1 Introduction
The notion of abstract elementary classes (AECs) was created by Shelah [Reference Shelah and Baldwin22] to encompass certain classes of models, including models of first-order theories. To develop the classification theory of AECs, notions like types, stability, and superstability were generalized to the AEC context. Superstability is a major topic in AECs because it is implied by categoricity and has good transfer properties (assuming tameness and a monster model). Early results could be found in [Reference Grossberg, VanDieren and Villaveces17, Reference Shelah23, Reference Shelah and Villaveces26–Reference VanDieren28], which were later extended by Boney, Grossberg, VanDieren, and Vasey.
Guided by the first-order case, Grossberg and Vasey [Reference Grossberg and Vasey18] provided several equivalent definitions of superstability for AECs: no long splitting chains, existence of a good frame, existence of a unique limit model, existence of a superlimit model, solvability, and that the union of an increasing chain of
$\lambda $
-saturated models is
$\lambda $
-saturated (where
$\lambda $
is high enough). Later Vasey [Reference Vasey38, Corollary 4.24] added the (expected) criterion of stability in a tail. However, their results had two drawbacks:
-
1. The cardinal threshold was high (the first Hanf number): if an AEC satisfies one of the criteria in
$\operatorname {LS}(\mathbf {K})$ or
${\operatorname {LS}(\mathbf {K})}^+$ , it does not necessarily imply any other criterion.
-
2. The cardinal jump was high between equivalent criteria: if an AEC satisfies one of the criteria in a (high-enough) cardinal
$\lambda $ , it was only known that other criteria hold in a much bigger cardinal (sometimes
$\beth _\omega (\lambda )$ ).
In this paper, we aim to refine the list of equivalent superstability criteria using known techniques in the literature. We reduce the cardinal threshold to
${\operatorname {LS}(\mathbf {K})}^+$
and the cardinal jump to a successor cardinal. The missing piece in [Reference Grossberg and Vasey18] was to show the uniqueness of limit models from no long splitting chains. The original approach used Galois-Morleyization and averages [Reference Boney and Vasey11] at the cost of a high cardinal jump. We observe that the tower approach from [Reference VanDieren and Vasey31] is cleaner and has no cardinal jump, which allows us to rewrite many results in the literature that involve the uniqueness of limit models.
On the other hand, the equivalent criteria of superstability in first-order theories have their strictly-stable analogues. For example, when the local character of forking
$\kappa (T)$
is uncountable, the union of an increasing chain of
$\lambda $
-saturated models is
$\lambda $
-saturated, provided that the chain has cofinality at least
$\kappa (T)$
. This motivates us to look for generalizations of the superstability criteria. It turns out that a key assumption is “continuity of nonsplitting,” which allows us to mimick the proofs of many superstable results. Such assumption was partially explored in [Reference Boney and VanDieren10, Reference Vasey38] but a full picture was yet to be seen. It seems that the study of stability is much more difficult without assuming continuity of nonsplitting: Vasey [Reference Vasey38] gave some eventual results of stability which were only applicable to high cardinals.
In the following, we provide an overview of the upcoming sections and highlight some key theorems: Section 2 states the global assumptions and preliminaries. Section 3 studies the properties of nonsplitting, which will be used to build good frames in Section 4. The idea of good frames was developed in [Reference Shelah24, IV Theorem 4.10], assuming categoricity and non-ZFC axioms, in order to deduce nice structural properties of an AEC. Later Boney and Grossberg [Reference Boney and Grossberg6] built a good frame from coheir with the assumption of tameness and extension property of coheir in ZFC. Vasey [Reference Vasey34, Section 5] further developed on coheir and [Reference Vasey32] managed to construct a good frame at a high categoricity cardinal (categoricity can be replaced by superstability and type locality, but the initial cardinal of the good frame is still high).
Another approach to building a good frame is via nonsplitting. It is in general not clear whether uniqueness or transitivity hold for nonsplitting (where models are ordered by universal extensions). To resolve this problem, Vasey [Reference Vasey33] constructed nonforking from nonsplitting, which has nicer properties: assuming superstability in
$K_{\mu }$
, tameness, and a monster model, nonforking gives rise to a good frame over the limit models in
$K_{\mu ^+}$
[Reference VanDieren and Vasey31, Corollary 6.14]. Later it was found that uniqueness of nonforking also holds for limit models in
$K_{\mu }$
[Reference Vasey35].
We will generalize the nonforking results by replacing the superstability assumption by continuity of nonsplitting. A key observation is that the extension property of nonforking still holds if we have continuity of nonsplitting and stability. This allows us to replicate extension, uniqueness, and transitivity properties. Since the assumption of continuity of nonsplitting applies to universal extensions only, we only get continuity and local character for universal extensions. Hence we can build an approximation of a good frame which is over the skeleton (see Definition 2.4) of long enough limit models ordered by universal extensions. We state the known result and our result for comparison.
Theorem 1.1. Let
$\mu \geq \operatorname {LS}(\mathbf {K})$
,
$\mathbf {K}$
have a monster model, be
$\mu $
-tame and stable in
$\mu $
. Let
$\chi $
be the local character of
$\mu $
-nonsplitting.
-
1. [Reference Vasey37, Corollary 13.16] If
$\mathbf {K}$ is
$\mu $ -superstable, then there exists a good frame over the skeleton of limit models in
$K_{\mu }$ ordered by
$\leq _u$ , except for symmetry.
-
2. (Corollary 4.13) If
$\mu $ is regular and
$\mathbf {K}$ has continuity of
$\mu $ -nonsplitting, then there exists a good
$\mu $ -frame over the skeleton of
$(\mu ,\geq \chi )$ -limit models ordered by
$\leq _u$ , except for symmetry. The local character is
$\chi $ in place of
${\aleph _0}$ .
In Section 5, we will deduce symmetry under extra stability assumptions. In Section 6, we will generalize known superstability results using the symmetry properties. Symmetry is an important property of a good frame that connects superstability and the uniqueness of limit models. To obtain symmetry for our frame, we look into the argument in [Reference VanDieren and Vasey31]. In [Reference VanDieren29, Reference VanDieren30], VanDieren defined a stronger version of symmetry called
$\mu $
-symmetry and proved its equivalence with the continuity of reduced towers. VanDieren and Vasey [Reference VanDieren and Vasey31, Lemma 4.6] noticed that a weaker version of symmetry is sufficient in one direction and deduced the weaker version of symmetry via superstability. To generalize these arguments, we replace superstability by continuity of nonsplitting and stability in a range of cardinals (the range depends on the no order property of
$\mathbf {K}$
, see Proposition 5.9). Then we can obtain a local version of
$\mu $
-symmetry, which implies symmetry of our frame for long enough limit models.
