On the spectra of cardinalities of branches of Kurepa trees
M\'ark Po\'or

TL;DR
This paper investigates the possible sets of branch cardinalities in Kurepa trees within models of ZFC + CH, providing conditions under which these sets can be realized consistently.
Contribution
It introduces a sufficient condition for sets of cardinals to be the branch cardinalities of Kurepa trees in certain models.
Findings
Identifies conditions for sets of cardinals to correspond to branch sizes in Kurepa trees.
Provides a method to realize these sets consistently in models of ZFC + CH.
Advances understanding of the structure of Kurepa trees in set theory.
Abstract
We are interested in the possible sets of cardinalities of branches of Kurepa trees in models of . In this paper we present a sufficient condition (for sets of cardinals) to be consistently the set of cardinalities of branches of Kurepa trees.
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\stackMath
On the spectra of cardinalities of branches of Kurepa trees
Márk Poór
Eötvös Loránd University, Institute of Mathematics, Pázmány Péter s. 1/c, 1117 Budapest, Hungary
Abstract.
We are interested in the possible sets of cardinalities of branches of Kurepa trees in models of . In this paper we present a sufficient condition (for sets of cardinals) to be consistently the set of cardinalities of branches of Kurepa trees.
The author was supported by the National Research, Development and Innovation Office – NKFIH, grant no. 104178, 124749 and 129211.
Supported by the ÚNKP-18-3 New National Excellence Program of the Ministry of Human Capacities.
1. Introduction
Kurepa trees are trees of height which have countable levels, but have more than -many cofinal branches. We are interested in how those sets of cardinals look like for which there is a model of such that the following holds:
[TABLE]
[TABLE]
[TABLE]
J.H. Silver showed that the existence of Kurepa trees is independent of [6]. R. Jin. and S. Shelah constructed a model of and , where Kurepa trees only with -many cofinal branches exist, and another with and Kurepa trees having exactly -many cofinal branches [2]. Moreover, in the latter there are no Jech-Kunen trees (a tree of height is a Jech-Kunen tree, iff each level is of power at most , with the cardinality of cofinal branches strictly between and ). In [3] they prove that it is consistent with that there are Jech-Kunen trees, but no Kurepa trees.
If for a given sequence of cardinals we have Kurepa trees () with each having -many cofinal branches, then is a Kurepa tree with -many cofinal branches. This means that the set of cardinalities of Kurepa trees is closed under taking limits of countable sequences. Similarly, it is not hard to see that this set is closed under taking limits of -long sequences too. Our goal is that for a given set of cardinals satisfying a slightly strengthened version of this necessary condition (in a countable transitive model [c.t.m.] of ) to construct a forcing extension where
[TABLE]
(where denotes the set of cofinal branches of ). Our main result, Theorem 3.2 implies that, for example, consistently there are Kurepa trees with -many (), - and -many cofinal branches (but there are no Kurepa trees with -many cofinal branches).
2. Preliminaries and notations
In this paper, all ordinals are von Neumann ordinals, and by the cardinality of a set (in symbols ) we mean the least ordinal such that there exists a bijection between and . For any function with domain , the following sequencelike notation will also symbol the set
[TABLE]
that is
[TABLE]
For a given set , and ordinal , will symbol the set of functions from to , i.e. . Similarly . We use the notation (for any set and cardinal ) , and similarly . Regarding forcing we refer to [1] and [4].
Definition 2.1**.**
A tree is a partially ordered set (poset) in which for each the set
[TABLE]
is well ordered by .
Definition 2.2**.**
The height of in the tree is the order type of
[TABLE]
Definition 2.3**.**
For each ordinal the ’th level of , or is
[TABLE]
The restriction of to is
[TABLE]
Definition 2.4**.**
The height of the tree , or is the least such that
[TABLE]
A tree of height , with for each is called an -tree.
From now on by branch we will mean cofinal branch.
Definition 2.5**.**
A branch of a tree is an ordered set (w.r.t. ) containing exactly one element of each , . For a given tree denotes the set of branches of .
Definition 2.6**.**
An -tree is a Kurepa tree if it has more than -many branches.
In this paper we will restrict our attention to trees that are downward closed subsets of , i.e. a set of -valued functions on countable ordinals, where
[TABLE]
and , iff extends as a function (). Then it is easy to see that for each , , and .) The following well-known lemma states that regarding our problem, we can assume that trees are of the form as in .
Lemma 2.7**.**
Suppose that is an -tree. Then there exists an -tree which is downward closed (i.e. is of the form ) with the same cardinality of branches, i.e.
[TABLE]
Proof.
First we need the following claim.
Claim 2.8**.**
Assume that is a tree such that , . Then there is an order-preserving mapping such that for the downward closed tree generated by
[TABLE]
the following holds. For each for the -th level of
[TABLE]
Proof.
Fix an injection . For each if then define for each the element to be the unique element in under , i.e.,
[TABLE]
and let
[TABLE]
Now define as follows.
[TABLE]
Now if () is the restriction of for some , i.e. f=F(t)\raisebox{-1.29167pt}{\upharpoonright}_{\omega\cdot(\gamma+1)}, and if is such that , then clearly . Using that if , we obtain by that
[TABLE]
therefore .
∎
Obviously .
The function given by the claim is an order-preserving embedding from to . Moreover, the fact that the -th level of is the -image of (by ) implies that for each cofinal branch and
[TABLE]
This means that (fixing ) by the order-preservation
[TABLE]
therefore , indeed.
∎
Before stating our main theorem, we need some technical preparations.
Definition 2.9**.**
Let the ordinal be given, and let . We define the mapping as follows
[TABLE]
[TABLE]
It can be easily seen that is an automorphism of the tree .
Definition 2.10**.**
A tree which is downward closed is said to be homogeneous if for each pair on the same level, is an automorphism of .
Definition 2.11**.**
A tree is normal if the following conditions hold
- •
each which is not on the top level of has at least two immediate successors in ,
- •
for each , and each (where ) there exists an element , ,
- •
for each limit (where ) and b\in\mathcal{B}(T\raisebox{-1.29167pt}{\upharpoonright}_{\alpha}), there is at most one common upper bound of in .
Definition 2.12**.**
The set
[TABLE]
is a notion of forcing with the partial order
[TABLE]
i.e. the condition extends the condition iff the tree is an end-extension of .
(It is easy to see that a -generic filter corresponds to a homogeneous subtree of of height .)
Definition 2.13**.**
A partial order is -closed, if whenever is a decreasing sequence (i.e. implies ) of length , then there exists a common lower bound , i.e. for each .
Lemma 2.14**.**
* is -closed.*
Proof.
If a decreasing sequence is given, then is a growing union of countable homogeneous normal trees. Since it is easy to check that the growing union of normal trees is normal, and homogeneity of a tree means that for ( are on the same level) , we are done. ∎
At some point we will make use of the following claim.
Claim 2.15**.**
Let be a homogeneous tree, , and , that is and are on the same level, and is on a higher level. Furthermore, assume that , i.e. is an extension of as a function. Then t\cup t^{\prime\prime}\raisebox{-1.29167pt}{\upharpoonright}_{\beta\setminus\alpha}\in T, that is, roughly speaking, and have the same extensions in .
Proof.
It is easy to check that F_{tt^{\prime}}(t^{\prime\prime})=t\cup t^{\prime\prime}\raisebox{-1.29167pt}{\upharpoonright}_{\beta\setminus\alpha}. ∎
We will make use of the next lemma which is [4, VII., Thm. 6.14.]
Lemma 2.16**.**
Let be a c.t.m. Suppose that the cardinal , and the sets () are given. Let be a -closed notion of forcing, be -generic over , , . Then .
Which has the following straightforward corollary.
Corollary 2.17**.**
If , then forcing with a -closed notion of forcing adds no new subsets of .
The next lemma is a corollary of the proof of Lemma 2.16. It is folklore. (Recall that for each element there is a canonical -name
[TABLE]
for which the evaluation of by
[TABLE]
whenever is -generic over , see [4, Ch VII, Definition 2.10].)
Lemma 2.18**.**
Let be a c.t.m. Suppose that the cardinal , and the sets () are given. Let be a -closed notion of forcing, , is a -name for which
[TABLE]
Then there is an extension , and a function such that
[TABLE]
Proof.
Let -generic over with . Then apply Lemma 2.16, set , and choose , such that . ∎
The lemma has the following straightforward application.
Corollary 2.19**.**
Forcing with a -closed notion of forcing adds no new sequences of type (for any ), that is, if denotes the generic filter, then
[TABLE]
For some technical reasons we will later use the following definition and lemma.
Definition 2.20**.**
If is a c.t.m., is a notion of forcing and are -names, then is a nice -name for a subset of if is of the form
[TABLE]
The next lemma is [4, Ch VII, Lemma 5.12].
Lemma 2.21**.**
Suppose that is a c.t.m., is a notion of forcing, are -names. Then there is a nice name for a subset of such that
[TABLE]
The following lemma is folklore, but for the sake of completeness we include the proof.
Lemma 2.22**.**
Let be a c.t.m., , be cardinals in , be a notion of forcing which is -cc. Then, whenever is generic over , and is such that
[TABLE]
then
[TABLE]
Proof.
By -cc, there are at most -many antichains in . This implies that since , there are at most -many nice names for subsets of in . Let denote the set of nice names for subsets of (where ). By Lemma 2.21 each subset is represented by a nice name . ∎
The following lemma can be found as [4, Ch. VII., Lemma 6.9]
Lemma 2.23**.**
Let be a cardinal in a c.t.m. , and be a poset which is -cc in . Then forcing with preserves cofinalities , i.e. if
[TABLE]
then whenever is -generic over ,
[TABLE]
in particular if is regular in , then preserves cardinals .
The following well-known fact can be found as [4, Ch. VIII. Lemma 3.4].
Lemma 2.24**.**
Suppose that is an -tree, is -closed. Then forcing with does not add any new branch to .
The next technical lemma will be later needed [4, Ch. VII. Lemma 7.11].
Lemma 2.25**.**
Let the poset given, and suppose that is a dense subset of . Then
- •
if is generic over , then the intersection is -generic over ,
[TABLE]
moreover, ,
- •
if is generic over , then the filter is -generic over ,
[TABLE]
moreover, .
3. The main result
In this section we will prove the following theorem. We will make use of some ideas from [2], where the authors proved among others, that it is consistent with that , and Kurepa trees only with -many branches exist.
Our main forcing object will be a two-step forcing iteration where we can isolate a dense closed subset. The first paper in which such argument arose was [5] (see the end of Section 3 of that paper. In fact, the forcings used by Kunen are essentially versions of the forcings used in this paper.)
Definition 3.1**.**
A set of ordinals is closed under taking -limits, iff whenever is an ordinal such that and is cofinal in , then .
Theorem 3.2**.**
Let be a c.t.m. of , and let be a set of ordinals such that , and the following holds (in ).
