for every
$b\in \Theta $
find a Silver function
$f^b$
with dom
$(f^ b)=b$
such that for no
$x\in \, ^\omega 2\cap \mathbf {V}$
,
$x\upharpoonright b =f^b$
. Then
$C=\langle f^b:b\in \Theta \rangle $
codes the
$\mathfrak {P}_2$
-set
$$ \begin{align*}A(C)= \,^\omega 2\setminus \bigcup\left\lbrace [f ^ b]: b\in \Theta\right\rbrace\end{align*} $$
which has the property
$^\omega 2\cap \mathbf {V}\subseteq A(C )$
.
(2) For every
$\gamma <\omega _2$
, in
$\mathbf {V}^{P_{\gamma +1}}$
we can choose a dense
$\Omega ^{\gamma +1} \subseteq \left [ \omega \right ] ^\omega $
such that for every
$b\in \Omega ^{\gamma +1} $
,
$\left [ b\right ] ^\omega \cap \mathbf {V}^{P_{\gamma }} =\emptyset $
, and then, as in (1), define a coding system
$C^{\gamma +1} =\langle f^b:b\in \Omega ^{\gamma +1} \rangle $
such that
$^\omega 2\cap \mathbb {V}^{P_\gamma }\subseteq A(C^{\gamma +1} )$
. In
$\mathbf {V}^{P_{\omega _2}}$
we have the coding system
$C=\langle f^b: b\in \Omega ^{\gamma +1} , \gamma <\omega _2\rangle $
defining the
$\mathfrak {P}_2$
-set
$A(C)$
which covers
$^\omega 2\cap \mathbf {V} $
.
Remark 5.1. In [Reference Cichoń, Rosłanowski, Steprāns and Weglorz2, Proposition 4.13] it is shown that Corollary 5.1(1) is false for Laver as well as for Miller forcing.
Now the precise crucial definition of how we extend our coding systems along the iteration is as follows.
Definition 5.1. Suppose that in
$\mathbf {V} $
we are given some coding tree
$T(\bar Z)$
. Recall that in particular
$\bar Z=\langle Z(\beta ): \beta \in Y\rangle $
is a partition of
$\omega $
into infinite sets,
$\bar t=\langle t_s: s\in \,^{<\omega }2\rangle $
canonically enumerates
$\mathrm {split}(T(\bar Z))$
and
$L(\beta )=\left \lbrace \vert t_s\vert : \vert s\vert \in Z(\beta )\right \rbrace $
. Let
$\dot x$
with fusion
$\bar S$
and limit
$p_\omega $
be associated with
$T(\bar Z)$
, so in particular
$\mathrm {supp} (p_\omega )=Y$
and
$T(\bar Z)=T(\dot x,\bar S)$
. Let
$A_n(\bar {S})=\langle a^s=a^s(\bar {S}): s\in \,^{n}2\rangle $
for
$n<\omega $
be the associated refining maximal antichains below
$p_\omega $
(see Remark 2.1(3)). Let
$\pi =\mathrm { id}_Y$
.
Let
$\mathcal {K} $
be the isomorphism class of names associated with
$T(\bar Z)$
. For
$\dot x' \in \mathcal {K}$
let
$\bar S'$
be the associated fusion,
$p_\omega '$
its limit,
$\pi ' :Y \rightarrow \mathrm {supp}(p_\omega ')$
the isomorphism and
$A_n(\bar {S}')=\langle a^s(\bar {S}'): s\in \,^{n}2\rangle $
the associated maximal antichains below
$p_\omega '$
. Moreover, let
$\dot {g}_\gamma \in \mathbf {V}^{P_{\gamma +1}}$
be the Silver real added by the iterand
$\dot Q_\gamma $
of the iteration.
Given any
$ {b}\subseteq \omega $
, define
$Z(b)=\left \lbrace \vert s\vert : \vert t_s\vert \in b\right \rbrace .$
Note that it may happen that there are s with
$\vert s\vert \in Z(b)$
but
$\vert t_s\vert \not \in b$
(see property (ii) of a coding tree).
For every
$\gamma <\omega _2$
, in
$\mathbf {V}^{P_{\gamma }}$
we have already defined
$\Omega ^{\gamma }(T(\bar Z))$
at the beginning of this section. Now let us define the coding system
$$ \begin{align*}C^{\gamma}(T(\bar Z))=\langle u^{\dot{b}}: \dot{b}\in \Omega^{\gamma}(T(\bar Z))\rangle\end{align*} $$
in
$\mathbf {V}^{P_{\gamma +1}}$
. For
$\dot {b}\in \Omega ^{\gamma }(T(\bar Z))$
,
$u^{\dot {b}}$
will be defined according to the three types of members of
$ \Omega ^{\gamma }(T(\bar Z))$
. Always,
$u^{\dot {b}}$
will be defined as a uniform tree with
$a^{u^{\dot {b}}}=\omega \setminus {\dot {b}}$
.