Theorem 1.2. Let
$\mu \geq \operatorname {LS}(\mathbf {K})$
,
$\mathbf {K}$
be
$\mu $
-tame and stable in
$\mu $
. Let
$\chi $
be the local character of
$\mu $
-nonsplitting.
-
1. [Reference VanDieren and Vasey31, Corollary 6.9] If
$\mathbf {K}$ is
$\mu $ -superstable, then it has
$\mu $ -symmetry.
-
2. (Corollary 5.13) If
$\mu $ is regular and
$\mathbf {K}$ has continuity of
$\mu $ -nonsplitting. There is
$\lambda <h(\mu )$ such that if
$\mathbf {K}$ is stable in every cardinal between
$\mu $ and
$\lambda $ , then
$\mathbf {K}$ has
$(\mu ,\chi )$ -symmetry.
The notions of continuity of nonsplitting and of local symmetry were already exploited in [Reference Boney and VanDieren10, Theorem 20] to obtain the uniqueness of long enough limit models (see Fact 6.1). They simply assumed the local symmetry while we used the argument in [Reference VanDieren and Vasey31] to deduce it from extra stability and continuity of nonsplitting (Corollary 6.2). On the other hand, [Reference Vasey38, Section 11] used continuity of nonsplitting to deduce that a long enough chain of saturated models of the same cardinality is saturated. There he assumed saturation of limit models and managed to satisfy this assumption using his earlier result with Boney [Reference Boney and Vasey11], which has a high cardinal threshold. Since we already have local symmetry under continuity of nonsplitting and extra stability, we immediately have uniqueness of long limit models, and hence Vasey’s argument can be applied to obtain the above result of saturated models (see Proposition 6.6; a comparison table of the approaches can be found in Remark 6.8(2)).
Vasey [Reference Vasey38, Lemma 11.6] observed that a localization of VanDieren’s result [Reference VanDieren29] can give: if the union of a long enough chain of
$\mu ^+$
-saturated models is
$\mu ^+$
-saturated, then local symmetry is satisfied. Assuming more tameness, we use this observation to obtain converses of our results (see Main Theorem 8.1(4)
$\Rightarrow $
(3)). In particular, local symmetry will lead to uniqueness of long limit models, which implies local character of nonsplitting (Main Theorem 8.1(3)
$\Rightarrow $
(1)). Despite the important observation by Vasey, he did not derive these implications.
Theorem 1.3. Let
$\mu>\operatorname {LS}(\mathbf {K})$
,
$\delta \leq \mu $
be regular,
$\mathbf {K}$
have a monster model, be
$(<\mu )$
-tame, stable in
$\mu $
and has continuity of
$\mu $
-nonsplitting. If any increasing chain of
$\mu ^+$
-saturated models of cofinality
$\geq \delta $
has a
$\mu ^+$
-saturated union, then
$\mathbf {K}$
has
$\delta $
-local character of
$\mu $
-nonsplitting.
In the original list inside [Reference Grossberg and Vasey18],
$(\lambda ,\xi )$
-solvability was considered for
$\lambda>\xi $
, which they showed to be an equivalent definition of superstability, with a huge jump of cardinal from no long splitting chains to solvability. Further developments in [Reference Vasey36] indicate that such solvability has downward transfer properties which seems too strong to be called superstability. We propose a variation where
$\lambda =\xi $
and will prove its equivalence with no long splitting chains in the same cardinal above
$\mu ^+$
(under continuity of nonsplitting and stability). At
$K_{\mu }$
, we demand
$(<\mu )$
-tameness for the equivalence to hold, up to a jump to the successor cardinal.
Theorem 1.4. Let
$\mu>\operatorname {LS}(\mathbf {K})$
,
$\mathbf {K}$
have a monster model, be
$(<\mu )$
-tame, stable in
$\mu $
.
-
1. [Reference Shelah and Villaveces26] If there is
$\lambda>\mu $ such that
$\mathbf {K}$ is
$(\lambda ,\mu )$ -solvable, then it is
$\mu $ -superstable.
-
2. [Reference Grossberg and Vasey18, Corollary 5.5] If
$\mu $ is high enough and
$\mathbf {K}$ is
$\mu $ -superstable, then there is some
$\lambda \geq \mu $ and some
$\lambda '<\lambda $ such that
$\mathbf {K}$ is
$(\lambda ,\lambda ')$ -solvable.
-
3. (Proposition 6.24) If
$\mathbf {K}$ has continuity of
$\mu $ -nonsplitting, then it is
$\mu $ -superstable iff it is
$(\mu ^+,\mu ^+)$ -solvable.
In Section 7, we will consider two characterizations of superstability: stability in a tail and the boundedness of the U-rank. Vasey [Reference Vasey38, Corollary 4.24] showed that stability in a tail is also an equivalent definition of superstability, but the starting cardinal of superstability
$(\lambda '(\mathbf {K}))^++\chi _1$
is only bounded above by the Hanf number of
$\mu $
(he also implicitly assumed continuity of nonsplitting in deriving his results). In contrast, we carry out a slightly different proof to obtain
$\mu ^{ }$
-superstability, assuming stability in unboundedly many cardinals below
$\mu ^{ }$
, and enough stability above
$\mu $
.
Theorem 1.5. Let
$\mu>\operatorname {LS}(\mathbf {K})$
with cofinality
${\aleph _0}$
,
$\mathbf {K}$
have a monster model, have continuity of nonsplitting, be
$\mu $
-tame, stable in both
$\mu $
and unboundedly many cardinals below
$\mu $
.
-
1. [Reference Vasey38, Corollary 4.14] If
$\mu \geq (\lambda '(\mathbf {K}))^++\chi _1$ , then
$ \mathbf {K}$ is
$\mu $ -superstable.
-
2. (Proposition 7.5) There is
$\lambda <h(\mu )$ such that if
$\mathbf {K}$ is stable in
$[\mu ,\lambda )$ , then it is
$\mu $ -superstable.