[TABLE]
If , then further assume that
[TABLE]
Then there is notion of forcing such that whenever is -generic over , then
[TABLE]
Remark 3.3**.**
If , then in the final model will be , thus is requiring condition to be true in the final model.
First we define . We will work in .
If , i.e. we would like to obtain a final model in which every Kurepa tree has more than branches, then define to be the following Lévy collapse
[TABLE]
Let
[TABLE]
and let
be the -name of the generic tree.
We have two distinct cases depending on whether . We will need the following sets defined for each ordinal in .
Definition 3.4**.**
Let the system
[TABLE]
of pairwise disjoint sets such that
- •
If , then
[TABLE]
- •
otherwise, if then
[TABLE]
Observe that if one collapses each cardinal greater than and less than (where each other cardinal remains a cardinal), then in that model .
Definition 3.5**.**
Let
,
,
be -names in for which
[TABLE]
with the pointwise extension order, i.e.
[TABLE]
and a name for the greatest element
[TABLE]
Such names exist by the maximal principle [4, II., Thm. 8.2]. Now after one adds a -generic filter over ,
will be decoded into a partial order with the reverse inclusion relation, with the largest element \mathchoice{\vphantom{\vtop{\halign{#\cr\hfil{\displaystyle\mathbb{1}{\mathbb{Q}{\alpha}}}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\displaystyle\mathbb{1}{\mathbb{Q}{\alpha}}}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\textstyle\mathbb{1}{\mathbb{Q}{\alpha}}}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\textstyle\mathbb{1}{\mathbb{Q}{\alpha}}}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\scriptstyle\mathbb{1}{\mathbb{Q}{\alpha}}}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\scriptstyle\mathbb{1}{\mathbb{Q}{\alpha}}}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}{\vtop{\halign{#\cr\hfil{\scriptscriptstyle\mathbb{1}{\mathbb{Q}{\alpha}}}\hfil\hfil\mathord{\scriptscriptstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}[F].
Remark 3.6**.**
After replacing
by \mathchoice{\vphantom{\vtop{\halign{#\cr\hfil{\displaystyle\mathbb{Q}{\alpha}}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\displaystyle\mathbb{Q}{\alpha}}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\textstyle\mathbb{Q}{\alpha}}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\textstyle\mathbb{Q}{\alpha}}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\scriptstyle\mathbb{Q}{\alpha}}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\scriptstyle\mathbb{Q}{\alpha}}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}{\vtop{\halign{#\cr\hfil{\scriptscriptstyle\mathbb{Q}_{\alpha}}\hfil\hfil\mathord{\scriptscriptstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}\cup\{\langle\emptyset,\mathbb{1}_{\mathbb{P}_{\alpha}}\rangle\} (if needed) we can assume that
[TABLE]
and
[TABLE]
Now we define -s () to be the following two step iterations as in [4, Ch VIII., §5.].
Definition 3.7**.**
[TABLE]
which is a notion of forcing with the following partial order
[TABLE]
and a (not necessarily unique) greatest element
[TABLE]
Definition 3.8**.**
For our fixed set let be the following countably supported product
[TABLE]
(where by we mean the set ) which is a partial order with the product ordering, i.e.
[TABLE]
For a set define
[TABLE]
Clearly for any partition , of ,
[TABLE]
Now we can define .
Definition 3.9**.**
[TABLE]
(Where by we mean the product the partial order, i.e. iff and .)
If and , then denotes its projection onto its -th coordinate.
From now on we fix an -generic filter over . For any set to be -s projection onto . Similarly, for any set let G\raisebox{-1.29167pt}{\upharpoonright}_{E}\subseteq\mathbb{R}\raisebox{-1.29167pt}{\upharpoonright}_{E} denote -s projection onto \mathbb{R}\raisebox{-1.29167pt}{\upharpoonright}_{E}. The following lemma [4, Ch VIII., Lemma 1.3] guarantees that G\raisebox{-1.29167pt}{\upharpoonright}_{E} is \mathbb{R}\raisebox{-1.29167pt}{\upharpoonright}_{E}-generic over .
Lemma 3.10**.**
Let be a product partial order, and fix a filter which is -generic over . Then is -generic over , is -generic over , and .
We will make use of the following too [4, Ch VIII., Thm. 1.4].
Lemma 3.11**.**
Let , be filters, then the following three conditions are equivalent.
- (1)
* is -generic over ,* 2. (2)
* is -generic over , and is -generic over ,* 3. (3)
* is -generic over , and is -generic over .*
Furthermore if holds, then
[TABLE]
The next definition, and lemma can help us to find an intermediate model between , and (for a fixed ordinal ).
Definition 3.12**.**
Let be a partial order in , and let
be a -name for a partial order. If the filter is -generic over , and H\subseteq\mathchoice{\vphantom{\vtop{\halign{#\cr\hfil{\displaystyle\mathbb{Q}}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\displaystyle\mathbb{Q}}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\textstyle\mathbb{Q}}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\textstyle\mathbb{Q}}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\scriptstyle\mathbb{Q}}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\scriptstyle\mathbb{Q}}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}{\vtop{\halign{#\cr\hfil{\scriptscriptstyle\mathbb{Q}}\hfil\hfil\mathord{\scriptscriptstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}[F]\in M[F], then
[TABLE]
We state [4, Ch VIII. Thm. 5.5]
Lemma 3.13**.**
*Let be a partial order in , and let
be a -name for a partial order. Let G\subseteq\mathbb{P}\ast\mathchoice{\vphantom{\vtop{\halign{#\cr\hfil{\displaystyle\mathbb{Q}}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\displaystyle\mathbb{Q}}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\textstyle\mathbb{Q}}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\textstyle\mathbb{Q}}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\scriptstyle\mathbb{Q}}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\scriptstyle\mathbb{Q}}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}{\vtop{\halign{#\cr\hfil{\scriptscriptstyle\mathbb{Q}}\hfil\hfil\mathord{\scriptscriptstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}} be a filter, , and let*
[TABLE]
If is \mathbb{P}\ast\mathchoice{\vphantom{\vtop{\halign{#\cr\hfil{\displaystyle\mathbb{Q}}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\displaystyle\mathbb{Q}}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\textstyle\mathbb{Q}}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\textstyle\mathbb{Q}}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\scriptstyle\mathbb{Q}}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\scriptstyle\mathbb{Q}}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}{\vtop{\halign{#\cr\hfil{\scriptscriptstyle\mathbb{Q}}\hfil\hfil\mathord{\scriptscriptstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}-generic over , then
- •
* is -generic over ,*
- •
H\subseteq\mathchoice{\vphantom{\vtop{\halign{#\cr\hfil{\displaystyle\mathbb{Q}}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\displaystyle\mathbb{Q}}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\textstyle\mathbb{Q}}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\textstyle\mathbb{Q}}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\scriptstyle\mathbb{Q}}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\scriptstyle\mathbb{Q}}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}{\vtop{\halign{#\cr\hfil{\scriptscriptstyle\mathbb{Q}}\hfil\hfil\mathord{\scriptscriptstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}[F]\in M[F]* is \mathchoice{\vphantom{\vtop{\halign{#\cr\hfil{\displaystyle\mathbb{Q}}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\displaystyle\mathbb{Q}}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\textstyle\mathbb{Q}}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\textstyle\mathbb{Q}}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\scriptstyle\mathbb{Q}}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\scriptstyle\mathbb{Q}}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}{\vtop{\halign{#\cr\hfil{\scriptscriptstyle\mathbb{Q}}\hfil\hfil\mathord{\scriptscriptstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}[F]-generic over ,*
- •
, and
- •
.
Using this, and having the filter which is -generic over , we can define
[TABLE]
(where is -generic over ) so that
[TABLE]
and
[TABLE]
holds.
Now we will verify some technical statements about the aforementioned partial orders.
Definition 3.14**.**
For each we define the subset as follows. Let \langle p,\mathchoice{\vphantom{\vtop{\halign{#\cr\hfil{\displaystyle q}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\displaystyle q}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\textstyle q}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\textstyle q}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\scriptstyle q}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\scriptstyle q}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}{\vtop{\halign{#\cr\hfil{\scriptscriptstyle q}\hfil\hfil\mathord{\scriptscriptstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}\rangle\in\mathbb{R}^{\bullet}_{\alpha}, iff (\langle p,\mathchoice{\vphantom{\vtop{\halign{#\cr\hfil{\displaystyle q}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\displaystyle q}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\textstyle q}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\textstyle q}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\scriptstyle q}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\scriptstyle q}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}{\vtop{\halign{#\cr\hfil{\scriptscriptstyle q}\hfil\hfil\mathord{\scriptscriptstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}\rangle\in\mathbb{R}_{\alpha}, and)
- (1)
, , 2. (2)
\mathchoice{\vphantom{\vtop{\halign{#\cr\hfil{\displaystyle q}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\displaystyle q}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\textstyle q}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\textstyle q}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\scriptstyle q}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\scriptstyle q}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}{\vtop{\halign{#\cr\hfil{\scriptscriptstyle q}\hfil\hfil\mathord{\scriptscriptstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}=\widehat{h} for a function (with ) mapping into the top level of , i.e. .
Definition 3.15**.**
Let to be the subset of consisting of elements satisfying for every .
Lemma 3.16**.**
Let . Then is dense in .
Proof.
Since for a fixed \langle p,\mathchoice{\vphantom{\vtop{\halign{#\cr\hfil{\displaystyle q}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\displaystyle q}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\textstyle q}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\textstyle q}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\scriptstyle q}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\scriptstyle q}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}{\vtop{\halign{#\cr\hfil{\scriptscriptstyle q}\hfil\hfil\mathord{\scriptscriptstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}\rangle\in\mathbb{R}_{\alpha}
[TABLE]
thus
[TABLE]
and by applying the maximal principle [4, II., Thm. 8.2] two times, there exist names and such that
[TABLE]
and
[TABLE]
Now, recall that is -closed (by Lemma 2.14), thus the set will not grow by an extension with a -generic filter. Moreover, the -closedness of allows us to apply Lemma 2.18, and we obtain a condition , and functions , () such that
[TABLE]
By - this means that determines
, i.e. letting
[TABLE]
By extending further if necessary, we may assume that has successor height, say , and, moreover, . Now, for each , choose extending (this is possible since is normal), and define a function with and for all . Then we are done, since
[TABLE]
∎
Fact 3.17**.**
For a set the restriction \mathbb{R}^{\bullet}\raisebox{-1.29167pt}{\upharpoonright}_{E}=\{r\raisebox{-1.29167pt}{\upharpoonright}_{E}:\ r\in\mathbb{R}^{\bullet}\} is a dense subset of \mathbb{R}\raisebox{-1.29167pt}{\upharpoonright}_{E}.
Lemma 3.18**.**
Let be a c.t.m. such that
[TABLE]
i.e. there are no new sequences of type consisting of elements of .