(I) Suppose
${\dot {b}}\in \Omega ^{\gamma }(T(\bar Z))$
is such that
${\dot {b}}\subseteq L(\beta ) $
for some
$\beta \in Y$
. In
$\mathbf {V}^{P_{\gamma }}$
, let
${\dot {b}}={\dot {b}}_0\cup {\dot {b}}_1 $
be a partition into two infinite sets. Levels of
$u^{\dot {b}}$
in
${\dot {b}}_0$
are defined in
$\mathbf {V} ^{P_{\gamma }}$
while levels in
${\dot {b}}_1$
are defined in
$\mathbf {V} ^{P_{\gamma +1}}$
. More precisely, for every
$l\in {\dot {b}}_0$
and
$t\in \,^l2$
, at first we define a next digit
$d(t)<2$
(in
$\mathbf {V}^{P_{\gamma }}$
). Then, in
$\mathbf {V}^{P_{\gamma +1}}$
, for every
$l\in {\dot {b}}_1$
and
$t\in \,^l2$
a next digit
$d(t)$
will be defined. Then, in
$\mathbf {V} ^{P_{\gamma +1}}$
,
$u^{\dot {b}}$
will be the non-empty uniform tree such that for every
$t\in u^{\dot {b}}$
, if
$\vert t\vert \in {\dot {b}}$
, then its next digit is
$d(t)$
.
-
(0) (On levels from
${\dot {b}}_0$
we make sure that we can deal with
$\dot {x}'\in \mathcal {K} $
with associated isomorphism
$\pi '$
such that
$\gamma \leq \pi '(\beta )$
.) On levels in
${\dot {b}}_0$
we define
$u^{\dot {b}}$
in
$\mathbf {V}^{P_{\gamma }}$
analogously as in Definition 4.1(1)(I): Suppose
$l\in {\dot {b}}_0$
and
$t\in \, ^{l}2$
. Then
$d(t)$
is determined as follows:-
• if
$t\not \in T(\bar Z)$
let
$d(t)=0 $
; -
• if
$t \in T(\bar Z)\setminus \mathrm {split}(T(\bar Z))$
let
$d(t)$
be such that
$t\,^\smallfrown \, d(t)\not \in T(\bar Z)$
; -
• if
$t\in \mathrm {split}(T(\bar Z))$
, thus
$t=t_s$
for some
$s\in \,^{<\omega }2$
, letting we stipulate that
$$ \begin{align*}i(t,\beta):= \vert \left\lbrace j\in \vert s\vert \cap Z(\beta) :s(j)=1\right\rbrace \vert \,\mathrm{mod}\,2,\end{align*} $$
$$ \begin{align*}d(t)= t_{s\,^\smallfrown\, i(t,\beta)}(\vert t\vert ) .\end{align*} $$
-
-
(1) (On levels from
${\dot {b}}_1$
, we take care of those
$\dot {x}'$
with
$\pi '(\beta ) <\gamma $
.) In
$\mathbf {V} ^{P_{\gamma +1}}$
we can choose a real
$\dot {h}:Z({\dot {b}}_1)\rightarrow 2$
such that
$\dot {h}\not \in \mathbf {V}^{P_{\gamma }}$
, e.g., if
$\dot {\rho } :Z({\dot {b}}_1 ) \rightarrow \omega $
is a bijection in
$\mathbf {V}^{P_{\gamma }}$
, let
$\dot {h}=\dot {g}_\gamma \circ \dot {\rho }$
.Now suppose
$l\in {\dot {b}}_1$
and
$t\in u^{\dot {b}}\cap \,^{l}2$
. Define
$d(t)<2$
as follows:-
• if
$t\not \in T(\bar Z)$
let
$d(t)=0 $
; -
• if
$t \in T(\bar Z)\setminus \mathrm {split}(T(\bar Z))$
let
$d(t) $
such that
$t\,^\smallfrown \, d(t)\not \in T(\bar Z)$
; -
• if
$t\in \mathrm {split}(T(\bar Z))$
, hence
$t=t_s$
for some s, and
$\vert s\vert =i\in Z({\dot {b}}_1)$
for some i, we let
$$ \begin{align*}d(t)= t_{s\,^\smallfrown\, h(i)}(\vert t\vert ) .\end{align*} $$
-
(II) Suppose
${\dot {b}}\in \Omega ^{\gamma }(T(\bar Z))$
is such that
$\forall \beta \in Y\; \vert {\dot {b}}\cap L(\beta ) \vert \leq 1$
and
${\dot {b}}\subseteq \bigcup \left \lbrace L(\beta ):\beta \in Y\right \rbrace $
. For
$b_*\subseteq {\dot {b}}$
let
$$ \begin{align*}Y(b_*)=\left\lbrace \beta\in Y: b_*\cap L(\beta)\ne \emptyset\right\rbrace .\end{align*} $$
Clearly,
$Y({\dot {b}})\in \mathbf {V}^{P_{\gamma }}$
is infinite. Note that for any
$\dot {x}'\in \mathcal {K}$
we have
$$ \begin{align*}Y( b_*) = \left\lbrace \beta\in Y: b_*\cap L'(\pi'(\beta)) \ne \emptyset\right\rbrace \!.\end{align*} $$