It was mentioned at the end of [Reference Grossberg and Vasey18] that the no tree property and the boundedness of a rank function could be generalized to AECs. Some partial answers were given in [Reference Grossberg and Mazari-Armida15] regarding the no tree property (assuming a simple independence relation). Here we prove that the boundedness of the U-rank (with respect to
$\mu $
-nonforking for limit models in
$K_{\mu }$
ordered by universal extensions) is equivalent to
$\mu $
-superstability (Corollary 7.14). We will need to extend our nonforking to longer types, using results from [Reference Boney and Vasey12]. Then we can quote a lot of known results from [Reference Boney and Grossberg6, Reference Boney, Grossberg, Kolesnikov and Vasey7, Reference Grossberg and Mazari-Armida15]. Our strategy of extending frames contrasts with [Reference Vasey32] which used a complicated axiomatic framework and drew technical results from [Reference Shelah24, III]. Here we directly construct a type-full good
$\mu $
-frame from nonforking and the known results apply (which are independent of the technical ones in [Reference Shelah24, Reference Vasey32]).
Theorem 1.6. Let
$\mu \geq \operatorname {LS}(\mathbf {K})$
be regular,
$\mathbf {K}$
have a monster model, be
$\mu $
-tame, stable in
$\mu $
and have continuity of
$\mu $
-nonsplitting. Let
$U(\cdot )$
be the U-rank induced by
$\mu $
-nonforking restricted to limit models in
$K_{\mu }$
ordered by
$\leq _u$
. The following are equivalent:
-
1.
$\mathbf {K}$ is
$\mu $ -superstable.
-
2.
$U(p)<\infty $ for all
$p\in \operatorname {gS}(M)$ and limit model
$M\in K_{\mu }$ .
In Section 8, we summarize all our results as two main theorems: one for the superstable case and one for the strictly stable case. We give two applications in algebra: those results were known but here we only rely on model-theoretic techniques.
2 Preliminaries
Throughout this paper, we assume the following:
Assumption 2.1.
-
1.
$\mathbf {K}$ is an AEC with
$AP$ ,
$JEP,$ and
$NMM$ .
-
2.
$\mathbf {K}$ is stable in some
$\mu \geq \operatorname {LS}(\mathbf {K})$ .
-
3.
$\mathbf {K}$ is
$\mu $ -tame.
-
4.
$\mathbf {K}$ satisfies continuity of
$\mu $ -nonsplitting (Definition 3.5).
-
5.
$\chi \leq \mu $ where
$\chi $ is the minimum local character cardinal of
$\mu $ -nonsplitting (see Definition 3.10).
$AP$
stands for amalgamation property,
$JEP$
for joint embedding property, and
$NMM$
for no maximal model. They allow the construction of a monster model. Given a model
$M\in K$
, we write
$\operatorname {gS}(M)$
the set of Galois types over M (the ambient model does not matter because of
$AP$
).
Definition 2.2. Let
$\lambda $
be an infinite cardinal.
-
1.
$\alpha \geq 2$ be an ordinal,
$\mathbf {K}$ is
$(<\alpha )$ -stable in
$\lambda $ if for any
$\Vert M \Vert =\lambda $ ,
$|\operatorname {gS}^{<\alpha }(M)|\leq \lambda $ . We omit
$\alpha $ if
$\alpha =2$ .
-
2.
$\mathbf {K}$ is
$\lambda $ -tame if for any
$N\in K$ , any
$p\neq q\in \operatorname {gS}(N)$ , there is
$M\leq N$ of size
$\lambda $ such that
$p\restriction M\neq q\restriction M$ .
Definition 2.3. Let
$\lambda \geq \operatorname {LS}(\mathbf {K})$
be a cardinal and
$\alpha ,\beta <\lambda ^+$
be regular. Let
$M\leq N$
and
$\Vert M \Vert =\lambda $
.
-
1. N is universal over M (
$M<_uN$ ) if
$M<N$ and for any
$\Vert N' \Vert =\Vert N \Vert $ , there is
$f:N'\xrightarrow [M]{}N$ .
-
2. N is
$(\lambda ,\alpha )$ -limit over M if
$\Vert N \Vert =\lambda $ and there exists
$\langle M_i:i\leq \alpha \rangle \subseteq K_\lambda $ increasing and continuous such that
$M_0=M$ ,
$M_\alpha =N$ and
$M_{i+1}$ is universal over
$M_i$ for
$i<\alpha $ . We call
$\alpha $ the length of N.
-
3. N is
$(\lambda ,\alpha )$ -limit if there exists
$\Vert M' \Vert =\lambda $ such that N is
$(\lambda ,\alpha )$ -limit over
$M'$ .
-
4. N is
$(\lambda ,\geq \beta )$ -limit (over M) if there exists
$\alpha \geq \beta $ such that (2) (resp. (3)) holds.
-
5. N is
$(\lambda ,\lambda ^+)$ -limit (over M) if
$\Vert N \Vert =\lambda ^+$ and we replace
$\alpha $ by
$\lambda ^+$ in (2) (resp. (3)).
-
6. Let
$\lambda _1\leq \lambda _2$ , then N is
$([\lambda _1,\lambda _2],\geq \beta )$ -limit (over M) if there exists
$\lambda \in [\lambda _1,\lambda _2]$ such that N is
$(\lambda ,\geq \beta )$ -limit (over M).
-
7. If
$\lambda>\operatorname {LS}(\mathbf {K})$ , we say M is
$\lambda $ -saturated if for any
$M'\leq M$ ,
$\Vert M' \Vert <\lambda $ ,
$M\vDash \operatorname {gS}(M')$ .
-
8. M is saturated if it is
$\Vert M \Vert $ -saturated.
In general, we do not know limit models or saturated models are closed under chains, so they do not necessary form an AEC. We adapt [Reference Vasey32, Definition 5.3] to capture such behaviours.
Definition 2.4. An abstract class
$\mathbf {K}_{\mathbf {1}}$
is a
$\mu $
-skeleton of
$\mathbf {K}$
if the following is satisfied:
-
1.
$\mathbf {K}_{\mathbf {1}}$ is a sub-AC of
$\mathbf {K}_{\boldsymbol {\mu }}$ :
$K_1\subseteq K_{\mu }$ and for any
$M,N\in K_1$ ,
$M\leq _{\mathbf {K}_{\mathbf {1}}} N$ implies
$M\leq _{\mathbf {K}} N$ .
-
2. For any
$M\in K_{\mu }$ , there is
$M'\in K_1$ such that
$M\leq _{\mathbf {K}} M'$ .
-
3. Let
$\alpha $ be an ordinal and
$\langle M_i:i<\alpha \rangle $ be
$\leq _{\mathbf {K}}$ -increasing in
$K_1$ . There exists
$N\in K_1$ such that for all
$i<\alpha $ ,
$M_i\leq _{\mathbf {K}_{\mathbf {1}}} N$ (the original definition requires strict inequality but it is immaterial under
$NMM$ ).