Then for any set ()
[TABLE]
Proof.
Note that our conditions imply that (), and holds also in . For the conclusion of the lemma these corollaries would be sufficient, but in our applications will always hold.
We will need the following lemma [4, Ch II. Thm. 1.6.] which we will refer to as the -system Lemma.
Lemma 3.19**.**
Let be an infinite cardinal, let be regular, and satisfy (). Assume that , and (). Then there is a , such that , and forms a -system, i.e. there is a kernel set such that
[TABLE]
From now on we will work in , therefore will stand for . Assume on the contrary that A\subseteq\mathbb{R}\raisebox{-1.29167pt}{\upharpoonright}_{E} is an antichain of size . We can apply Lemma 3.19 for the set of supports of the antichain (with , and ), since each support is countable (by ) and in (by ). Hence we can assume w.l.o.g. that is a -system with the kernel
[TABLE]
that is for each , ,
[TABLE]
and if , then there is at most one for which
[TABLE]
By Fact 3.17 we can assume that A\subseteq\mathbb{R}^{\bullet}\raisebox{-1.29167pt}{\upharpoonright}_{E}. Therefore for each we can define the function with , , and a sequence such that the following holds
[TABLE]
Since
[TABLE]
by , (in ), we obtain that there is an element , and a set , , such that
[TABLE]
Now we will apply the -system Lemma for the system of countable sets, hence there is a subset such that
[TABLE]
where is countable. Now
[TABLE]
where this latter set has size at most , because by
[TABLE]
Therefore we can obtain a subset of size such that
[TABLE]
Now it is straightforward to check that if , then and are compatible.
∎
Next we prove that for each the has -many branches in . With a slight abuse of notation, from now on we will identify each branch with the corresponding function from to , i.e. the following holds
[TABLE]
Lemma 3.20**.**
Let be fixed. Then the following holds in .
[TABLE]
Proof.
Using , there is a filter which is -generic over , and a filter H_{\alpha}\subseteq\mathchoice{\vphantom{\vtop{\halign{#\cr\hfil{\displaystyle\mathbb{Q}{\alpha}}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\displaystyle\mathbb{Q}{\alpha}}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\textstyle\mathbb{Q}{\alpha}}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\textstyle\mathbb{Q}{\alpha}}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\scriptstyle\mathbb{Q}{\alpha}}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\scriptstyle\mathbb{Q}{\alpha}}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}{\vtop{\halign{#\cr\hfil{\scriptscriptstyle\mathbb{Q}{\alpha}}\hfil\hfil\mathord{\scriptscriptstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}[F_{\alpha}] which is \mathchoice{\vphantom{\vtop{\halign{#\cr\hfil{\displaystyle\mathbb{Q}{\alpha}}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\displaystyle\mathbb{Q}{\alpha}}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\textstyle\mathbb{Q}{\alpha}}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\textstyle\mathbb{Q}{\alpha}}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\scriptstyle\mathbb{Q}{\alpha}}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\scriptstyle\mathbb{Q}{\alpha}}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}{\vtop{\halign{#\cr\hfil{\scriptscriptstyle\mathbb{Q}{\alpha}}\hfil\hfil\mathord{\scriptscriptstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}[F_{\alpha}]-generic over . Now by the very definition of the name
, , \mathchoice{\vphantom{\vtop{\halign{#\cr\hfil{\displaystyle\mathbb{Q}{\alpha}}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\displaystyle\mathbb{Q}{\alpha}}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\textstyle\mathbb{Q}{\alpha}}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\textstyle\mathbb{Q}{\alpha}}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\scriptstyle\mathbb{Q}{\alpha}}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\scriptstyle\mathbb{Q}{\alpha}}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}{\vtop{\halign{#\cr\hfil{\scriptscriptstyle\mathbb{Q}_{\alpha}}\hfil\hfil\mathord{\scriptscriptstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}[F_{\alpha}] is the notion of forcing
[TABLE]
where a condition is stronger than iff for each , is an end-extension of as functions, i.e.,
[TABLE]
It is straightforward to see that a generic filter adds branches, where the pairwise distinct new branches (by genericity) are corresponding to elements of , thus
[TABLE]
The following lemma proves that the inequality also holds in .
Before stating that lemma we state that for functions where having the same domain, by we mean the pointwise addition modulo .
Lemma 3.21**.**
Denoting the set of branches explicitly added by the filter by
[TABLE]
the following will hold. For each branch , there is an ordinal , and branches such that
[TABLE]
that is, for each .
This is a statement in , assume that it doesn’t hold, and let be a counterexample, and a name for it. Then there is an element that forces (in ) that is a counterexample, i.e.
[TABLE]
By Lemma 3.16 we can assume that .
Working in , first we will need the following Claim.
Claim 3.22**.**
There exist a decreasing sequence of conditions in , and a strictly increasing sequence of countable ordinals , such that the following conditions hold
- (i)
* from ,* 2. (ii)
the height of is , 3. (iii)
the function maps its domain onto -s top level, i.e.,
[TABLE] 4. (iv)
for each , , there exists such that
[TABLE]
Claim 3.23**.**
For the sequences given by Claim 3.22 there exist a countable ordinal and a lower bound () in , where
[TABLE]
and
[TABLE]
[TABLE]
Moreover, for each there exists , , such that
[TABLE]
Before proving these claims first we show that Claim 3.22 and 3.23 finish the proof of Lemma 3.21. Suppose that Claim 3.23 gives the lower bound for the decreasing sequence (given by Claim 3.22). Then for a generic filter with , determines the levels of the generic tree T^{\prime}=\cup\{p:\ \exists\mathchoice{\vphantom{\vtop{\halign{#\cr\hfil{\displaystyle q}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\displaystyle q}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\textstyle q}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\textstyle q}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\scriptstyle q}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\scriptstyle q}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}{\vtop{\halign{#\cr\hfil{\scriptscriptstyle q}\hfil\hfil\mathord{\scriptscriptstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}\ \ \langle p,\mathchoice{\vphantom{\vtop{\halign{#\cr\hfil{\displaystyle q}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\displaystyle q}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\textstyle q}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\textstyle q}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\scriptstyle q}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\scriptstyle q}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}{\vtop{\halign{#\cr\hfil{\scriptscriptstyle q}\hfil\hfil\mathord{\scriptscriptstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}\rangle\in G^{\prime}\} below , i.e. .
Therefore, towards a contradiction, suppose that is an arbitrary filter that is generic over with , and holds with . In is a branch through the generic tree , and \dot{b}[G^{\prime}]\raisebox{-1.29167pt}{\upharpoonright}_{\gamma_{\infty}} must be an element of . Claim 3.23 states that there exist , , such that
[TABLE]
Also by the construction of (Claim 3.23) , and (and recall that is decreasing, hence is increasing). This means that there is a finite such that , and . Then condition from Claim 3.22 implies that there is a such that
[TABLE]
In particular (using that )
[TABLE]
which contradicts .
For the proof of Claim 3.22 and 3.23 we will need the following technical preparations.
Claim 3.24**.**
Suppose that is a limit ordinal and is a countable homogeneous normal tree of height , and , is a countable set of branches of . Then
[TABLE]
is a countable homogeneous normal tree of height , where T\raisebox{-1.29167pt}{\upharpoonright}_{\xi}=T^{\prime}.
Proof.
Define the set as
[TABLE]
and
[TABLE]
i.e. we add some branches to and obtain a tree of height , in other words
[TABLE]
First we have to check that every element of which we added is indeed a branch of , i.e.,
[TABLE]
Fixing an arbitrary element from , , and first we can assume that . Observe that for each , using that contains , u_{1}\raisebox{-1.29167pt}{\upharpoonright}_{\delta}, u_{2}\raisebox{-1.29167pt}{\upharpoonright}_{\delta}, , u_{2k+1}\raisebox{-1.29167pt}{\upharpoonright}_{\delta}, by the homogeneity of
[TABLE]
Now, if t,b\raisebox{-1.29167pt}{\upharpoonright}_{\beta}\in T, we can use Claim 2.15 to get that t\cup b\raisebox{-1.29167pt}{\upharpoonright}_{\beta\setminus\operatorname{dom}(t)}\in T^{\prime}.
is obviously countable, and the normality will follow from the fact that is normal, we only have to check that for each there is greater than , i.e., . Indeed, if is not on the top level of then choosing an arbitrary , we have by the construction that t\cup u\raisebox{-1.29167pt}{\upharpoonright}_{\xi\setminus\operatorname{dom}(t)}\in T.
For the homogeneity of , fix , , , we have to check that is in . We can assume that since otherwise , and the homogeneity of implies that F_{cd}(t)=F_{\left(c\raisebox{-0.90417pt}{\upharpoonright}_{\operatorname{dom}(t)}\right)\left(d\raisebox{-0.90417pt}{\upharpoonright}_{\operatorname{dom}(t)}\right)}(t)\in T^{\prime}. Therefore , and t=t^{\prime}\cup b\raisebox{-1.29167pt}{\upharpoonright}_{\xi\setminus\operatorname{dom}(t)} for some , . Second, if , then letting , can be considered as
[TABLE]
(where t\raisebox{-1.29167pt}{\upharpoonright}_{\delta}\in T^{\prime} by ), hence again by the homogeneity of we have
[TABLE]
This means that the only remaining case is when , that is, , and are of the form
[TABLE]
[TABLE]
Now, if the ordinals are not equal, then letting , we can view t=t^{\prime}\cup b\raisebox{-1.29167pt}{\upharpoonright}_{\xi\setminus\operatorname{dom}(t^{\prime})} as t=t\raisebox{-1.29167pt}{\upharpoonright}_{\delta}\cup b\raisebox{-1.29167pt}{\upharpoonright}_{\xi\setminus\delta}, and similarly
[TABLE]
[TABLE]
Then
[TABLE]
we would only need that . By implies , therefore is a homogeneous tree, indeed.
∎
Moreover, we obtain the following.
Corollary 3.25**.**
Let , be a decreasing sequence in . Then there is a common lower bound of the sequence.
Proof.
Let , are such that . Let (which is in by Lemma 2.14), and to be the function such that , assigning . Then is a branch of .
If the ’s are strictly decreasing (and thus the ’s are strictly increasing) must be a limit ordinal, and then for obtaining we can apply Claim 3.24 with
[TABLE]
, .
∎
Corollary 3.26**.**
For , , with the countable ordinals there exists an extension of with .
Proof.
The statement holds also for , if . We apply induction on , and assume that .