(See Definition 4.1(2).) Let
$\dot {\delta } $
be the largest accumulation point of
$Y(\dot {b})$
. It is easy to choose
$ {\dot {b}}_0$
and
${\dot {b}}_1$
in
$\mathbf {V}^{P_{\gamma }}$
such that
${\dot {b}}_0$
and
${\dot {b}}_1$
are disjoint infinite subsets of
${\dot {b}}$
with
$\sup (Y({\dot {b}}_0))=\sup (Y({\dot {b}}_1)) =\dot \delta $
and
$\dot \delta \not \in Y({\dot {b}}_0\cup {\dot {b}}_1)$
. As in Case I, on the levels in
${\dot {b}}_0$
,
$u^{\dot {b}}$
will be defined in
$\mathbf {V}^{P_{\gamma }}$
, while its levels in
${\dot {b}}_1$
will be defined in
$\mathbf {V}^{P_{\gamma +1}}$
, by defining a next-digit-function d on
$\bigcup _{l\in {\dot {b}}_0}\,^l2$
,
$\bigcup _{l\in {\dot {b}}_1}\,^l2$
,
$\bigcup _{l\in \dot {b}\setminus (\dot {b}_0\cup \dot {b}_1})\,^l2$
, respectively, and then letting
$u^{\dot {b}}$
be the non-empty tree such that for every
$t\in u^{\dot {b}}$
with
$\vert t\vert \in {\dot {b}}$
, its next digit is
$d(t)$
.
-
(0) (On levels from
${\dot {b}}_0$
we take care of
$\dot x'\in \mathcal {K}$
such that we have
$\sup (\pi [Y({\dot {b}}_0)]) \geq \gamma +1 $
, and hence
$\sup (\pi [Y({\dot {b}}_0)])>\gamma +1 $
.) We define
$u^{\dot {b}}$
on levels from
${\dot {b}}_0$
as in Definition 4.1(II): Given
$t\in \,^l2$
with
$l\in {\dot {b}}_0$
,
$d(t)$
is determined such that-
• if
$t\not \in T(\bar Z)$
or
$t\in \mathrm {split}(T(\bar Z))$
then
$d(t)= 0 $
; -
• if
$t \in T(\bar Z)\setminus \mathrm {split}(T(\bar Z))$
,
$d(t)<2$
is defined such that
$t\,^\smallfrown \, d(t)\not \in T(\bar Z)$
.
-
-
(1) (On levels from
${\dot {b}}_1$
we take care of
$\dot x'\in \mathcal {K}$
such that we have
$\sup (\pi ' [Y({\dot {b}}_0)]) \leq \gamma $
.) Then
$Z({\dot {b}}_1)\in \mathbf {V}^{P_{\gamma }}$
(see the beginning of the present definition for its definition). We choose
$\dot h:{Z({\dot {b}}_1)}\rightarrow 2$
in
$\mathbf {V}^{P_{\gamma +1}}$
such that for no
$x:Z({\dot {b}}_1)\rightarrow 2$
in
$ \mathbf {V}^{P_{\gamma }}$
we have
$x =\dot h$
. Now suppose
$l\in {\dot {b}}_1$
and
$t\in \,^l2$
. Then
$d(t)$
is defined as follows:-
• if
$t\not \in T(\bar Z)$
let
$d(t)= 0 $
; -
• if
$t \in T(\bar Z)\setminus \mathrm {split}(T(\bar Z))$
let
$d(t)<2$
such that
$t\,^\smallfrown \, d(t)\not \in T(\bar Z)$
; -
• if
$t\in \mathrm {split}(T(\bar Z))$
, thus
$t=t_s$
for some
$s\in \,^{<\omega }2$
(and
$\vert s\vert \in Z({\dot {b}}_1 )$
), we let
$$ \begin{align*}d(t)= t_{s\,^\smallfrown\, \dot h(\vert s\vert)}(\vert t\vert ) .\end{align*} $$
-
-
(2) For
$t\in u^{\dot {b}}$
with
$\vert t\vert \in {\dot {b}}\setminus ({\dot {b}}_0\cup {\dot {b}}_1)$
, its next digit in
$u^{\dot {b}}$
is irrelevant, we require that it is
$0$
.
(III) Suppose
${\dot {b}}\in \Omega ^{\gamma }(T(\bar Z))$
is such that
${\dot {b}}\cap \bigcup \left \lbrace L(\beta ):\beta \in Y\right \rbrace =\emptyset $
. Hence no splitnode of
$T(\bar Z)$
has its length in
${\dot {b}}$
. We define
$u^{\dot {b}}$
in
$\mathbf {V}^{P_{\gamma }} $
such that for every
$l\in {\dot {b}}$
and
$t\in \,^l2\cap u^{\dot {b}}$
the following hold:
-
• if
$t\not \in T(\bar Z) t\,^\smallfrown \, 0\in u^{\dot {b}}$
; -
• if
$t \in T(\bar Z) $
let
$t\,^\smallfrown \, i\in u^{\dot {b}}$
such that
$t\,^\smallfrown \, i\not \in T(\bar Z)$
.