We say
$\mathbf {K}_{\mathbf {1}}$
is a
$(\geq \mu )$
-skeleton of
$\mathbf {K}$
if the above items hold for
$K_{\geq \mu }$
in place of
$K_{\mu }$
.
By [Reference Shelah24, II Claim 1.16], limit models in
$\mu $
with
$\leq _{\mathbf {K}}$
form a
$\mu $
-skeleton of
$\mathbf {K}$
. Similarly let
$\alpha <\mu ^+$
be regular, then
$(\geq \mu ,\geq \alpha )$
-limits form a
$(\geq \mu )$
-skeleton of
$\mathbf {K}$
.
On the other hand, good frames were developed by Shelah [Reference Shelah24] for AECs in a range of cardinals. Vasey [Reference Vasey32] defined good frames over a coherent abstract class. We specialize the abstract class to a skeleton of an AEC.
Definition 2.5. Let
$\mathbf {K}$
be an AEC, and let
$\mathbf {K}_{\mathbf {1}}$
be a
$\mu $
-skeleton of
$\mathbf {K}$
. We say a nonforking relation is a good
$\mu $
-frame over the skeleton of
$\mathbf {K}_{\mathbf {1}}$
if the following holds:
-
1. The nonforking relation is a binary relation between a type
$p\in \operatorname {gS}(N)$ and a model
$M\leq _{\mathbf {K}_{\mathbf {1}}}N$ . We say p does not fork over M if the relation holds between p and M. Otherwise we say p forks over M.
-
2. Invariance: if
$f\in \operatorname {Aut}(\mathfrak {C})$ and p does not fork over M, then
$f(p)$ does not fork over
$f[M]$ .
-
3. Monotonicity: if
$p\in \operatorname {gS}(N)$ does not fork over M and
$M\leq _{\mathbf {K}_{\mathbf {1}}}M'\leq _{\mathbf {K}_{\mathbf {1}}} N$ for some
$M'\in K_1$ , then
$p\restriction M'$ does not fork over M while p itself does not fork over
$M'$ .
-
4. Existence: if
$M\in K_1$ and
$p\in \operatorname {gS}(M)$ , then p does not fork over M.
-
5. Extension: if
$M\leq _{\mathbf {K}_{\mathbf {1}}}N\leq _{\mathbf {K}_{\mathbf {1}}}N'$ and
$p\in \operatorname {gS}(N)$ does not fork over M, then there is
$q\in \operatorname {gS}(N')$ such that
$q\supseteq p$ and q does not fork over M.
-
6. Uniqueness: if
$p,q\in \operatorname {gS}(N)$ do not fork over M and
$p\restriction M=q\restriction M$ , then
$p=q$ .
-
7. Transitivity: if
$M_0\leq _{\mathbf {K}_{\mathbf {1}}}M_1\leq _{\mathbf {K}_{\mathbf {1}}}M_2$ ,
$p\in \operatorname {gS}(M_2)$ does not fork over
$M_1$ ,
$p\restriction M_1$ does not fork over
$M_0$ , then p does not fork over
$M_0$ .
-
8. Local character
${\aleph _0}$ : if
$\delta $ is an ordinal of cofinality
$\geq {\aleph _0}$ ,
$\langle M_i:i\leq \delta \rangle $ is
$\leq _{\mathbf {K}_{\mathbf {1}}}$ -increasing and continuous, then there is
$i<\delta $ such that p does not fork over
$M_i$ .
-
9. Continuity: Let
$\delta $ is a limit ordinal and
$\langle M_i:i\leq \delta \rangle $ be
$\leq _{\mathbf {K}_{\mathbf {1}}}$ -increasing and continuous. If for all
$1\leq i<\delta $ ,
$p_i\in \operatorname {gS}(M_i)$ does not fork over
$M_0$ and
$p_{i+1}\supseteq p_i$ , then
$p_\delta $ does not fork over
$M_0$ .
-
10. Symmetry: let
$M\leq _{\mathbf {K}_{\mathbf {1}}}N$ ,
$b\in |N|$ ,
$\operatorname {gtp}(b/M)$ do not fork over M,
$\operatorname {gtp}(a/N)$ do not fork over M. There is
$N_a\geq _{\mathbf {K}_{\mathbf {1}}}M$ such that
$\operatorname {gtp}(b/N_a)$ do not fork over M.
If the above holds for a
$(\geq \mu )$
-skeleton
$\mathbf {K}_{\mathbf {1}}$
, then we say the nonforking relation is a good
$(\geq \mu )$
-frame over the skeleton
$\mathbf {K}_{\mathbf {1}}$
. If
$\mathbf {K}_{\mathbf {1}}$
is itself an AEC (in
$\mu $
), then we omit “skeleton.” Let
$\alpha <\mu ^+$
be regular. We say a nonforking relation has local character
$\alpha $
if we replace “
${\aleph _0}$
” in item (8) by
$\alpha $
.
Remark 2.6.
-
1. In this paper,
$\mathbf {K}_{\mathbf {1}}$ will be the
$(\mu ,\geq \alpha )$ -limit models for some
$\alpha <\mu ^+$ , with
$\leq _{\mathbf {K}_{\mathbf {1}}}=\leq _u$ (the latter is in
$\mathbf {K})$ .
-
2. In Fact 7.20, we will draw results of a good frame over longer types, where we allow the types in the above definition to be of arbitrary length. Extension property will have an extra clause that allows extension of a shorter type to a longer one that still does not fork over the same base.
-
3. Some of the properties of a good frame imply or simply one another. Instead of using a minimalistic formulation (for example, in [Reference Vasey37, Definition 17.1]), we keep all the properties because sometimes it is easier to deduce a certain property first.
3 Properties of nonsplitting
Let
$p\in \operatorname {gS}(N)$
,
$f:N\rightarrow N'$
, we write
$f(p):=\operatorname {gtp}(f^+(d)/f(N))$
where
$f^+$
extends f to include some
$d\vDash p$
in its domain.
Proposition 3.1. Such
$f^+$
exists by
$AP$
and
$f(p)$
is independent of the choice of
$f^+$
.
Proof Pick
$a\in N_1\geq $
realizing p, use
$AP$
to obtain
$f_1^+:a\mapsto c$
extending f (enlarge
$N_1$
if necessary so that
$f_1^+(N_1)$
contains
$f(N)$
).