For , if is the desired extension for , then let p^{\prime}=p^{\prime\prime}\cup\{t\mathbin{\mathchoice{\text{\raisebox{3.44444pt}{\smallfrown}}}{\text{\raisebox{3.44444pt}{\smallfrown}}}{\text{\raisebox{3.44444pt}{\smallfrown}}}{\text{\raisebox{3.44444pt}{\smallfrown}}}}i:\ i\in\{0,1\},t\in p\}, and for each let g^{\prime}(x)=g(x)\mathbin{\mathchoice{\text{\raisebox{3.44444pt}{\smallfrown}}}{\text{\raisebox{3.44444pt}{\smallfrown}}}{\text{\raisebox{3.44444pt}{\smallfrown}}}{\text{\raisebox{3.44444pt}{\smallfrown}}}}0.
Finally, for limit choose a sequence such that . By induction choose a decreasing sequence
[TABLE]
in so that . Now applying Claim 3.25 will work. ∎
Proof.
(Claim 3.22) We are given (the height of ), and we will apply induction.
Assume that , and are defined. Let be an enumeration of the set (recall that is countable using from Definition 3.5). Now we construct a decreasing sequence
[TABLE]
below in , and a sequence (where each ) such that
- (1)
if , then
[TABLE] 2. (2)
for we have .
Suppose that the -s, and the -s are defined for , and let . Then using that and ,
[TABLE]
hence there is a countable ordinal , and a condition , such that
[TABLE]
By Lemma 3.16, we can assume that , and let . Also by further extension (using Corollary 3.26) we can assume that
[TABLE]
As we obtained the decreasing sequence under and the -s, we can define to be a lower bound of the ’s as follows. Using Corollary 3.25 first we define , to be a lower bound of the ’s.
Define as follows. For each let . For ensuring , for each we can pick pairwise distinct elements from , and define . Now we have checked .
It remained to check that , , (defined by the equalities and ) satisfy . If , then and together implies from . follows from the fact that holds for ’s first coordinate . ∎
Proof.
(Claim 3.23) So suppose that , fulfills our requirements . Let which is a countable homogeneous normal tree of height
[TABLE]
by Lemma 2.14, and because the sequence of -s is strictly increasing. We define the function as follows.
[TABLE]
and for each define to be , which is a function, since -s form a decreasing sequence in . By , , which is not an element of , since (by ).
Apply Claim 3.24 with , , , and let be the given tree (that is
[TABLE]
where
[TABLE]
we are done. ∎
∎
We will need the following basic lemmas.
Lemma 3.27**.**
Suppose that , is a set, is a c.t.m., such that
[TABLE]
i.e., there is no new sequence of type consisting of elements of . Let () be a tree of height with countable levels. Then extending by a filter F\subseteq\mathbb{R}\raisebox{-1.29167pt}{\upharpoonright}_{E} which is \mathbb{R}\raisebox{-1.29167pt}{\upharpoonright}_{E}-generic adds no new branches to , i.e.
[TABLE]
Proof.
As \mathbb{R}^{\bullet}\raisebox{-1.29167pt}{\upharpoonright}_{E} is dense in \mathbb{R}\raisebox{-1.29167pt}{\upharpoonright}_{E} (by Corollary 3.17), forcing with one yields exactly the same extensions as forcing with the other (by Lemma 2.25), we only have to show that \mathbb{R}^{\bullet}\raisebox{-1.29167pt}{\upharpoonright}_{E} is -closed in to apply Lemma 2.24. Our conditions together with by Corollary 3.25 imply that for each decreasing sequence (of type ) in \mathbb{R}^{\bullet}\raisebox{-1.29167pt}{\upharpoonright}_{E} belonging to has a lower bound. (In fact first we can find such a lower bound only in and we can restrict the obtained condition, or we can also refer to the fact that these partial functions are countably supported). Then Lemma 2.24 gives the desired result.
∎
Lemma 3.28**.**
Let be a set, be a c.t.m. such that
[TABLE]
i.e. there is no new sequence of type consisting of elements of . Then extending by a filter F\subseteq\mathbb{R}\raisebox{-1.29167pt}{\upharpoonright}_{E} which is \mathbb{R}\raisebox{-1.29167pt}{\upharpoonright}_{E}-generic adds no new sequences of type consisting of elements of , i.e.
[TABLE]
Proof.
Again, (similarly to the proof of Lemma 3.27) we have that each decreasing -sequence in \mathbb{R}^{\bullet}\raisebox{-1.29167pt}{\upharpoonright}_{E} belonging to has a lower bound (by Corollary 3.25). Then apply Corollary 2.19. ∎
Recall that is -generic over . In the next lemma we will prove that if , then there is no Kurepa tree in with branches.
Lemma 3.29**.**
Let () be a tree of height with countable levels, and let be an ordinal so that
[TABLE]
Then .
The proof of this lemma will take a lot of effort. From Lemma 3.31 to Lemma 3.55 we will find two models, each containing , but exactly the greater containing all branches of . From Claim 3.56 to Lemma 3.59 we will see that the homogeneity of our generic -s imply that the larger of the two models cannot contain all the branches, contradicting our previous arguments.
Fix an ordinal such that
[TABLE]
We will derive a contradiction by finding a suitable intermediate model containing , arguing that the residual forcing still adds new branches.
First we would like to find an intermediate extension () which is small enough, but .
Before that we prove that .
Claim 3.30**.**
[TABLE]
This implies that does not collapse, that is,
[TABLE]
Moreover,
[TABLE]
in particular
[TABLE]
Proof.
We have two cases depending on . The filter is -generic, where either (if ), or (otherwise, ). Now as is -closed, both and are -closed (Corollary 3.25 and each condition is countably supported). This means that by Corollary 2.19 forcing with , or does not add new sequences. Then recalling Corollary 3.17 we obtain that is dense in , and is dense in , we are done (by Lemma 2.25). ∎
Lemma 3.31**.**
Suppose that be a c.t.m. where
[TABLE]
and for our inaccessible from (and )
[TABLE]
Then
[TABLE]
Proof.
Suppose that is a an antichain of size . First we can apply the -system lemma (Lemma 3.19) for the system , since is countable by , and for any infinite ordinal
[TABLE]
by the fact that is inaccessible in . Therefore we can assume that is a -system, let denote its kernel. Since is countable, and is inaccessible in , implies that . Therefore, there is an ordinal such that . This and the definition of imply that for each , \operatorname{ran}(a\raisebox{-1.29167pt}{\upharpoonright}_{K})\subseteq\delta. But the derived system A^{\prime}=\{a\raisebox{-1.29167pt}{\upharpoonright}_{K}:\ a\in A\}\subseteq\delta^{K}, gives an upper bound
[TABLE]
for the cardinality of . This contradicts the fact that is an antichain of size . ∎
Lemma 3.32**.**
In the final model,
[TABLE]
In general, cardinals and cofinalities greater than or equal to are preserved, where
- •
, if , that is, we forced with too,
- •
, if .
Proof.
For proving , by Definition 3.4 it is enough to show that if then only cardinals strictly between and are collapsed, and if , then no cardinals are collapsed.
In both cases, Corollary 3.30 states that is not collapsed. Now if , then we had forced with , otherwise only with . In the first case, by Lemma 3.11, can be identified with I\times G\raisebox{-1.29167pt}{\upharpoonright}_{C}, and we can consider this extension as first extending with , and then with G\raisebox{-1.29167pt}{\upharpoonright}_{C}. Therefore, in both case it is enough to show that
- (1)
adding the filter which is generic over destroys exactly cardinals in . 2. (2)
extending (resp., ) by G\raisebox{-1.29167pt}{\upharpoonright}_{C} doesn’t collapse cardinals greater than (resp., )
Note that by Corollary 3.30, is absolute. For the first claim, collapses every cardinal between and , because the generic filter gives surjections from onto each . Lemma 3.31 gives that is -cc in , thus by Lemma 2.23 cardinals and cofinalities greater than or equal to remain cardinal in .
For , we can apply Lemma 3.18 for (with , and too, because of Corollary 3.30), and we obtain that is -cc in in each case. Then Lemma 2.23 implies that cardinals (and cofinalities) greater than or equal to are still cardinals (and cofinalities) after forcing. This completes the proof of . ∎
Lemma 3.33**.**
If is a forcing extension, ( is generic over ), where is a notion of forcing (smaller than )M, then holds in for .
Moreover, in the case when we used the inaccessible cardinal (i.e. ), then
[TABLE]
Proof.
Let . By Lemma 2.22, if is a cardinal in , then
[TABLE]
This yields that
[TABLE]
by in . Therefore we also obtain that in for (because ).
Moreover, if M[G]=M[I][G\raisebox{-1.29167pt}{\upharpoonright}_{C}] and is inaccessible in , then is still a strong limit in . In this case, because , and obviously has the -cc in , we have that Lemma 2.23 guarantees that is still in . This yields the conclusion that remains inaccessible in . ∎
For finding our desired model which contains as an element, but cannot contain all of its branches (because is smaller there), we need to extend by filters containing less information than what and G\raisebox{-1.29167pt}{\upharpoonright}_{C}\subseteq\mathbb{R}\raisebox{-1.29167pt}{\upharpoonright}_{C}=\mathbb{R} give. First we are to find a model between and extracting minimal information from the extension by .
We would like to consider the notion of forcing as a product.
Definition 3.34**.**
For defined in , and a set of ordinals let
[TABLE]
Then clearly
[TABLE]
Furthermore, for any filter define
[TABLE]
Claim 3.35**.**
There exists an ordinal such that
[TABLE]
Proof.
First, using Lemma 3.28
[TABLE]
also implying that is absolute. Moreover, is a cardinal in M[G\raisebox{-1.29167pt}{\upharpoonright}_{C}]\subseteq M[G], because \mathbb{R}=\mathbb{R}\raisebox{-1.29167pt}{\upharpoonright}_{C} has the -cc in (by Lemma 3.18). Now Lemma 3.31 states that
[TABLE]
Applying Lemma 2.21 in M[G\raisebox{-1.29167pt}{\upharpoonright}_{C}], there is a nice -name \sigma\in M[G\raisebox{-1.29167pt}{\upharpoonright}_{C}] for a subset of for which
[TABLE]
where \dot{T}\in M[G\raisebox{-1.29167pt}{\upharpoonright}_{C}] is a -name for T\in M[G\raisebox{-1.29167pt}{\upharpoonright}_{C}][I]. Here (where each is an antichain in , and of size by ). Note that \operatorname{cf}^{M[G\raisebox{-0.90417pt}{\upharpoonright}_{C}]}(\kappa)=\kappa by Lemma 2.23 (because \mathbb{R}\raisebox{-1.29167pt}{\upharpoonright}_{C} is -cc by Lemma 3.18). This means that for each there is an ordinal such that
[TABLE]
Define
[TABLE]
Then clearly A_{f}\subseteq\mathbb{L}\raisebox{-1.29167pt}{\upharpoonright}_{\mu} for each . Since \mathbb{L}\simeq\mathbb{L}\raisebox{-1.29167pt}{\upharpoonright}_{\mu}\times\mathbb{L}\raisebox{-1.29167pt}{\upharpoonright}_{\kappa\setminus\mu}, and if I\raisebox{-1.29167pt}{\upharpoonright}_{\mu}=I\cap\mathbb{L}\raisebox{-1.29167pt}{\upharpoonright}_{\mu}, I\raisebox{-1.29167pt}{\upharpoonright}_{\kappa\setminus\mu}=I\cap\mathbb{L}\raisebox{-1.29167pt}{\upharpoonright}_{\kappa\setminus\mu} are filters given by in the components, then the tree depends only on coordinates in , i.e. on I\raisebox{-1.29167pt}{\upharpoonright}_{\mu}. Therefore
[TABLE]
as desired.