This completes the definition of
$C^{\gamma } (T(\bar Z))=\langle u^{\dot {b}}: {\dot {b}}\in \Omega ^{\gamma }(T(\bar Z))\rangle $
.
Finally, we define a coding system
$C(T(\bar Z))$
in
$\mathbf {V}^{P_{\omega _2} }$
as
$$ \begin{align*}C(T(\bar Z))=\langle u^{\dot{b}}: {\dot{b}}\in \bigcup \left\lbrace \Omega^{\gamma} (T(\bar Z)):\gamma<\omega_2\right\rbrace \rangle.\end{align*} $$
The following main lemma shows that the coding system constructed in Definition 5.1 has the desired property.
Lemma 5.2. If
$T(\bar Z)$
,
$\mathcal {K}$
and
$C(T(\bar Z))$
are as in Definition 5.1,
$\dot x'\in \mathcal {K}$
is arbitrary and
$\bar S'$
is the associated fusion with limit
$p_\omega '$
, then
$$ \begin{align*}p_\omega' \Vdash_{P_{\omega_2}} \,\, \forall b\in \bigcup \left\lbrace \Omega^{\gamma } (T(\bar Z)):\gamma<\omega_2\right\rbrace\quad \dot x'\not\in [u^b].\end{align*} $$
Proof. Let
$q\leq p_\omega '$
,
$\gamma <\omega _2$
and
$\dot {b}$
be arbitrary such that
$$ \begin{align*}q\Vdash_{P_{\omega_2}} \dot{b}\in\Omega^{\gamma } (T(\bar Z)).\end{align*} $$
We have to find
$q'\leq q$
such that
$q'\Vdash _{P_{\omega _2}}\; \dot {x}'\not \in \left [ u^{\dot {b}}\right ] $
.
Wlog we may assume that q forces that t is a split node of
$T(\bar {Z})$
whenever
$\vert t\vert \in \dot {b}$
and t is a possible initial segment of
$\dot {x} '$
below condition q, as otherwise, by the definition of
$u^{\dot {b}} $
, some
$q'\leq q$
clearly forces that
$\dot {x}'\upharpoonright \vert t\vert +1\not \in u^{\dot {b}} $
. It follows that we can ignore Case III.
We need to consider the different cases in Definition 5.1.
Case I:
Wlog we may assume that
$q\Vdash _{P_{\omega _2}} \dot {b}\subseteq L(\beta )$
, for some
$\beta \in Y$
, hence we have
$\pi '(\beta )\in \mathrm {supp}(p_\omega ')$
and in
$\mathbf {V}^{P_{\gamma }}$
we have fixed the partition
$\dot {b}=\dot {b}_0\cup \dot {b}_1 $
for the definition of
$u^{\dot {b}}$
.
Let
$G_{\gamma +1}$
be a
$P_{\gamma +1} $
-generic filter containing
$q\upharpoonright \gamma +1 $
. Moreover, we let
$b=\dot {b}[G_{\gamma }]$
and
$b_i=\dot {b}_i[G_{\gamma }]$
for
$i<2$
.
Subcase I(0): We have
$\pi '(\beta )\geq \gamma $
. We can proceed essentially as in the first case of the proof of Lemma 4.1 with
${b}_0$
in place of b, as
$\dot {g}_{\pi '(\beta )}$
is generic over
$\mathbf {V}[G_{\gamma }]$
, where we have defined the levels from
${b}_0$
of
$u^{{b}}$
.
In
$ \mathbf {V}\left [ G_{\gamma }\right ] $
, we obtain the partial evaluations
$T^q\left [ G_{\gamma } \right ] $
and
$R^q\left [ G_{\gamma } \right ] $
of the trees
$T^q=T^q(\dot {x}')$
and
$R^q=R^q(\dot {x}')$
defined accordingly for
$\dot {x} '$
and
$q$
as in the proof of Lemma 4.1, i.e.,:
$$ \begin{align*}R^q [ G_{\gamma } ] = \lbrace s\in\, ^{<\omega}2: a^s(\bar{S}')\left[ G_{\gamma}\right] \upharpoonright\gamma\in G_{\gamma}\,\wedge\, \lnot \,\,q\left[ G_{\gamma}\right] \perp_{P_{\omega_2}/G_{\gamma}} a^s(\bar{S}')\left[ G_{\gamma}\right] \rbrace\end{align*} $$
and
$T^q\left [ G_{\gamma }\right ]$
is the subtree of
$T^q$
generated by all
$t_s\in T^q$
where
$s\in R^q [ G_{\gamma } ]$
.