Suppose
$b\in N_2$
realizes p and there is
$f_2^+:b\mapsto d$
extending f. Extend
$N_2$
so that
$f_2^+$
is an isomorphism. We need to find
$h:d\mapsto c$
which fixes
$f(N)$
. Since
$a,b\vDash p$
, by
$AP$
there is
$N_3\ni b$
and
$g:N_1\xrightarrow [N]{}N_3$
that maps a to b. Extend g to an isomorphism
$N_1'\cong _N N_3\geq N_2$
. By
$AP$
again, obtain
$f_1^{++}$
of domain
$N_1'$
extending
$f_1^+$
. Therefore,
$d\in f(N_2^+)$
and
$f_1^{++}\circ g^{-1}\circ \operatorname {id}_{N_2}\circ (f_2^+)^{-1}(d)=c$
. Hence we can take
$h:= f_1^{++}\circ g^{-1}\circ \operatorname {id}_{N_2}\circ (f_2^+)^{-1}:f_2^+(N_2)\xrightarrow [f(N)]{}f_1^{++}(N_1') $
.
Definition 3.2. Let
$M,N\in K$
,
$p\in \operatorname {gS}(N)$
.
$p \ \mu $
-splits over M if there exists
$N_1,N_2$
of size
$\mu $
such that
$M\leq N_1,N_2\leq N$
and
$f:N_1\xrightarrow [M]{}N_2$
such that
$f(p)\restriction N_2\neq p\restriction N_2$
.
Proposition 3.3 (Monotonicity of nonsplitting).
Let
$M,N\in K_{\mu }$
,
$p\in \operatorname {gS}(N)$
do not
$\mu $
-split over M. For any
$M_1,N_1$
with
$M\leq M_1\leq N_1\leq N$
, we have
$p\restriction N_1$
does not
$\mu $
-split over
$M_1$
.
Proposition 3.4. Let
$M,N\in K$
,
$M\in K_{\mu }$
and
$p\in \operatorname {gS}(N)$
.
$p \ 5\mu $
-splits over M iff
$p \ (\geq \mu )$
-splits over M (the witnesses
$N_1,N_2$
can be in
$K_{\geq \mu }$
).
Proof We sketch the backward direction: pick
$N_1,N_2\in K_{\geq \mu }$
witnessing
$p \ (\geq \mu )$
-splits over M. By
$\mu $
-tameness and Löwenheim–Skolem axiom, we may assume
$N_1,N_2\in K_{\mu }$
.
Definition 3.5. Let
$\chi $
be a regular cardinal.
-
1. A chain
$\langle M_i:i\leq \delta \rangle $ is u-increasing if
$M_{i+1}>_uM_i$ for all
$i<\delta $ .
-
2.
$\mathbf {K}$ satisfies continuity of
$\mu $ -nonsplitting if for any limit ordinal
$\delta $ ,
$\langle M_i:i\leq \delta \rangle \subseteq K_{\mu }$ u-increasing and continuous,
$p\in \operatorname {gS}(M_\delta )$ ,
$$ \begin{align*} p\restriction M_i \text{ does not } \mu\text{-split over } M_0 \text{ for } i < \delta \Rightarrow p \text{ does not } \mu\text{-split over } M_0. \end{align*} $$
-
3.
$\mathbf {K}$ has
$\chi $ -weak local character of
$\mu $ -nonsplitting if for any limit ordinal
$\delta \geq \chi $ ,
$\langle M_i:i\leq \delta \rangle \subseteq K_{\mu }$ u-increasing and continuous,
$p\in \operatorname {gS}(M_\delta )$ , there is
$i<\delta $ such that
$p\restriction M_{i+1}$ does not
$\mu $ -split over
$M_i$ .
-
4.
$\mathbf {K}$ has
$\chi $ -local character of
$\mu $ -nonsplitting if the conclusion in (3) becomes: p does not
$\mu $ -split over
$M_i$ .
We call any
$\delta $
that satisfies (3) or (4) a (weak) local character cardinal.
Remark 3.6. When defining the continuity of nonsplitting, we can weaken the statement by removing the assumption that p exists and replacing
$p\restriction M_i$
by
$p_i$
increasing. This is because we can use [Reference Boney4, Proposition 5.2] to recover p. In details, we can use the weaker version of continuity and weak uniqueness (Proposition 3.12) to argue that the
$p_i$
’s form a coherent sequence. p can be defined as the direct limit of the
$p_i$
’s.
The following lemma connects the three properties of
$\mu $
-nonsplitting:
Lemma 3.7 [Reference Boney, Grossberg, VanDieren and Vasey9, Lemma 11(1)].
If
$\mu $
is regular,
$\mathbf {K}$
satisfies continuity of
$\mu $
-nonsplitting and has
$\chi $
-weak local character of
$\mu $
-nonsplitting, then it has
$\chi $
-local character of
$\mu $
-nonsplitting.
Proof Let
$\delta $
be a limit ordinal of cofinality
$\geq \chi $
,
$\langle M_i:i\leq \delta \rangle $
u-increasing and continuous. Suppose
$p\in \operatorname {gS}(M_\delta )$
splits over
$M_i$
for all
$i<\delta $
. Define
$i_0:=0$
. By
$\delta $
regular and continuity of
$\mu $
-nonsplitting, build an increasing and continuous sequence of indices
$\langle i_k:k<\delta \rangle $
such that
$p\restriction M_{i_{k+1}} \ \mu $
-splits over
$M_{i_k}$
. Notice that
$M_{i_{k+1}}>_uM_{i_k}$
. Then applying
$\chi $
-weak local character to
$\langle M_{i_k}:k<\delta \rangle $
yields a contradiction.
From stability (even without continuity of nonsplitting), it is always possible to obtain weak local character of nonsplitting. Shelah sketched the proof and alluded to the first-order analog, so we give details here.
Lemma 3.8 [Reference Shelah23, Claim 3.3(2)].
If
$\mathbf {K}$
is stable in
$\mu $
(which is in Assumption 2.1), then for some
$\chi \leq \mu $
, it has weak
$\chi $
-local character of
$\mu $
-nonsplitting.
Proof Pick
$\chi \leq \mu $
minimum such that
$2^{\chi }>\mu $
. Suppose we have
$\langle M_i:i\leq \chi \rangle $
u-increasing and continuous and
$d\vDash p\in \operatorname {gS}(M_\chi )$
such that for all
$i<\chi $
,
$p\restriction M_{i+1} \ \mu $
-splits over
$p\restriction M_i$
. Then for
$i<\chi $
, we have
$N_i^1$
and
$N_i^2$
of size
$\mu $
,
$M_i\leq N_i^1,N_i^2\leq M_{i+1}$
,
$f_i:N_i^1\cong _{M_i}N_i^2$
and
$f_i(p)\restriction N_i^2\neq p\restriction N_i^2$
. We build
$\langle M_i':i\leq \chi \rangle $
and
$\langle h_\eta :M_{l(\eta )}\xrightarrow [M_0]{}M_{l(\eta )}'\,|\,\eta \in 2^{\leq \chi }\rangle $
both increasing and continuous with the following requirements:
-
1.