∎
Definition 3.36**.**
In the case when G=I\times G\raisebox{-1.29167pt}{\upharpoonright}_{C} (because ) we define M^{\prime\prime}=M[I\raisebox{-1.29167pt}{\upharpoonright}_{\mu}], and if G=G\raisebox{-1.29167pt}{\upharpoonright}_{C}, then let .
Note that in each case
[TABLE]
Claim 3.37**.**
There exists a set , such that
[TABLE]
and
[TABLE]
Proof.
First, since is -closed (in ), M^{\omega}\cap M[I\raisebox{-1.29167pt}{\upharpoonright}_{\mu}]=M^{\omega}\cap M. Therefore one can apply Lemma 3.18 and obtain that in (and M[I\raisebox{-1.29167pt}{\upharpoonright}_{\mu}], resp.). \mathbb{R}\raisebox{-1.29167pt}{\upharpoonright}_{C} has no antichain of size (, resp.). We get that
[TABLE]
By Lemma 2.21, there is a nice \mathbb{R}\raisebox{-1.29167pt}{\upharpoonright}_{C}-name for a subset of such that
[TABLE]
(where is a \mathbb{R}\raisebox{-1.29167pt}{\upharpoonright}_{C}-name for T\in M^{\prime\prime}[G\raisebox{-1.29167pt}{\upharpoonright}_{C}], and ). Define , and
[TABLE]
Since each is countable (by the very definition of , and \mathbb{R}\raisebox{-1.29167pt}{\upharpoonright}_{C} is a projection) and because (in ), we have that
[TABLE]
(Note that, since we worked in we only have that .)
Recall that G\raisebox{-1.29167pt}{\upharpoonright}_{C} can be identified with the product G\raisebox{-1.29167pt}{\upharpoonright}_{S}\times G\raisebox{-1.29167pt}{\upharpoonright}_{C\setminus S}. Now \sigma[G\raisebox{-1.29167pt}{\upharpoonright}_{C}] depends only on G\raisebox{-1.29167pt}{\upharpoonright}_{C}’s projection onto \mathbb{R}\raisebox{-1.29167pt}{\upharpoonright}_{S}, G\raisebox{-1.29167pt}{\upharpoonright}_{S}, and there is a corresponding \mathbb{R}\raisebox{-1.29167pt}{\upharpoonright}_{S}-name such that
[TABLE]
∎
In the beginning of Lemma 3.29, our condition was equality
[TABLE]
(where ), and our goal is to find a model between and M^{\prime\prime}[G\raisebox{-1.29167pt}{\upharpoonright}_{S}] with , and
[TABLE]
implying that
[TABLE]
(since each branch corresponds to a function from to ).
Now working in , we are to show that \mathbb{R}\raisebox{-1.29167pt}{\upharpoonright}_{S}, which is a product of two-step iterations is isomorphic to a two-step iteration of products. \mathbb{R}\raisebox{-1.29167pt}{\upharpoonright}_{S} was a product restricted to the countably supported elements, each coordinate is a two-step iteration \mathbb{P}_{\gamma}\ast\mathchoice{\vphantom{\vtop{\halign{#\cr\hfil{\displaystyle\mathbb{Q}{\gamma}}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\displaystyle\mathbb{Q}{\gamma}}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\textstyle\mathbb{Q}{\gamma}}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\textstyle\mathbb{Q}{\gamma}}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\scriptstyle\mathbb{Q}{\gamma}}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\scriptstyle\mathbb{Q}{\gamma}}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}{\vtop{\halign{#\cr\hfil{\scriptscriptstyle\mathbb{Q}{\gamma}}\hfil\hfil\mathord{\scriptscriptstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}. Recall that an element r\in\mathbb{R}\raisebox{-1.29167pt}{\upharpoonright}_{S}\subseteq\prod_{\gamma\in S}\mathbb{R}_{\gamma} has coordinates of the form r_{\gamma}=\langle p_{\gamma},\mathchoice{\vphantom{\vtop{\halign{#\cr\hfil{\displaystyle q{\gamma}}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\displaystyle q_{\gamma}}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\textstyle q_{\gamma}}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\textstyle q_{\gamma}}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\scriptstyle q_{\gamma}}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\scriptstyle q_{\gamma}}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}{\vtop{\halign{#\cr\hfil{\scriptscriptstyle q_{\gamma}}\hfil\hfil\mathord{\scriptscriptstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}\rangle\in\mathbb{R}_{\gamma} (). Here
is a -name for which
[TABLE]
by , .
Definition 3.38**.**
Let \mathbb{P}\raisebox{-1.29167pt}{\upharpoonright}_{S} be
[TABLE]
We will construct a partial order that has a dense subset isomorphic to \mathbb{R}^{\bullet}\raisebox{-1.29167pt}{\upharpoonright}_{S}.
Definition 3.39**.**
Define \mathchoice{\vphantom{\vtop{\halign{#\cr\hfil{\displaystyle\mathbb{Q}\raisebox{-1.29167pt}{}{S}}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\displaystyle\mathbb{Q}\raisebox{-1.29167pt}{}{S}}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\textstyle\mathbb{Q}\raisebox{-1.29167pt}{}{S}}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\textstyle\mathbb{Q}\raisebox{-1.29167pt}{}{S}}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\scriptstyle\mathbb{Q}\raisebox{-0.90417pt}{}{S}}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\scriptstyle\mathbb{Q}\raisebox{-0.90417pt}{}{S}}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}{\vtop{\halign{#\cr\hfil{\scriptscriptstyle\mathbb{Q}\raisebox{-0.64583pt}{}_{S}}\hfil\hfil\mathord{\scriptscriptstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}\in M to be the \mathbb{P}\raisebox{-1.29167pt}{\upharpoonright}_{S}-name so that
[TABLE]
Definition 3.40**.**
Let (\mathbb{P}\raisebox{-1.29167pt}{\upharpoonright}_{S}\ast\mathchoice{\vphantom{\vtop{\halign{#\cr\hfil{\displaystyle\mathbb{Q}\raisebox{-1.29167pt}{}{S}}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\displaystyle\mathbb{Q}\raisebox{-1.29167pt}{}{S}}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\textstyle\mathbb{Q}\raisebox{-1.29167pt}{}{S}}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\textstyle\mathbb{Q}\raisebox{-1.29167pt}{}{S}}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\scriptstyle\mathbb{Q}\raisebox{-0.90417pt}{}{S}}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\scriptstyle\mathbb{Q}\raisebox{-0.90417pt}{}{S}}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}{\vtop{\halign{#\cr\hfil{\scriptscriptstyle\mathbb{Q}\raisebox{-0.64583pt}{}{S}}\hfil\hfil\mathord{\scriptscriptstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}})^{\bullet} be the following subset of the two-step iteration \mathbb{P}\raisebox{-1.29167pt}{\upharpoonright}_{S}\ast\mathchoice{\vphantom{\vtop{\halign{#\cr\hfil{\displaystyle\mathbb{Q}\raisebox{-1.29167pt}{}{S}}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\displaystyle\mathbb{Q}\raisebox{-1.29167pt}{}{S}}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\textstyle\mathbb{Q}\raisebox{-1.29167pt}{}{S}}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\textstyle\mathbb{Q}\raisebox{-1.29167pt}{}{S}}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\scriptstyle\mathbb{Q}\raisebox{-0.90417pt}{}{S}}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\scriptstyle\mathbb{Q}\raisebox{-0.90417pt}{}{S}}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}{\vtop{\halign{#\cr\hfil{\scriptscriptstyle\mathbb{Q}\raisebox{-0.64583pt}{}{S}}\hfil\hfil\mathord{\scriptscriptstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}. Define \langle p,\mathchoice{\vphantom{\vtop{\halign{#\cr\hfil{\displaystyle q}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\displaystyle q}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\textstyle q}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\textstyle q}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\scriptstyle q}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\scriptstyle q}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}{\vtop{\halign{#\cr\hfil{\scriptscriptstyle q}\hfil\hfil\mathord{\scriptscriptstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}\rangle to be an element of (\mathbb{P}\raisebox{-1.29167pt}{\upharpoonright}_{S}\ast\mathchoice{\vphantom{\vtop{\halign{#\cr\hfil{\displaystyle\mathbb{Q}\raisebox{-1.29167pt}{}{S}}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\displaystyle\mathbb{Q}\raisebox{-1.29167pt}{}{S}}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\textstyle\mathbb{Q}\raisebox{-1.29167pt}{}{S}}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\textstyle\mathbb{Q}\raisebox{-1.29167pt}{}{S}}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\scriptstyle\mathbb{Q}\raisebox{-0.90417pt}{}{S}}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\scriptstyle\mathbb{Q}\raisebox{-0.90417pt}{}{S}}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}{\vtop{\halign{#\cr\hfil{\scriptscriptstyle\mathbb{Q}\raisebox{-0.64583pt}{}_{S}}\hfil\hfil\mathord{\scriptscriptstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}})^{\bullet}, iff
- (1)
\mathchoice{\vphantom{\vtop{\halign{#\cr\hfil{\displaystyle q}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\displaystyle q}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\textstyle q}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\textstyle q}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\scriptstyle q}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\scriptstyle q}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}{\vtop{\halign{#\cr\hfil{\scriptscriptstyle q}\hfil\hfil\mathord{\scriptscriptstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}=\widehat{h} for some with , 2. (2)
, 3. (3)
for each .