Wlog we may assume that in
$ \mathbf {V}\left [ G_{\gamma }\right ] $
there are
$k, v,m_0,n_0,l,n_1,m_1$
such that
$q\upharpoonright \pi '(\beta ) $
forces (with respect to the forcing
$P_{\omega _2}/G_{\gamma } $
) the following:
-
•
$v\in \, ^k2$
, stem
$(q(\pi '(\beta )))=\,\mathrm {stem}(p_\omega '(\pi '(\beta ))^v)$
and
$m_0=\vert \mathrm {stem}(q(\pi '(\beta )))\vert $
; -
• at step
$n_0$
of the fusion
$\bar {S}'$
that produced
$p_\omega '$
,
$\pi '(\beta )$
was active for the kth time; -
•
$l\in b_0$
is bigger than
$\max \left \lbrace \vert t_s\vert : s\in \,^{n_0}2\right \rbrace $
; -
•
$n_1\in Z(\beta )$
is such that
$\forall \,t\in T^q\left [ G_{\gamma }\right ] \cap \,^l2\,\exists \,s\in R^q\left [ G_{\gamma }\right ] \cap \, ^{n_1}2\quad t=t_s$
; -
•
$\dot {c}(p_\omega ', \pi '(\beta ))(n_1) =m_1,$
and -
• for every
$s\in R^q\left [ G_{\gamma }\right ] $
with
$\vert s\vert =n_1$
, we have
$$ \begin{align*}q\left[ G_{\gamma}\right] \wedge a^s(\bar{S} ')\left[ G_{\gamma}\right]\;\Vdash_{P_{\omega_2}/G_{\gamma}} \, \dot{x}'(\vert t_s\vert)=t_{s\,^ \smallfrown \, \dot{g}_{\pi'(\beta)} (m_1)}( \vert t_s\vert).\end{align*} $$
Analogously as in the proof of Lemma 4.1, in
$\mathbf {V}$
we find
$\hat {q}\leq q $
in
$P_{\omega _2}$
such that
$\hat q\upharpoonright \pi '(\beta )$
forces all these facts and
$\hat q\upharpoonright \pi '(\beta )$
also decides
$k,v,m_0,n_0,l, n_1,m_1$
as above. Moreover,
$$ \begin{align*}\hat q\upharpoonright\pi'(\beta)\Vdash_{P_{\pi'(\beta)} }\, \mathrm{com}(\hat{q}(\pi'(\beta)))\cap m_1+1= \left\lbrace m_0\right\rbrace\end{align*} $$
and
$\hat q\upharpoonright \pi '(\beta )$
decides
$\hat {q}(\pi '(\beta ))\upharpoonright m_1+1$
, thus
$\dot g_{\pi '(\beta )} \upharpoonright (m_1+1 \setminus \lbrace m_0\rbrace ) $
, say as
$\langle g(0),...,g(m_0 -1),g(m_0 +1),...,g(m_1)\rangle $
.
Now we can find
$s_0,s_1\in R^{\hat {q}}\cap \, ^{n_1} 2$
such that
$s_i(n_0)=i$
and
$s_0(j) =s_1(j)$
for every
$j\in Z(\beta )\cap n_1\setminus \left \lbrace n_0\right \rbrace $
, and hence
$$ \begin{align*}i( t_{s_0\,^\smallfrown\,g(m_1)},\beta)\ne i(t_{s_1\,^\smallfrown\,g(m_1)},\beta).\end{align*} $$
We know that both
$t_{s_0}$
and
$t_{s_1}$
are splitnodes of
$T(\bar {Z})$
of length l. We can choose j such that
$i(t_{s_j\,^\smallfrown \,g(m_1)},\beta ) \ne g(m_1)$
and a common extension
$q'$
of
$\hat {q}$
and
$a^{s_j}(\bar {S}' ) $
. We conclude
$$ \begin{align*}q'\Vdash_{P_{\omega_2} }\; t_{s_j\,^\smallfrown\,g(m_1)} \upharpoonright \vert t_{s_j} \vert +1\subset \dot x '\end{align*} $$
and hence
$q'\Vdash _{P_{\omega _2}}\; \dot x'\not \in \left [ u^{\dot {b}}\right ],$
by the definition of
$u^{\dot {b}}$
.