$h_{\langle \rangle }:=\operatorname {id}_{M_0}$ and
$M_0':= M_0$ .
-
2. For
$\eta \in 2^{<\chi }$ ,
$h_{\eta ^\frown 0}\restriction N_{l(\eta )}^2=h_{\eta ^\frown 1}\restriction N_{l(\eta )}^2$ .

We specify the successor step: suppose
$l(\nu )=i$
and
$h_\nu $
has been constructed. By
$AP$
, obtain:
-
1.
$h:N_i^2\rightarrow M^*\geq M_i'$ with
$h\supseteq h_\nu $ .
-
2.
$h_{\nu ^\frown 0}:M_{i+1}\rightarrow M^{**}\geq M^*$ with
$h_{\nu ^\frown 0}\supseteq h$ .
-
3.
$g_0:M_{i+1}\rightarrow M_{hf_i}\geq M^*$ with
$g_0\supseteq h\circ f_i$ .
-
4.
$g_1:M_{hf_i}\rightarrow M_{i+1}'\geq M^{**}$ with
$g_1\circ g_0=h_{\nu ^\frown 0}$ .
Define
$h_{\nu ^\frown 1}:= g_1\circ g_0:M_{i+1}\rightarrow M_{i+1}'$
. By diagram chasing,
$h_{\nu ^\frown 1}\restriction M_i=g_1\circ g_0\restriction M_i=g_1\circ h\circ f_i\restriction M_i=g_1\circ h\restriction M_i=h\restriction M_i=h_\nu \restriction M_i$
. On the other hand,
$h_{\nu ^\frown 0}\restriction M_i=h\restriction M_i=h_\nu \restriction M_i$
. Therefore the maps are increasing. Now
$h_{\nu ^\frown 1}\restriction N_i^2=g_1\circ g_0\restriction N_i^2=h_{\nu ^\frown 0}\restriction N_i^2$
by item (4) in our construction.
For
$\eta \in 2^{\chi }$
, extend
$h_\eta $
so that its range includes
$M_\chi '$
and its domain includes d. We show that
$\{\operatorname {gtp}(h_\eta (d)/M_{\chi }'):\eta \in 2^{\chi }\}$
are pairwise distinct. For any
$\eta \neq \nu \in 2^{\chi }$
, pick the minimum
$i<\chi $
such that
$\eta [i]\neq \nu [i]$
. Without loss of generality, assume
$\eta [i]=0$
,
$\nu [i]=1$
. Using the diagram above (see the comment before Proposition 3.1),

This contradicts the stability in
$\mu $
.
Proposition 3.9. If
$\mu $
is regular, then for some
$\chi \leq \mu $
,
$\mathbf {K}$
has the
$\chi $
-local character of
$\mu $
-nonsplitting.
Proof By Lemma 3.8 and uniqueness of limit models of the same cofinality,
$\mathbf {K}$
has
$\mu $
-weak local character of
$\mu $
-nonsplitting. By Lemma 3.7 (together with continuity of
$\mu $
-nonsplitting in Assumption 2.1),
$\mathbf {K}$
has
$\mu $
-local character of
$\mu $
-nonsplitting. Hence
$\chi $
exists and
$\chi \leq \mu $
.
From now on, we fix the following.
Definition 3.10.
$\chi $
is the minimum local character cardinal of
$\mu $
-nonsplitting.
$\chi \leq \mu $
if either
$\mu $
is regular (by the previous proposition), or
$\mu $
is greater than some regular stability cardinal
$\xi $
where
$\mathbf {K}$
has continuity of
$\xi $
-nonsplitting and is
$\xi $
-tame (by Lemma 6.7).
Remark 3.11. Without continuity of nonsplitting, it is not clear whether there can be gaps between the local character cardinals: Definition 3.5(4) might hold for
$\delta =\aleph _0$
and
$\delta =\aleph _2$
but not
$\delta =\aleph _1$
. In that case defining
$\chi $
as the minimum local character cardinal might not be useful. Similar obstacles form when we only know a particular
$\lambda $
is a local character cardinal but not necessary those above
$\lambda $
.
Meanwhile, weak local character cardinals close upwards and we can eliminate the above situation by assuming continuity of nonsplitting: if we know
$\chi $
is the minimum local character cardinal, then it is also a weak local character cardinal, so are all regular cardinals between
$[\chi ,\mu ^+)$
. By the proof of Lemma 3.7, the regular cardinals between
$[\chi ,\mu ^+)$
are all local character cardinals.
We now state the existence, extension, weak uniqueness, and weak transitivity properties of
$\mu $
-nonsplitting. The original proof for weak uniqueness assumes
$\Vert M \Vert =\mu $
but it is not necessary; while that for extension and for weak transitivity assume all models are in
$K_{\mu }$
; but under tameness we can just require
$\Vert M \Vert =\Vert N \Vert $
.
Proposition 3.12. Let
$M_0<_u M\leq N$
where
$\Vert M_0 \Vert =\mu $
.
-
1. [Reference Shelah23, Claim 3.3(1)] (Existence) If
$p\in \operatorname {gS}(N)$ , there is
$N_0\leq N$ of size
$\mu $ such that p does not
$\mu $ -split over
$N_0$ .
-
2. [Reference Grossberg and VanDieren16, Theorem 6.2] (Weak uniqueness) If
$p,q\in \operatorname {gS}(N)$ both do not
$\mu $ -split over
$M_0$ , and
$p\restriction M=q\restriction M$ , then
$p=q$ .
-
3. [Reference Grossberg and VanDieren16, Theorem 6.1] (Extension) Suppose
$\Vert M \Vert =\Vert N \Vert $ . For any
$p\in \operatorname {gS}(M)$ that does not
$\mu $ -split over
$M_0$ , there is
$q\in \operatorname {gS}(N)$ extending p such that q does not
$\mu $ -split over
$M_0$ .
-
4. [Reference Vasey33, Proposition 3.7] (Weak transitivity) Suppose
$\Vert M \Vert =\Vert N \Vert $ . Let
$M^*\leq M_0$ and
$p\in \operatorname {gS}(N)$ . If p does not
$\mu $ -split over
$M_0$ while
$p\restriction M$ does not
$\mu $ -split over
$M^*$ , then p does not
$\mu $ -split over
$M^*$ .