Claim 3.41**.**
(\mathbb{P}\raisebox{-1.29167pt}{\upharpoonright}_{S}\ast\mathchoice{\vphantom{\vtop{\halign{#\cr\hfil{\displaystyle\mathbb{Q}\raisebox{-1.29167pt}{}{S}}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\displaystyle\mathbb{Q}\raisebox{-1.29167pt}{}{S}}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\textstyle\mathbb{Q}\raisebox{-1.29167pt}{}{S}}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\textstyle\mathbb{Q}\raisebox{-1.29167pt}{}{S}}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\scriptstyle\mathbb{Q}\raisebox{-0.90417pt}{}{S}}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\scriptstyle\mathbb{Q}\raisebox{-0.90417pt}{}{S}}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}{\vtop{\halign{#\cr\hfil{\scriptscriptstyle\mathbb{Q}\raisebox{-0.64583pt}{}{S}}\hfil\hfil\mathord{\scriptscriptstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}})^{\bullet}* is a dense subset of \mathbb{P}\raisebox{-1.29167pt}{\upharpoonright}_{S}\ast\mathchoice{\vphantom{\vtop{\halign{#\cr\hfil{\displaystyle\mathbb{Q}\raisebox{-1.29167pt}{}{S}}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\displaystyle\mathbb{Q}\raisebox{-1.29167pt}{}{S}}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\textstyle\mathbb{Q}\raisebox{-1.29167pt}{}{S}}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\textstyle\mathbb{Q}\raisebox{-1.29167pt}{}{S}}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\scriptstyle\mathbb{Q}\raisebox{-0.90417pt}{}{S}}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\scriptstyle\mathbb{Q}\raisebox{-0.90417pt}{}{S}}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}{\vtop{\halign{#\cr\hfil{\scriptscriptstyle\mathbb{Q}\raisebox{-0.64583pt}{}{S}}\hfil\hfil\mathord{\scriptscriptstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}.*
Proof.
Fix \langle p,\mathchoice{\vphantom{\vtop{\halign{#\cr\hfil{\displaystyle q}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\displaystyle q}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\textstyle q}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\textstyle q}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\scriptstyle q}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\scriptstyle q}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}{\vtop{\halign{#\cr\hfil{\scriptscriptstyle q}\hfil\hfil\mathord{\scriptscriptstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}\rangle\in\mathbb{P}\raisebox{-1.29167pt}{\upharpoonright}_{S}\ast\mathchoice{\vphantom{\vtop{\halign{#\cr\hfil{\displaystyle\mathbb{Q}\raisebox{-1.29167pt}{}{S}}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\displaystyle\mathbb{Q}\raisebox{-1.29167pt}{}{S}}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\textstyle\mathbb{Q}\raisebox{-1.29167pt}{}{S}}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\textstyle\mathbb{Q}\raisebox{-1.29167pt}{}{S}}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\scriptstyle\mathbb{Q}\raisebox{-0.90417pt}{}{S}}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\scriptstyle\mathbb{Q}\raisebox{-0.90417pt}{}{S}}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}{\vtop{\halign{#\cr\hfil{\scriptscriptstyle\mathbb{Q}\raisebox{-0.64583pt}{}_{S}}\hfil\hfil\mathord{\scriptscriptstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}. Similarly to the proof of Lemma 3.16, first recall that
[TABLE]
and p\in\mathbb{P}\raisebox{-1.29167pt}{\upharpoonright}_{S}, which is -closed, as being then countable supported product of -closed posets (Lemma 2.14). Now a suitable extension of determines \operatorname{dom}(\mathchoice{\vphantom{\vtop{\halign{#\cr\hfil{\displaystyle q}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\displaystyle q}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\textstyle q}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\textstyle q}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\scriptstyle q}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\scriptstyle q}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}{\vtop{\halign{#\cr\hfil{\scriptscriptstyle q}\hfil\hfil\mathord{\scriptscriptstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}})\in[S]^{<\omega_{1}}. Then we can extend so that for each \alpha\in\operatorname{dom}(\mathchoice{\vphantom{\vtop{\halign{#\cr\hfil{\displaystyle q}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\displaystyle q}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\textstyle q}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\textstyle q}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\scriptstyle q}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\scriptstyle q}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}{\vtop{\halign{#\cr\hfil{\scriptscriptstyle q}\hfil\hfil\mathord{\scriptscriptstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}) it forces a value for \operatorname{supp}(\mathchoice{\vphantom{\vtop{\halign{#\cr\hfil{\displaystyle q}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\displaystyle q}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\textstyle q}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\textstyle q}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\scriptstyle q}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\scriptstyle q}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}{\vtop{\halign{#\cr\hfil{\scriptscriptstyle q}\hfil\hfil\mathord{\scriptscriptstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}(\alpha))\in[X_{\alpha}]^{<\omega_{1}}, and by picking a further extension we can assume that determines (\mathchoice{\vphantom{\vtop{\halign{#\cr\hfil{\displaystyle q}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\displaystyle q}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\textstyle q}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\textstyle q}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\scriptstyle q}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\scriptstyle q}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}{\vtop{\halign{#\cr\hfil{\scriptscriptstyle q}\hfil\hfil\mathord{\scriptscriptstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}(\alpha))(\beta)\in 2^{<\omega_{1}} for each \alpha\in\operatorname{dom}(\mathchoice{\vphantom{\vtop{\halign{#\cr\hfil{\displaystyle q}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\displaystyle q}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\textstyle q}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\textstyle q}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\scriptstyle q}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\scriptstyle q}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}{\vtop{\halign{#\cr\hfil{\scriptscriptstyle q}\hfil\hfil\mathord{\scriptscriptstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}), \beta\in\operatorname{supp}(\mathchoice{\vphantom{\vtop{\halign{#\cr\hfil{\displaystyle q}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\displaystyle q}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\textstyle q}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\textstyle q}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\scriptstyle q}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\scriptstyle q}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}{\vtop{\halign{#\cr\hfil{\scriptscriptstyle q}\hfil\hfil\mathord{\scriptscriptstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}(\alpha)). Therefore we obtain a function with \langle p^{\prime\prime},\widehat{h}\rangle\in\mathbb{P}\raisebox{-1.29167pt}{\upharpoonright}_{S}\ast\mathchoice{\vphantom{\vtop{\halign{#\cr\hfil{\displaystyle\mathbb{Q}\raisebox{-1.29167pt}{}{S}}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\displaystyle\mathbb{Q}\raisebox{-1.29167pt}{}{S}}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\textstyle\mathbb{Q}\raisebox{-1.29167pt}{}{S}}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\textstyle\mathbb{Q}\raisebox{-1.29167pt}{}{S}}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\scriptstyle\mathbb{Q}\raisebox{-0.90417pt}{}{S}}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\scriptstyle\mathbb{Q}\raisebox{-0.90417pt}{}{S}}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}{\vtop{\halign{#\cr\hfil{\scriptscriptstyle\mathbb{Q}\raisebox{-0.64583pt}{}_{S}}\hfil\hfil\mathord{\scriptscriptstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}, and
[TABLE]
After a further extension we can assume that , and for each the tree has a top level, and the function maps its domain into the top level of . ∎
Claim 3.42**.**
(\mathbb{P}\raisebox{-1.29167pt}{\upharpoonright}_{S}\ast\mathchoice{\vphantom{\vtop{\halign{#\cr\hfil{\displaystyle\mathbb{Q}\raisebox{-1.29167pt}{}{S}}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\displaystyle\mathbb{Q}\raisebox{-1.29167pt}{}{S}}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\textstyle\mathbb{Q}\raisebox{-1.29167pt}{}{S}}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\textstyle\mathbb{Q}\raisebox{-1.29167pt}{}{S}}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\scriptstyle\mathbb{Q}\raisebox{-0.90417pt}{}{S}}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\scriptstyle\mathbb{Q}\raisebox{-0.90417pt}{}{S}}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}{\vtop{\halign{#\cr\hfil{\scriptscriptstyle\mathbb{Q}\raisebox{-0.64583pt}{}_{S}}\hfil\hfil\mathord{\scriptscriptstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}})^{\bullet}* is isomorphic to \mathbb{R}^{\bullet}\raisebox{-1.29167pt}{\upharpoonright}_{S}.*
Proof.
Simply assign to the pair , where , and for each . ∎
As G\raisebox{-1.29167pt}{\upharpoonright}_{S}\cap\mathbb{R}^{\bullet}\raisebox{-1.29167pt}{\upharpoonright}_{S} is \mathbb{R}^{\bullet}\raisebox{-1.29167pt}{\upharpoonright}_{S}-generic over (Lemma 2.25), Claims 3.41, 3.42 (and applying Lemma 2.25 again) imply that there are generic filters F\raisebox{-1.29167pt}{\upharpoonright}_{S}\subseteq\mathbb{P}\raisebox{-1.29167pt}{\upharpoonright}_{S}, H\raisebox{-1.29167pt}{\upharpoonright}_{S}\subseteq\mathchoice{\vphantom{\vtop{\halign{#\cr\hfil{\displaystyle\mathbb{Q}\raisebox{-1.29167pt}{}{S}}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\displaystyle\mathbb{Q}\raisebox{-1.29167pt}{}{S}}\hfil\hfil\mathord{\displaystyle\sim}\hfil\crcr\kern 2.0pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\textstyle\mathbb{Q}\raisebox{-1.29167pt}{}{S}}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\textstyle\mathbb{Q}\raisebox{-1.29167pt}{}{S}}\hfil\hfil\mathord{\textstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}{\vphantom{\vtop{\halign{#\cr\hfil{\scriptstyle\mathbb{Q}\raisebox{-0.90417pt}{}{S}}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}\vtop{\halign{#\cr\hfil{\scriptstyle\mathbb{Q}\raisebox{-0.90417pt}{}{S}}\hfil\hfil\mathord{\scriptstyle\sim}\hfil\crcr\kern 2.1pt\nointerlineskip\cr}}}{\vtop{\halign{#\cr\hfil{\scriptscriptstyle\mathbb{Q}\raisebox{-0.64583pt}{}_{S}}\hfil\hfil\mathord{\scriptscriptstyle\sim}\hfil\crcr\kern 1.5pt\nointerlineskip\cr}}}[F\raisebox{-1.29167pt}{\upharpoonright}_{S}] such that
[TABLE]
We could consider the model M^{\prime\prime}[F\raisebox{-1.29167pt}{\upharpoonright}_{S}] in which the trees () are already existing elements, but at that moment we have not added the branches yet, implying that is small.
Definition 3.43**.**
In M^{\prime\prime}[F\raisebox{-1.29167pt}{\upharpoonright}_{S}] we define
[TABLE]
and
[TABLE]
Define be the filter so that
[TABLE]
holds.
We will have the following crucial lemma.
Lemma 3.44**.**
If is a c.t.m. such that (and ) then
[TABLE]
Proof.
The proof is a straightforward application of the -system lemma, and (which holds by Claim 3.30). Assume that is an antichain. Then, since the -s are countably supported , and by , there is a subset of size where
[TABLE]
Now, since implies that (and )
[TABLE]
Thus one can find -many elements of such that any two of them coincide on (which is the intersection of their domains). ∎
Our next goal is to find a subset , such that , and adding the branches indexed by the elements of to M^{\prime\prime}[F\raisebox{-1.29167pt}{\upharpoonright}_{S}] will result in a model that contains the tree .