Subcase I(1): We have
$\pi ' (\beta )<\gamma $
. In the intermediate model
$\mathbf {V}[G_\gamma ]$
we have the restricted tree of possibilities for
$\dot {x} '$
,
$T^q(\dot {x}')[G_\gamma ]$
(which is analogously defined as in Subcase I(0) above). We know that every
$$ \begin{align*}t\in T^q(\dot{x}')[G_\gamma] \cap \,\bigcup\left\lbrace \,^l 2: l\in b_1\right\rbrace\end{align*} $$
is a splitnode of
$T(\bar {Z})$
, hence
$t=t_s$
for some
$$ \begin{align*}s\in \bigcup\left\lbrace \,^{i}2: i\in {Z(b_1)}\right\rbrace .\end{align*} $$
Now we have that if
$l\in L(\beta )$
,
$t\in T^q(\dot {x}')[G_\gamma ]$
,
$\vert t\vert =l$
and
$t=t_s$
, then
$t_s$
is no longer a splitnode of
$T^q(\dot {x}')[G_\gamma ]$
, and its successive digit is
$$ \begin{align*}t_{s\,^\smallfrown\, g_{\beta'}\circ \dot{c}(p_\omega',\beta')[G_\gamma](\vert s\vert)} (l).\end{align*} $$
If in addition
$l\in b_1$
, we have
$\vert s\vert \in Z(b_1)$
. Note that by property (i) of a coding tree,
$\vert s\vert $
does not depend on t (but on l of course).
Hence in
$\mathbf {V}[G_{\gamma }]$
we can define a function
$F=F(\dot {x}'):Z(b_1)\rightarrow 2$
as follows: Given
$i\in Z(b_1)$
, we let
$$ \begin{align*}F(i)= g_{\beta'}\circ \dot{c}(p_\omega',\beta')[G_\gamma](i) .\end{align*} $$
For the definition of
$u^b$
on levels in
$b_1$
in the present case we applied some new function
$h=\dot {h}[G_{\gamma +1}]\in \mathbf {V}\left [ G_{\gamma +1}\right ]\setminus \mathbf {V}\left [ G_{\gamma }\right ]$
,
$h:Z(b_1)\rightarrow 2$
. Hence clearly we have
$$ \begin{align}\exists ^\infty i\in Z(b_1)\quad h(i)\neq F (i).\end{align} $$
By construction we can find s with
$\vert s\vert =i$
,
$i\in Z(b_1)$
,
$h(i)\ne F(i)$
and
$t_s\in T^q(\dot {x}')[G_\gamma ]$
. Letting
$l=\vert t_s\vert $
, hence
$l\in b_1$
, we get that
$$ \begin{align*}t_{s\,^\smallfrown\,F(i)} (l)\ne t_{s\,^\smallfrown\, h(i)} (l).\end{align*} $$
Hence, as
$t_{s\,^{\smallfrown } F(i)} \in T^q(\dot {x}')[G_\gamma ]$
, there exists
$q'\leq q$
in
$P_{\omega _2}$
(with
$q'\upharpoonright \gamma \in G_\gamma $
) such that
$$ \begin{align*}q'\Vdash_{P_{\omega_2}}\,\,\dot x'\upharpoonright l+1 =t_{s\,^{\smallfrown} F(i)} \upharpoonright l+1,\end{align*} $$
and therefore, by the definition of
$u^{\dot {b}}$
on levels in
$\dot {b}_1$
, we conclude
$$ \begin{align*}q'\Vdash_{P_{\omega_2}}\,\,\dot x'\not\in [u^{\dot{b}}].\end{align*} $$
Note that the proof here, in particular the correct choice of s that leads to a contradiction, is quite subtle. On the one hand we have that for every
$t\in T^q(\dot {x}')[G_\gamma ]$
such that
$t=t_{s'}$
for some
$s'$
of same length as s we have
$$ \begin{align*}t_{s'\,^\smallfrown\, F(i)} (\vert t_{s'}\vert)\ne t_{s'\,^\smallfrown\, h(i)} (\vert t_{s'}\vert).\end{align*} $$
Note that this does not imply that, letting
$l=\vert t_s\vert $
and
$l'=\vert t_{s'} \vert $
,
$$ \begin{align*}t_{s\,^\smallfrown\, F(i)} (l)=t_{s'\,^\smallfrown\,F(i)} (l')\end{align*} $$
(which is generally false). On the other hand, such other
$s'$
might be useless for our purpose as not necessarily
$\vert t_{s'}\vert \in b_1$
(see the remark after the the definition of
$Z(b)$
at the beginning of Definition 5.1).
Case II:
Let
$\dot {b}\in \Omega ^{\gamma }(T(\bar Z))$
,
$\dot \delta $
and
$\dot {b}_0,\dot {b}_1\subseteq \dot {b}$
be as there, and let
$G_{\gamma +1}$
be a
$P_{\gamma +1}$
-generic filter containing
$q\upharpoonright \gamma +1$
. Let
$b,\delta , b_0,b_1$
be the evaluations of
$\dot {b}_0,\dot {b}_1,\dot {b}$
by
$G_{\gamma }$
, respectively.