Proof
-
1. We skip the proof, which has the same spirit as that of Lemma 3.8.
-
2. By stability in
$\mu $ , we may assume that
$\Vert M \Vert =\mu $ . Suppose
$p\neq q$ , by tameness in
$\mu $ we may find
$M'\in K_{\mu }$ such that
$M\leq M'\leq N$ and
$p\restriction M'\neq q\restriction M'$ . By
$M_0<_uM$ and
$M_0<N$ , we can find
$f:M'\xrightarrow [M_0]{}M$ . Using nonsplitting twice, we have
$p\restriction f(M')=f(p)$ and
$q\restriction M'=f(q)$ . But
$f(M')\leq M$ implies
$p\restriction f(M')=q\restriction f(M')$ . Hence
$f(p)=f(q)$ and
$p=q$ .
-
3. By universality of M, find
$f:N\xrightarrow [M_0]{}M$ . We can set
$q:= f^{-1}(p\restriction f(N))$ .
-
4. Let
$q:= p\restriction M$ . By extension, obtain
$q'\supseteq q$ in
$\operatorname {gS}(N)$ such that
$q'$ does not
$\mu $ -split over
$M^*$ . Now
$p\restriction M=q\restriction M=q'\restriction M$ and both
$p,q'$ do not
$\mu $ -split over
$M_0$ (for
$q'$ use monotonicity, see Proposition 3.3). By weak uniqueness,
$p=q'$ and the latter does not
$\mu $ -split over
$M^*$ .
Transitivity does not hold in general for
$\mu $
-nonsplitting. The following example is sketched in [Reference Baldwin2, Example 19.3].
Example 3.13. Let T be the first-order theory of a single equivalence relation E with infinitely many equivalence classes and each class is infinite. Let
$M\leq N$
where N contains (representatives of) two more classes than M. Let d be an element. Then
$\operatorname {\mathrm {tp}}(d/N)$
splits over M iff
$dEa$
for some element
$a\in N$
but
$\neg dEb$
for any
$b\in M$
. Meanwhile, suppose
$M_0\leq M$
both of size
$\mu $
, then
$M_0<_u M$
iff M contains
$\mu $
-many new classes and each class extends
$\mu $
many elements. Now require
$M_0<_u M$
while N contains only an extra class than M, say witnessed by d, then
$\operatorname {\mathrm {tp}}(d/N)$
cannot split over M. Also
$\operatorname {\mathrm {tp}}(d/M)$
does not split over
$M_0$
because d is not equivalent to any elements from M. Finally
$\operatorname {\mathrm {tp}}(d/N)$
splits over
$M_0$
because it contains two more classes than
$M_0$
(one must be from M).
The same argument does not work if also
$M<_u N$
because N would contain two more classes than M and they will witness
$\operatorname {\mathrm {tp}}(d/N)$
splits over M. Baldwin originally assigned it as [Reference Baldwin2, Exercise 12.9] but later [Reference Baldwin3] retracted the claim.
Question 3.14. When models are ordered by
$\leq _u$
:
-
1. Does uniqueness of
$\mu $ -nonsplitting hold? Namely, let
$M<_uN$ both in
$K_{\mu }$ ,
$p,q\in \operatorname {gS}(N)$ both do not
$\mu $ -split over M,
$p\restriction M=q\restriction M$ , then
$p=q$ .
-
2. Does transitivity of
$\mu $ -nonsplitting hold? Namely, let
$M_0<_uM<_uN$ all in
$K_{\mu }$ ,
$p\in \operatorname {gS}(N)$ does not
$\mu $ -split over M and
$p\restriction M$ does not
$\mu $ -split over
$M_0$ , then p does not
$\mu $ -split over
$M_0$ .
In Assumption 2.1, we assumed continuity of
$\mu $
-nonsplitting. One way to obtain it is to assume superstability which is stronger. Another way is to assume
$\omega $
-type locality.
Definition 3.15.
-
1. [Reference Grossberg and Zhang14, Definition 7.12] Let
$\lambda \geq \operatorname {LS}(\mathbf {K})$ ,
$\mathbf {K}$ is
$\lambda $ -superstable if it is stable in
$\lambda $ and has
${\aleph _0}$ -local character of
$\lambda $ -nonsplitting.
-
2. [Reference Baldwin2, Definition 11.4] Types in
$\mathbf {K}$ are
$\omega $ -local if: for any limit ordinal
$\alpha $ ,
$\langle M_i:i\leq \alpha \rangle $ increasing and continuous,
$p,q\in \operatorname {gS}(M)$ and
$p\restriction M_i=q\restriction M_i$ for all
$i<\alpha $ , then
$p=q$ .
Proposition 3.16. Let
$\mathbf {K}$
satisfy Assumption 2.1 except for the continuity of
$\mu $
-nonsplitting. It will satisfy the continuity of
$\mu $
-nonsplitting if either:
-
1.
$\mathbf {K}$ is
$\mu $ -superstable; or
-
2. types in
$\mathbf {K}$ are
$\omega $ -local.
Proof For (1), it suffices to prove that for any regular
$\lambda \geq {\aleph _0}$
,
$\lambda $
-local character implies continuity of
$\mu $
-nonsplitting over chains of cofinality
$\geq \lambda $
. Let
$\langle M_i:i\leq \lambda \rangle $
be u-increasing and continuous. Suppose
$p\in \operatorname {gS}(M_\lambda )$
satisfies
$p\restriction M_i$
does not
$\mu $
-split over
$M_0$
for all
$i<\lambda $
. By
$\lambda $
-local character, p does not
$\mu $
-split over some
$M_i$
. If
$i=0$
we are done. Otherwise, we have
$M_0<_uM_i<_uM_{i+1}<_uM_\lambda $
. By assumption,
$p\restriction M_{i+1}$
does not
$\mu $
-split over
$M_0$
. By weak transitivity (Proposition 3.12), p does not
$\mu $
-split over
$M_0$
as desired.