We will see that adding the branches indexed by will result a model which cannot contain all the branches , because there will not be large enough (i.e. in M^{\prime\prime}[F\raisebox{-1.29167pt}{\upharpoonright}_{S}][K\raisebox{-1.29167pt}{\upharpoonright}_{{Z\cup(\bigcup\{X_{\delta}:\ \delta\in S,\delta<\alpha\})}}])). From now on we will work in M^{\prime\prime}[F\raisebox{-1.29167pt}{\upharpoonright}_{S}] to prove that forcing with \mathbb{Q}\raisebox{-1.29167pt}{\upharpoonright}_{Z\cup\bigcup\{X_{\delta}:\ \delta\in S,\delta<\alpha\}} will have these aforementioned properties.
Claim 3.45**.**
There exists a set of size at most , i.e.
[TABLE]
such that
[TABLE]
Proof.
Since T\subseteq M^{\prime\prime}[F\raisebox{-1.29167pt}{\upharpoonright}_{S}] (in fact, , by Claim 3.30), and T\in M^{\prime\prime}[F\raisebox{-1.29167pt}{\upharpoonright}_{S}][K], applying Lemma 2.21 gives that there is a nice -name in M^{\prime\prime}[F\raisebox{-1.29167pt}{\upharpoonright}_{S}] for a subset of , such that
[TABLE]
is a nice name i.e. is of the form
[TABLE]
where each is an antichain, and each is of size at most by Lemma 3.44. Let
[TABLE]
where . Then clearly depends only on K\raisebox{-1.29167pt}{\upharpoonright}_{Z}, thus
[TABLE]
∎
Definition 3.46**.**
Let denote the set
[TABLE]
Obviously
[TABLE]
let denote M^{\prime\prime}[F\raisebox{-1.29167pt}{\upharpoonright}_{S}][K\raisebox{-1.29167pt}{\upharpoonright}_{Y}].
Lemma 3.47**.**
N=M^{\prime\prime}[F\raisebox{-1.29167pt}{\upharpoonright}_{S}][K\raisebox{-1.29167pt}{\upharpoonright}_{Y}]* contains , but there are branches in which are not contained in .*
Proof.
The next lemma is the key for verifying that , where our assumption was that
[TABLE]
thus .
Now we have two cases depending on whether is empty, or not. If the set is empty, then since and either holds, implying , or thus . Therefore Claims 3.48 and 3.51 will finish the proof of Lemma 3.47.
Claim 3.48**.**
If , then
[TABLE]
Proof.
First we will need that this case holds above in M^{\prime\prime}[F\raisebox{-1.29167pt}{\upharpoonright}_{S}]. It suffices to prove the following claim.
Subclaim 3.49**.**
M^{\prime\prime}[F\raisebox{-1.29167pt}{\upharpoonright}_{S}]* can be obtained by a single forcing extension of , where the poset has cardinalityM less than .*
Proof.
Since we defined to be (Definition 3.36), we have that M^{\prime\prime}[F\raisebox{-1.29167pt}{\upharpoonright}_{S}] is a forcing extension of , where we forced with the set \mathbb{P}\raisebox{-1.29167pt}{\upharpoonright}_{S}. In order to show that
[TABLE]
first, in (because by Definition 2.12), hence |\mathbb{P}\raisebox{-1.29167pt}{\upharpoonright}_{S}|\leq|\mathbb{P}_{\hom}|^{\omega}\cdot|\omega_{1}^{\omega}=\omega_{1}. ∎
Recall that each , given by the -generic filter is a subtree in of height , thus is of size . In M^{\prime\prime}[F\raisebox{-1.29167pt}{\upharpoonright}_{S}] and Definition 3.5 (recalling that is generic) give us that \mathbb{Q}\raisebox{-1.29167pt}{\upharpoonright}_{Y} is of size
[TABLE]
We have to determine . Let
[TABLE]
Since , and by 3.37
[TABLE]
which gives . Now we will show that . Recall that by our assumptions .
Subclaim 3.50**.**
[TABLE]
[TABLE]
Proof.
First observe that as , states that , and then the condition in Theorem 3.2 implies that
[TABLE]
∎
Now , but by Lemma 3.32 (which is in fact of ), thus
[TABLE]
Letting denote , note that , by Lemma 3.32. Using ,
[TABLE]
Recalling that is -cc in M^{\prime\prime}[F\raisebox{-1.29167pt}{\upharpoonright}_{S}] (Lemma 3.44), Lemma 2.22 states that
[TABLE]
For calculating this cardinal in M^{\prime\prime}[F\raisebox{-1.29167pt}{\upharpoonright}_{S}] we have two cases.
- •
If is limit (and thus \operatorname{cf}^{M^{\prime\prime}[F\raisebox{-0.90417pt}{\upharpoonright}_{S}]}(\sigma)\leq\omega_{1}), then first recall that no cofinalities were collapsed in our case. Using the conditions for in Theorem 3.2 we have that , therefore for . This case using that , and by the in M^{\prime\prime}[F\raisebox{-1.29167pt}{\upharpoonright}_{S}] above for
[TABLE]
- •
If is a successor, then . Hence using again that holds above (and that )
[TABLE]
We get that and the above estimations give
[TABLE]
∎
Claim 3.51**.**
If
[TABLE]
Proof.
First recall that
[TABLE]
Since each is of size ,
[TABLE]
by . Then \mathbb{Q}\raisebox{-1.29167pt}{\upharpoonright}_{Z} is trivially -cc, and by Lemma .
[TABLE]
We can calculate (\omega_{1}^{\omega_{1}})^{M^{\prime\prime}[F\raisebox{-0.90417pt}{\upharpoonright}_{S}]}, because by Lemma 3.33 (with M^{\prime}=M^{\prime\prime}[F\raisebox{-1.29167pt}{\upharpoonright}_{S}]) is inaccessible in M^{\prime\prime}[F\raisebox{-1.29167pt}{\upharpoonright}_{S}], therefore
[TABLE]
as desired. ∎
This finishes the proof of Lemma 3.47, since it follows from that , and .
∎
Next we prove that M^{\prime\prime}[F\raisebox{-1.29167pt}{\upharpoonright}_{S}][K]=M^{\prime\prime}[F\raisebox{-1.29167pt}{\upharpoonright}_{S}][K\raisebox{-1.29167pt}{\upharpoonright}_{Y}][K\raisebox{-1.29167pt}{\upharpoonright}_{X\setminus Y}]=N[K\raisebox{-1.29167pt}{\upharpoonright}_{X\setminus Y}] will contain each branch of from the final model .
Claim 3.52**.**
[TABLE]
Proof.
First, is either M[G\raisebox{-1.29167pt}{\upharpoonright}_{C}], or M[I][G\raisebox{-1.29167pt}{\upharpoonright}_{C}], and G\raisebox{-1.29167pt}{\upharpoonright}_{C}\simeq G\raisebox{-1.29167pt}{\upharpoonright}_{S}\times G\raisebox{-1.29167pt}{\upharpoonright}_{C\setminus S}, I\simeq I\raisebox{-1.29167pt}{\upharpoonright}_{\mu}\times I\raisebox{-1.29167pt}{\upharpoonright}_{\kappa\setminus\mu}, and is either , or M[I\raisebox{-1.29167pt}{\upharpoonright}_{\mu}]. Also recall that M^{\prime\prime}[G\raisebox{-1.29167pt}{\upharpoonright}_{S}]=M^{\prime\prime}[F\raisebox{-1.29167pt}{\upharpoonright}_{S}][K] by from Definition 62. This means that
[TABLE]
and our final model is either M^{\prime\prime}[G\raisebox{-1.29167pt}{\upharpoonright}_{C}], or M^{\prime\prime}[G\raisebox{-1.29167pt}{\upharpoonright}_{C}][I\raisebox{-1.29167pt}{\upharpoonright}_{\kappa\setminus\mu}].
For this forcing extensions (i.e. extension by G\raisebox{-1.29167pt}{\upharpoonright}_{C\setminus S} and I\raisebox{-1.29167pt}{\upharpoonright}_{\kappa\setminus\mu}) we would like to apply Lemmas 2.19, 3.27 to ensure that none of them add new branches to . First, the \mathbb{R}\raisebox{-1.29167pt}{\upharpoonright}_{C\setminus S}-generic filter G\raisebox{-1.29167pt}{\upharpoonright}_{C\setminus S} doesn’t add new branches by Lemma 3.27 (applying with M^{\prime}=M^{\prime\prime}[F\raisebox{-1.29167pt}{\upharpoonright}_{S}][K]). Second, \mathbb{L}\raisebox{-1.29167pt}{\upharpoonright}_{\kappa\setminus\mu}\in M is -closed in . But there are no new -sequences in M^{\prime\prime}[G\raisebox{-1.29167pt}{\upharpoonright}_{C}]\subseteq M[G] by Corollary 3.30, thus it can be easily seen that
[TABLE]
Now we can apply Lemma 2.19, thus forcing with the -closed \mathbb{L}\raisebox{-1.29167pt}{\upharpoonright}_{\kappa\setminus\mu} adds no new branches to . ∎
As we got that all branches of are contained in M^{\prime\prime}[F\raisebox{-1.29167pt}{\upharpoonright}_{S}][K]=M^{\prime\prime}[F\raisebox{-1.29167pt}{\upharpoonright}_{S}][K\raisebox{-1.29167pt}{\upharpoonright}_{Y}][K\raisebox{-1.29167pt}{\upharpoonright}_{X\setminus Y}], and T\in N=M^{\prime\prime}[F\raisebox{-1.29167pt}{\upharpoonright}_{S}][K\raisebox{-1.29167pt}{\upharpoonright}_{Y}], it remains to show that the extension of with the filter K\raisebox{-1.29167pt}{\upharpoonright}_{X\setminus Y} adds more than -many branches, which would contradict .
Claim 3.53**.**
There exists an ordinal such that .
Proof.
By the definition of , we have
[TABLE]
and
[TABLE]
and we know that
[TABLE]
Now assume on the contrary that (hence also ), then the equality would hold, which contradicts to the fact that extension with K\raisebox{-1.29167pt}{\upharpoonright}_{X\setminus Y} adds branches to (because of Lemma 3.47). ∎
Therefore the fact that for (by Lemma 3.33), and
[TABLE]
(together with ) implies that for every
[TABLE]
Definition 3.54**.**
Let be denoted by , and let
[TABLE]
Then states
[TABLE]
Claim 3.55**.**
There is a set , of size such that decomposing \mathbb{Q}\raisebox{-1.29167pt}{\upharpoonright}_{X^{\prime}} into the product \mathbb{Q}\raisebox{-1.29167pt}{\upharpoonright}_{X^{\prime\prime}}\times\mathbb{Q}\raisebox{-1.29167pt}{\upharpoonright}_{X^{\prime}\setminus X^{\prime\prime}}, and thus obtaining the filters K\raisebox{-1.29167pt}{\upharpoonright}_{X^{\prime\prime}}, K\raisebox{-1.29167pt}{\upharpoonright}_{X^{\prime}\setminus X^{\prime\prime}},
[TABLE]
that is, all branches of are in the model N[K\raisebox{-1.29167pt}{\upharpoonright}_{X^{\prime\prime}}].