Subcase II(0): We have
$\sup (\pi '[Y(b_0)])>\gamma $
. We proceed essentially as in the second case in the proof of Lemma 4.1. We argue in
$\mathbf {V} [G_{\gamma }]$
. The set
$$ \begin{align*}\left\lbrace \beta\in Y(b_0): \pi'(\beta)>\gamma\right\rbrace\end{align*} $$
is infinite, so we can find a large enough
$\beta \in Y(b_0)$
,
$l\in b_0\cap L(\beta )$
,
$n\in Z(\beta )$
and
$s\in \, ^n2$
,
$s_0,s_1\in \,^n2 \cap R^q(\dot {x}')[G_{\gamma }]$
such that
$\vert t_s \vert =l$
and
$$ \begin{align*}s_0\upharpoonright n \cap \left\lbrace \nu \in Y: \pi'(\nu)\in \left( \gamma, \pi'(\beta)\right)\right\rbrace \ne s_1\upharpoonright n \cap \left\lbrace \nu \in Y: \pi'(\nu)\in \left( \gamma, \pi'(\beta)\right)\right\rbrace .\end{align*} $$
As in the proof of Lemma 4.1 we can apply properties (i) and (ii) of the coding tree
$T(\bar Z)$
to find
$q'\leq q$
,
$j<2$
and
$t\in \,^l2$
compatible with
$t_{s_j}$
such that t is not a splitnode of
$T(\bar Z)$
and
$$ \begin{align*}q'\Vdash_{P_{\omega_2}}\,\, \dot x'\upharpoonright l=t,\end{align*} $$
and hence
$$ \begin{align*}q'\Vdash_{P_{\omega_2}}\,\, \dot x'\upharpoonright l+1\not\in [ u^{\dot{b}}].\end{align*} $$
Subcase II(1): This is similar to Subcase I(1). We have
$\sup (\pi ' [Y(b_0)])\leq \gamma $
. In the intermediate model
$\mathbf {V}[G_\gamma ]$
we have the restricted tree of possibilities for
$\dot {x} '$
,
$T^q(\dot {x}')[G_\gamma ]$
(which is analogously defined as in Subcase I(0) above). We know that every
$t\in T^q(\dot {x}')[G_\gamma ] \cap \,\bigcup \left \lbrace \,^l 2: l\in b_1\right \rbrace $
is a splitnode of
$T(\bar {Z})$
, hence
$t=t_s$
for some
$s\in \bigcup \left \lbrace \,^{i}2: i\in {Z(b_1)}\right \rbrace $
.
Now we have that if
$\beta \in \mathrm {dom}(p_\omega )$
,
$\beta '=\pi '(\beta )<\gamma $
,
$l\in L(\beta )$
,
$t\in T^q(\dot {x}')[G_\gamma ]$
,
$\vert t\vert =l$
and
$t=t_s$
, then
$t_s$
is no longer a splitnode of
$T^q(\dot {x}')[G_\gamma ]$
, and its successive digit is
$$ \begin{align*}t_{s\,^\smallfrown\, g_{\beta'}\circ \dot{c}(p_\omega',\beta')[G_\gamma](\vert s\vert)} (l).\end{align*} $$
If in addition
$l\in b_1$
, we have
$\vert s\vert \in Z(b_1)$
.
By what we noticed so far, in
$\mathbf {V}[G_{\gamma }]$
we can define a function
$F=F(\dot {x}'):Z(b_1)\rightarrow 2$
as follows: Given
$i\in Z(b_1)$
, letting
$\beta '\in \mathrm {supp}(p_\omega ')$
with
$i\in Z(\beta ')$
, we let
$$ \begin{align*}F(i)= g_{\beta'}\circ \dot{c}(p_\omega',\beta')[G_\gamma](i) .\end{align*} $$
As
$h\ne F$
(
$h=\dot {h}[G_{\gamma +1}]$
where
$\dot {h}$
is from the Definition 5.1(II)(1)) there is
$i\in Z(b_1)$
such that
$h(i)\ne F(i)$
. By construction we can find s and
$\beta '$
with
$\vert s\vert =i$
and
$i\in Z(\beta ')$
such that
$\vert t_s\vert \in b_1$
and
$t_s\in T^q(\dot {x}')[G_\gamma ]$
. We get that
$$ \begin{align*}t_{s\,^\smallfrown\,F(i)} (\vert t_s\vert)\ne t_{s\,^\smallfrown\, h(i)} (\vert t_s\vert).\end{align*} $$
Hence, letting
$l:=\vert t_s\vert $
, as
$t_{s\,^{\smallfrown } F(i)} \in T^q(\dot {x}')[G_\gamma ]$
there exists
$q'\leq q$
in
$P_{\omega _2}$
(with
$q'\upharpoonright \gamma \in G_\gamma $
) such that
$$ \begin{align*}q'\Vdash_{P_{\omega_2}}\,\,\dot x'\upharpoonright l+1 =t_{s\,^{\smallfrown} F(i)} \upharpoonright l+1,\end{align*} $$
and therefore, by the definition of
$u^{\dot {b}}$
on levels in
$\dot {b}_1$
, we conclude
$$ \begin{align*}q'\Vdash_{P_{\omega_2}}\,\,\dot x'\not\in [u^{\dot{b}}].\end{align*} $$
Case III:
In this case we have that no splitnode of
$T(\bar Z)$
has its length in
$\dot {b}$
, and therefore, as we have noticed already,
$$ \begin{align*}p_\omega'\Vdash_{P_{\omega_2}}\,\,\dot x'\not\in [u^b]\end{align*} $$
follows immediately by the definition of
$u^b$
.