For (2), let
$\langle M_i:i\leq \lambda \rangle $
and p as above. By assumption
$p\restriction M_1$
does not
$\mu $
-split over
$M_0$
and
$M_1>_uM_0$
. By extension (Proposition 3.12), there is
$q\supseteq p\restriction M_1$
in
$\operatorname {gS}(M_\lambda )$
such that q does not
$\mu $
-split over
$M_0$
. By monotonicity, for
$2\leq i<\lambda $
,
$q\restriction M_i$
does not
$\mu $
-split over
$M_0$
. Now
$(q\restriction M_i)\restriction M_1=p\restriction M_1=(p\restriction M_i)\restriction M_1$
, we can use weak uniqueness (Proposition 3.12) to inductively show that
$q\restriction M_i=p\restriction M_i$
for all
$i<\lambda $
. By
$\omega $
-locality,
$p=q$
and the latter does not
$\mu $
-split over
$M_0$
as desired.
Once we have continuity of
$\mu $
-nonsplitting in
$K_{\mu }$
, it automatically works for
$K_{\geq \mu }$
.
Proposition 3.17. Let
$\delta $
be a limit ordinal,
$\langle M_i:i\leq \delta \rangle \subseteq K_{\geq \mu }$
be u-increasing and continuous,
$p\in \operatorname {gS}(M_\delta )$
. If for all
$i<\delta $
,
$p\restriction M_i$
does not
$\mu $
-split over
$M_0$
, then p also does not
$\mu $
-split over
$M_0$
.
Proof The statement is vacuous when
$M_0\in K_{>\mu }$
so we assume
$M_0\in K_{\mu }$
. By cofinality argument we may also assume
$\operatorname {cf}(\delta )\leq \mu $
. Suppose
$p \ \mu $
-splits over
$M_0$
and pick witnesses
$N^a$
and
$N^b$
of size
$\mu $
. Using stability, define another u-increasing and continuous chain
$\langle N_i:i\leq \delta \rangle \subseteq K_{\mu }$
such that:
-
1. For
$i\leq \delta $ ,
$N_i\leq M_i$ .
-
2.
$N_\delta $ contains
$N^a$ and
$N^b$ .
-
3.
$N_0:= M_0$ .
-
4. For
$i\leq \delta $ ,
$|N_i|\supseteq |M_i|\cap (|N^a|\cup |N^b|)$ .
By assumption each
$p\restriction M_i$
does not
$\mu $
-split over
$M_0$
, so by monotonicity
$p\restriction N_i$
does not
$\mu $
-split over
$N_0=M_0$
. By continuity of
$\mu $
-nonsplitting,
$p\restriction N_\delta $
does not
$\mu $
-split over
$N_0$
, contradicting item (2) above.
4 Good frame over
$(\geq \chi )$
-limit models except symmetry
As seen in Proposition 3.12,
$\mu $
-nonsplitting only satisfies weak transitivity but not transitivity, which is a key property of a good frame. We will adapt [Reference Vasey33, Definitions 3.8 and 4.2] to define nonforking from nonsplitting to solve this problem.
Definition 4.1. Let
$M\leq N$
in
$K_{\geq \mu }$
and
$p\in \operatorname {gS}(N)$
.
-
1. p (explicitly) does not
$\mu $ -fork over
$(M_0,M)$ if
$M_0\in K_{\mu }$ ,
$M_0<_uM$ and p does not
$\mu $ -split over
$M_0$ .
-
2. p does not
$\mu $ -fork over M if there exists
$M_0$ satisfying (1).
We call
$M_0$
the witness to
$\mu $
-nonforking over M.
The main difficulty of the above definition is that different
$\mu $
-nonforkings over M may have different witnesses. For extension, the original approach in [Reference Vasey33] was to work in
$\mu ^+$
-saturated models. Later [Reference VanDieren and Vasey31, Proposition 5.1] replaced it by superstability in an interval, which works for
$K_{\geq \mu }$
. We weaken the assumption to stability in an interval and continuity of
$\mu $
-nonsplitting, and use a direct limit argument similar to that of [Reference Boney4, Theorem 5.3].
Proposition 4.2 (Extension).
Let
$M\leq N\leq N'$
in
$K_{\geq \mu }$
. If
$\mathbf {K}$
is stable in
$[\Vert N \Vert ,\Vert N' \Vert ]$
and
$p\in \operatorname {gS}(N)$
does not
$\mu $
-fork over M, then there is
$q\supseteq p$
in
$\operatorname {gS}(N')$
such that q does not
$\mu $
-fork over M.
Proof Since p does not
$\mu $
-fork over M, we can find witness
$M_0\in K_{\mu }$
such that
$M_0<_uM$
and p does not
$\mu $
-split over
$M_0$
. If
$\Vert N \Vert =\Vert N' \Vert $
, we can use extension of nonsplitting (Proposition 3.12) to obtain (the unique)
$q\in \operatorname {gS}(N')$
extending p which does not
$\mu $
-split over
$M_0$
. By definition q does not
$\mu $
-fork over M.
If
$\Vert N \Vert {\kern-1.5pt}<{\kern-1.5pt}\Vert N' \Vert $
, first we assume
$N<_uN'$
and resolve
$N'{\kern-1.5pt}={\kern-1.5pt}\bigcup \{N_i:i{\kern-1.5pt}\leq{\kern-1.5pt} \alpha +1\}$
u-increasing and continuous where
$N_0=N$
,
$\Vert N_{\alpha } \Vert =\Vert N' \Vert $
,
$N_{\alpha +1}=N'$
. The construction is possible by stability in
$[\Vert N \Vert ,\Vert N' \Vert ]$
. We will define a coherent sequence
$\langle p_i:i\leq \alpha \rangle $
such that
$p_i$
is a nonsplitting extension of p in
$\operatorname {gS}(N_i)$
. The first paragraph gives the successor step. For limit step
$\delta \leq \alpha $
, we take the direct limit to obtain an extension
$p_\delta $
of
$\langle p_i:i<\delta \rangle $
. Since all previous
$p_i$
does not
$\mu $
-split over
$M_0$
, by Proposition 3.17,
$p_\delta $
also does not
$\mu $
-split over
$M_0$
. After the construction has finished, we obtain
$q:= p_{\alpha }$
a nonsplitting extension of p in
$\operatorname {gS}(N')$
. Since
$M_0<_uM\leq N'$
, we still have q does not
$\mu $
-fork over M.
In the general case where
$N\leq N'$
, extend
$N'<_u N"$
with
$\Vert N" \Vert =\Vert N' \Vert $
. Then we can extend p to a nonforking
$q"\in \operatorname {gS}(N")$
and use monotonicity to obtain the desired q.
Corollary 4.3. Let
$M_0<_u M\leq N'$
with
$M_0\in K_{\mu }$
. If
$\mathbf {K}$
is stable in
$[\Vert M \Vert ,\Vert N' \Vert ]$
and
$p\in \operatorname {gS}(M)$
does not