Proof.
In M^{\prime\prime}[F\raisebox{-1.29167pt}{\upharpoonright}_{S}][K] (and in ) there are branches of (by Claim 3.52, and ), and since each branch corresponds to a function from to , the set of branches can be identified with a function , which can be identified with a subset of . Let be the set in N[K\raisebox{-1.29167pt}{\upharpoonright}_{X\setminus Y}] coding the branches of . Then using Lemma 2.21, there is a nice \mathbb{Q}\raisebox{-1.29167pt}{\upharpoonright}_{X\setminus Y}-name in such that
[TABLE]
(where is a \mathbb{Q}\raisebox{-1.29167pt}{\upharpoonright}_{X\setminus Y}-name in for ). Now if
[TABLE]
where each A_{\langle\mu,\nu\rangle}\subseteq\mathbb{Q}\raisebox{-1.29167pt}{\upharpoonright}_{X\setminus Y} is an antichain, we have (because by Lemma 3.44
[TABLE]
that for each . Define
[TABLE]
and note that
[TABLE]
Then clearly B=\sigma[K\raisebox{-1.29167pt}{\upharpoonright}_{X\setminus Y}], and for any \mathbb{Q}\raisebox{-1.29167pt}{\upharpoonright}_{X\setminus Y}-generic filter
[TABLE]
This completes the proof the Claim. ∎
Claim 3.56**.**
(Let be given by Claim 3.55.) There is a set such that (in ) \mathbb{Q}\raisebox{-1.29167pt}{\upharpoonright}_{X^{\prime\prime}} is isomorphic to \mathbb{Q}\raisebox{-1.29167pt}{\upharpoonright}_{X^{\prime\prime\prime}}. (This gives rise to the split up of \mathbb{Q}\raisebox{-1.29167pt}{\upharpoonright}_{X^{\prime}} to the product
[TABLE]
Proof.
Since is given by Lemma 3.55 , and recall that
[TABLE]
and
[TABLE]
by . For each , , using the facts
[TABLE]
[TABLE]
there is a set of size . Letting , it follows that for each
[TABLE]
therefore (by , and the absoluteness of this definition, Lemma 3.30)
[TABLE]
as desired. ∎
The next point will be the key in the proof of Lemma 3.29, where we will make use of the homogeneity of the -s similarly as in [2].
Lemma 3.57**.**
Whenever is a \mathbb{Q}\raisebox{-1.29167pt}{\upharpoonright}_{X^{\prime\prime}}-name such that b[K\raisebox{-1.29167pt}{\upharpoonright}_{X^{\prime\prime}}]\in N[K\raisebox{-1.29167pt}{\upharpoonright}_{X^{\prime\prime}}] is a branch through , b[K\raisebox{-1.29167pt}{\upharpoonright}_{X^{\prime\prime}}]\notin N, then there exists a \mathbb{Q}\raisebox{-1.29167pt}{\upharpoonright}_{X^{\prime\prime}}-name such that (in )
[TABLE]
For the proof we will need the following claim.
Claim 3.58**.**
For any fixed element q\in\mathbb{Q}\raisebox{-1.29167pt}{\upharpoonright}_{X^{\prime\prime}}, and filter which is \mathbb{Q}\raisebox{-1.29167pt}{\upharpoonright}_{X^{\prime\prime}}-generic over , there is another \mathbb{Q}\raisebox{-1.29167pt}{\upharpoonright}_{X^{\prime\prime}}-generic filter (over ) such that , and
[TABLE]
Proof.
Recall that each is a homogeneous normal tree given by forcing with , normality implies that for each , , there exists an element with . Now by the definition of \mathbb{Q}\raisebox{-1.29167pt}{\upharpoonright}_{X^{\prime\prime}} , using -s countability, we can assume that there exists a countable ordinal such that implies that .
Let . Let be such that , and for each . For each , we define
[TABLE]
which is an automorphism of if we restrict to it, because is homogeneous. Now we define an automorphism of the poset \mathbb{Q}\raisebox{-1.29167pt}{\upharpoonright}_{X^{\prime\prime}}.
[TABLE]
[TABLE]
such that
[TABLE]
i.e. (considering \mathbb{Q}\raisebox{-1.29167pt}{\upharpoonright}_{X^{\prime\prime}} as the countable support product of -s) we applied an automorphism on some coordinates (coordinates in ). Obviously , by and . It is straightforward to check that is indeed an automorphism, since for a pair q_{1},q_{2}\in\mathbb{Q}\raisebox{-1.29167pt}{\upharpoonright}_{X^{\prime\prime}}
[TABLE]
Now letting
[TABLE]
we obtain a filter containing . It remained to check that is generic over . Suppose that is a dense subset of \mathbb{Q}\raisebox{-1.29167pt}{\upharpoonright}_{X^{\prime\prime}}. Then is also a dense subset (note that ), and if , then and .
∎
Proof.
(Lemma 3.57) Now suppose that b[K\raisebox{-1.29167pt}{\upharpoonright}_{X^{\prime\prime}}] is a new branch of , and q\in K\raisebox{-1.29167pt}{\upharpoonright}_{X^{\prime\prime}} forces that, i.e.
[TABLE]
Now using Claim 3.58, for any filter which is generic over , there exists a generic filter containing , such that
[TABLE]
This means that
[TABLE]
and by and ,
[TABLE]
thus
[TABLE]
Since was arbitrary, we have that
[TABLE]
Finally, applying the maximal principle [4, II., Thm. 8.2], there exists a name such that
[TABLE]
∎
Now, if \psi:\mathbb{Q}\raisebox{-1.29167pt}{\upharpoonright}_{X^{\prime\prime}}\to\mathbb{Q}\raisebox{-1.29167pt}{\upharpoonright}_{X^{\prime\prime\prime}} is an isomorphism (provided by Claim 3.56), (and denotes the induced operation between the \mathbb{Q}\raisebox{-1.29167pt}{\upharpoonright}_{X^{\prime\prime}}-names and \mathbb{Q}\raisebox{-1.29167pt}{\upharpoonright}_{X^{\prime\prime\prime}}-names) then by our previous lemma clearly
[TABLE]
The next Lemma completes the proof of Lemma 3.29, since Lemma 3.55 guarantees that N[K\raisebox{-1.29167pt}{\upharpoonright}_{X^{\prime\prime}}] contains all branches of .
Lemma 3.59**.**
For given by Lemma 3.57
[TABLE]
Before the proof recall the facts implying that Lemma 3.59 completes the proof of 3.29. By Lemma 3.55 each branch of which is in already appears in N[K\raisebox{-1.29167pt}{\upharpoonright}_{X^{\prime\prime}}], that is
[TABLE]
But \psi^{*}(b^{\prime})[K\raisebox{-1.29167pt}{\upharpoonright}_{X^{\prime\prime\prime}}] is a new branch in N[K\raisebox{-1.29167pt}{\upharpoonright}_{X^{\prime\prime}}][K\raisebox{-1.29167pt}{\upharpoonright}_{X^{\prime\prime\prime}}] which is a model between and , a contradiction.
Proof.
Whenever J\subseteq\mathbb{Q}\raisebox{-1.29167pt}{\upharpoonright}_{X^{\prime\prime\prime}} is generic over we have that is a new branch of (i.e. not in ), therefore we claim the following.
Claim 3.60**.**
[TABLE]
Proof.
Assume on the contrary, that is a counterexample. But then for each there exists such that whenever decides , then
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from which
[TABLE]
Now, since we defined in , determines , thus forces that is not a new branch, a contradiction. ∎
But this claim is true even in N[K\raisebox{-1.29167pt}{\upharpoonright}_{X^{\prime\prime}}].
Claim 3.61**.**
[TABLE]
Proof.
For a fixed let and given by Claim 3.60 for which holds. Let J\subseteq\mathbb{Q}\raisebox{-1.29167pt}{\upharpoonright}_{X^{\prime\prime\prime}} be generic over N[K\raisebox{-1.29167pt}{\upharpoonright}_{X^{\prime\prime}}]. If for some , then is also generic over , and by Lemma 3.60,
[TABLE]
Statement is absolute between transitive models, thus
[TABLE]
We conclude that choosing the same -s and works. ∎
Let B\in N[K\raisebox{-1.29167pt}{\upharpoonright}_{X^{\prime\prime}}] denote the set of branches of in N[K\raisebox{-1.29167pt}{\upharpoonright}_{X^{\prime\prime}}]. Now if q\in\mathbb{Q}\raisebox{-1.29167pt}{\upharpoonright}_{X^{\prime\prime\prime}} forces that is not a new branch
[TABLE]
then in a fixed generic filter K\raisebox{-1.29167pt}{\upharpoonright}_{X^{\prime\prime\prime}} containing
[TABLE]
This implies that there exists a branch , such that , which is forced by some q^{\prime}\in K\raisebox{-1.29167pt}{\upharpoonright}_{X^{\prime\prime\prime}},
[TABLE]
i.e. determines , which contradicts to Claim 3.61.
∎
Remark 3.62**.**
By further forcing we could prescribe to be any cardinal greater than or equal to the cardinalities of branches of Kurepa trees.
Question 3.63**.**
In Theorem 3.2 can we drop the condition that for if
[TABLE]
then must be contained in ? Or is it true that the existence of a Kurepa tree with -many branches (with either countable or ) implies not only the inequality (Konig’s inequality), but the existence of a Kurepa tree with exactly branches?
Acknowledgment
I would like to thank my masters thesis supervisor Péter Komjáth for presenting me the problem, and also giving hints and ideas, providing some useful advice.
I also gratefully thank to the referee for constructive comments and recommendations, also pointing to possible simplifications of otherwise tedious technical arguments.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] Goldstern, Martin: Tools for your forcing constructions. H. Judah, editor, Set Theory of the Reals, volume 6 of Israel Mathematical Conference Proceedings (1992), pp. 305–360.
- 2[2] Jin, R., Shelah, S.: Planting Kurepa Trees And Killing Jech-Kunen Trees By Using One Inaccessible Cardinal. Fundamenta Mathematicae, vol. 141 (1992), pp 287-296.
- 3[3] Jin, R., Shelah, S.: A Model In Which There Are Jech-Kunen Trees But There Are No Kurepa Trees. Israel Journal of Mathematics, 84 (1993), pp. 1-16.
- 4[4] Kunen, Kenneth: Set theory: An introduction to independence proofs. Studies in logic and foundations of mathematics, vol. 102., North–Holland Publishing Co, 1983.
- 5[5] Kunen, Kenneth: Saturated Ideals. J. Symbolic Logic 43 (1978), no. 1, 65–76.
- 6[6] Silver, J.: The independence of Kurepa hypothesis and two cardinal conjecture in model theory. 1971 Axiomatic Set Theory (Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., 1967) pp 383-390 Amer. Math. Soc., Providence, R.I.