We have completed the proof of Lemma 5.2.
6 Conclusion
By Lemma 5.2 we obtain our main result.
Theorem 6.1. If
$\mathbf {V}\models \mathrm {ZFC+CH}$
and
$P_{\omega _2}$
is the CS-iteration of Silver forcing
$\mathbb {SI}$
of length
$\omega _2$
, then
$$ \begin{align*}\mathbf{V}^{P_{\omega_2}}\models\quad \mathrm{cov}(\mathfrak{C}_2) <\;\mathrm{cov}(\mathfrak{P}_2).\end{align*} $$
Proof. Given any
$P_{\omega _2}$
-name
$\dot x$
and
$p\in P_{\omega _2}$
such that
$$ \begin{align*}p\Vdash_{P_{\omega_2}}\, \dot x\in \, ^\omega 2\setminus \mathbf{V},\end{align*} $$
by Theorem 3.1 we obtain a fusion sequence
$\bar {S}$
with limit
$p_\omega \leq p$
and a tree
$T=T(\dot x, \bar S)$
with properties (0), (1), and (2), which is the tree of possibilities of
$\dot x$
below
$p_\omega $
. Let
$\mathcal {K}$
be the isomorphism type of
$\dot x$
as defined in Definition 4.2. Moreover, every
$\dot x'\in \mathcal {K}$
produces the same tree.
In Definition 4.1 we have defined a coding system
$C^0(T)=\langle u^b: b\in \Omega ^0\rangle $
depending only on
$\mathcal {K}$
such that, by Lemma 4.1, in
$\mathbf {V}$
$$ \begin{align*}p_\omega\Vdash _{P_\alpha}\, \forall b\in \Omega^0 \quad \dot x\not\in \left[ u^b\right] .\end{align*} $$
In Definition 5.1, for every
$0<\gamma <\omega _2$
of countable cofinality we have defined a coding system
$C^{\gamma }(T)=\langle u^b:b\in \Omega ^{\gamma }(T)\rangle $
in
$\mathbf {V}^{P_{\gamma +1}}$
, where
$\Omega ^{\gamma }(T)\subseteq \left [ \omega \right ] ^\omega $
is dense in
$\mathbf {V}^{P_\gamma }$
with
$\Omega ^{\gamma }(T)\cap \bigcup _{\beta <\gamma }\mathbf {V}^{P_\beta }=\emptyset $
, such that by Lemma 5.2 we have
$$ \begin{align*}p_\omega \Vdash_{P_{\omega_2}} \,\, \forall b\in \bigcup \left\lbrace \Omega^\gamma (T):0<\gamma<\omega_2\right\rbrace\quad \dot x\not\in [u^b].\end{align*} $$
For
$\gamma <\omega _2$
of uncountable cofinality,
$\Omega ^{\gamma }(T)$
and
$C^{\gamma }(T)$
were defined by taking the union of the
$\Omega ^{\beta }(T)$
,
$C^{\beta }(T)$
, respectively, for
$\beta <\gamma $
. Hence in
$\mathbf {V}^{P_{\omega _2}}$
we have the
$\mathfrak {C}_2$
-set
$A(C)$
, where
$C=\langle u^b:b\in \Omega ^{\gamma }, \gamma <\omega _2\rangle $
, such that
$$ \begin{align*}p_\omega \Vdash_{P_{\omega_2}} \,\, \dot x\in A(C).\end{align*} $$
As
$A(C)$
only depends on T, and as by CH in the ground model there are only
$\aleph _1$
-many isomorphism types
$\mathcal {K}$
, we have proved that in
$\mathbf {V}^{P_{\omega _2}}$
,
$^\omega 2\setminus \mathbf {V}$
can be covered by
$\aleph _1$
-many
$\mathfrak {C}_2$
-sets. By Corollary 5.1(2) we know that
$^\omega 2\cap \mathbf {V}$
is even in
$\mathfrak {P}_2$
. Hence cov
$(\mathfrak {C}_2)=\aleph _1$
holds in
$\mathbf {V}^{P_{\omega _2}}$
.
As we have explained in the introduction,
$\mathbf {V}^{P_{\omega _2}}\models \;\mathrm {cov}(v^0)=\aleph _2$
, where
$v^0$
is the Silver ideal. As
$\mathfrak {P}_2\subseteq v^0$
, we have
$\mathbf {V}^{P_{\omega _2}}\models \;\mathrm {cov}(\mathfrak {P}_2)=\aleph _2$
.
Acknowledgments
I would like to thank Martin Goldstern for bringing Grigorieff’s paper to my attention. Moreover, I am indebted to the referee for his several very careful and sharp reports which forced me to clear up several obscure or faulty arguments.
Funding
The author would like to thank the DFG for partial support (Grant No. SP683/5-1).
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