Model theory for a compact cardinal
Abstract.
We would like to develop classification theory for , a complete theory in when is a compact cardinal. We already have bare bones stability theory and it seemed we can go no further. Dealing with ultrapowers (and ultraproducts) naturally we restrict ourselves to “ a complete ultrafilter on , probably regular”. The basic theorems of model theory work and can be generalized (like Łos theorem), but can we generalize deeper parts of model theory?
In particular, can we generalize stability enough to generalize [Sh:c, Ch.VI]? Let us concentrate on saturation in the local sense (types consisting of instances of one formula). We prove that at least we can characterize the ’s (of cardinality for simplicity) which are minimal for appropriate cardinal in each of the following two senses. One is generalizing Keisler order which measures how saturated are ultrapowers. Another ask: Is there an theory of cardinality such that for every model of of cardinality , the reduct of is saturated. Moreover, the two versions of stable used in the characterization are different. Further we succeed to connect our investigation with the logic introduced in [Sh:797] proving it satisfies several parallel of classical theorems on first order logic, strengthening the thesis that it is a natural logic. In particular, two models are equivalent iff for some sequence of complete ultrafilters, the iterated ultrapowers by it of those two models are isomorphic.
Key words and phrases:
model theory, infinitary logics, compact cardinals, ultrapowers, ultra limits, stability, saturated models, classification theory, isomorphic ultralimits2010 Mathematics Subject Classification:
Primary: 03C45; Secondary: 03C30, 03C55Anotated Content
§0 Introduction, pg. ‣ 0. Introduction
§(0A) Background and results, (label v), pg.0(A)
§(0B) Preliminaries, (label w,x), pg. 0(B)
§1 Basic Stability, (label y), pg. 1

[We try to sort out several natural generalizations of “ is stable” and give examples to show they are different.]
§2 Saturation of ultrapowers, (label a), pg. 2

[We characterize the ’s which are minimal in several senses, where is a complete theory with no model of cardinality . First, there is of cardinality such that for every is locally saturated. Second, the ultrapower is locally saturated for every model of and complete regular ultrafilter on . We also give an example to show that those two properties are not equivalent. Above, “locally” means types involving instances of just one formula . Omitting this (but still we restrict ourselves to the case ) we get a parallel characterization.]
§3 On , the logic interpolating and , (label d), pg.3

[We characterize equivalence of by having isomorphic ultralimits by a sequence of length of complete ultrafilters. This logic, , is from [Sh:797] except that here we restrict ourselves to is a compact cardinal. We also define spec ial model of complete theory , for strong limit of cofinality and prove existence and uniqueness.]
0. Introduction
0(a). Background and results
In Winter 2012, I have tried to explain in a model theory class, a position I held for long: model theory can extensively deal with classes and a.e.c. however while we can generalize basic model theory to classes, , see [Dec 85], we cannot do considerably more. The latter logics are known to have downward LST theorems and various connections to large cardinals and consistency results, and only rudimentary stability theory (see [Sh:300a]). Note that, e.g. if there is such that iff is isomorphic to for some ordinal such that ; hence if then has a model of cardinality and every model of of cardinality is isomorphic to . It folows that, e.g. for every second order sentence , there is which is categorical in the cardinal iff and ; so the categoricity spectrum is not so nice. Such views have been quite general  see Väänänen’s book [Vää11].
This work is dedicated to starting to try to disprove this for the logic for a compact cardinal. Still Łos theorem on ultraproducts was known to generalize so let us review the background in this direction.
In the sixties, ultraproducts were very central in model theory. Recall Kochen uses iteration on taking ultrapowers (on a well ordered index set) to characterize elementary equivalence. Gaifman [Gai74] uses ultrapowers on complete ultrafilters iterated along linear ordered index set. Keisler [Kei63] uses general , see below, Definition 0.19(1) for . Keisler, assuming an instance of GCH characterizes elementary equivalence by proving; for any two models (of vocabulary of cardinality and) of cardinality , they have isomorphic ultrapowers, even for some ultrafilter on iff are elementarily equivalent. Shelah [Sh:13] proves this in ZFC (but the ultrafilter is on ).
HodgesShelah [HoSh:109] is closer to the present work and see there on earlier works, it deals with isomorphic ultrapowers (and isomorphic reduced powers) for the complete ultrafilter (and filter) case, but note that having isomorphic ultrapowers by complete ultrafilters is not an equivalence relation. In particular assume is a compact cardinal and little more (we can get it by forcing over a universe with a supercompact cardinal and a class of measurable cardinals). Then two models have isomorphic ultrapowers for some complete ultrafilter iff in all relevant games the isomorphism player does not lose. Those relevant games are of length and deal with the reducts to a subvocabulary of cardinality .
The characterization [HoSh:109] of having isomorphic ultrapowers by complete ultrafilters, necessarily is not so “nice” because this relation is not an equivalence relation. Hence having isomorphic ultrapowers is not connected to having the same theory in some logic. But [Sh:797] suggests a logic with some good properties (like well ordering not characterizable, interpolation) maximal under such properties. We may wonder, do we have a characterization of models being equivalent?
In §3 we characterize equivalence of models by having isomorphic iterated ultrapowers of length . We then prove some generalizations of classical model theoretic theorems, like the existence and uniqueness of special models in when is strong limit of cofinality . All this seems to strengthen the thesis of [Sh:797] that is a natural logic.
Let us turn to another direction, now for the logic itself. We are mainly interested in generalizations of [Sh:c, Ch.VI], on Keisler order and saturation of ultrapowers, see history there and recent works with Malliaris ([MiSh:996], [MiSh:997], [MiSh:998]).
In particular it is proved there that:
Theorem 0.1.
Assume is a complete first order countable theory.
1) The following conditions are equivalent:

if is a regular ultrafilter on and is a model of then is saturated

there is a first order theory such that: is locally saturated (i.e. for types for any

is stable without the

like but .
2) The following conditions are equivalent:

if , where is a limit ordinal and for each is a regular ultrafilter on a cardinal then for any (equivalently some) model of is saturated where is ultralimit of by (i.e. is increasing continuous,

there is a first order theory such that: is saturated

is superstable without the

like (b) but .
3) The following conditions are equivalent:

like (b) but

is stable without the
The main topic of §1, §2 is generalizing such results replacing first order logic with , so “countable” is replaced by “of cardinality ”. More specifically, one aim is to characterize the complete theories such that for some theory extending , for every model of , the reduct of the model is (locally) saturated, such will be called (locally) minimal.
Note that of Theorem 0.1 is close to Keisler order (on first order complete ’s) which Keisler [Kei67] introduced and started to investigate; it is a characterization of the minimal ones. There is much more to be said on this order.
Parallelly, of Theorem 0.1 is related to the partial orders really investigated in [Sh:c, Ch.VI] but introduced in [Sh:500], see more on them in DzamonjaShelah [DjSh:692], ShelahUsvyatsov [ShUs:844] and lately MalliarisShelah (citeMiSh:1051); related is BaldwinGrossbergShelah [BGSh:570].
But in our context trying to generalize Theorem 0.1, i.e. the minimal case was hard enough. In fact, there is a problem already in generalizing stable. In §1 we suggest some reasonable definitions and try to map their relations. Note that those generalizations are really very different in the present context (though equivalent for the first order case). Some are satisfied by some “unstable” ’s categorical in all relevant ’s; some “unstable” versions imply maximal number of models up to isomorphism in relevant cardinalities, and some “stable ’s” have an intermediate behaviour (i.e. ).
To get sufficient conditions on for having many models we may consider the tree and try to combine it with the identities for (see [Sh:74]) which is a kind of relevant indiscernible, we hope to deal with this in [Sh:F1396].
Originally we were interested in generalizing the characterization of the theories mentioned in Keisler order , where is bigger if for fewer regular ultrafilters on and/or the cardinal is saturated for some (equivalent any) model of .
Earlier version was flawed but we succeed in characterizing the minimal ones, see §2. Later we get also the characterization of the minimal ones, but we use a different version of stable.
Of course, before all this we have to define saturation and local saturation. This is straightforward “unfortunately” two wonderful properties true in the first order case are missing: existence and uniqueness.
The main achievement is in §2: first (in 2.29), a characterization of the (locally) minimal theories as stable with n.c.p. under reasonable definitions (see Definition 2.7). But unlike the first order case, some stable theories (even just theories of one equivalence relation) are maximal. In fact we get two characterizations: one for the local version (dealing with types containing only for one , various ’s) and another for the global one (naturally for theories ). Second (in 2.30), we characterize the minimal as definably stable with the n.c.p.
We may hope this will help us to resolve the categoricity spectrum. It is natural to try to first prove: having long linear orders implies many models. But this is not so  see 1.13; so the situation has a marked difference from the first order case. We hope to continue this in [Sh:F1396].
This work was presented in a lecture in MAMLS meeting, Fall 2012 and in courses in The Hebrew University, Spring 2012 and 2013.
We thank Doron Shafrir for (in late 2013) proofreading, pointing out several problematic claims (subsequently some were withdrawn, some changed, some given a full proof) and rewriting the proof of 2.15(3).
We thank the referee for many helpful remarks.
Discussion 0.2.
1) We may wonder, for a compact cardinal what about theories?
2) Recall the logic from [HoSh:271, §2], that is, given two compact cardinal , a logic is defined and proved to be “nice”, e.g. it is compact for , has interpolation, has downward property down to and the upward property for models of cardinality .
3) On the classical results on see e.g. [Dic85]; on “when for given there are and such that ”, see HodgesShelah [HoSh:109].
4) Recently close works are MalliarisShelah [MiSh:999] which deals with complete ultrafilters (on sets and relevant Boolean algebras) on the way to understanding the amount of saturation of ultrapowers by regular ultrafilters. On reduced power, a work in preparation is [Sh:F1403].
5) Concerning dependent (nonelementary) classes, see also KaplanLaviShelah [KpLaSh:1055].
6) Is the lack of uniqueness of saturation a sign this is a bad choice? It does not seem so to me.
7) If we insist on “union on increasing countable chain” is an extension, we can restrict ourselves to , but what about unions of length ? If we restrict our logic as in for all those we may get close to a.e.c., or get an interesting new logic with EM models (as indicated in [Sh:797], [Sh:893]).
8) Presently, our intention here is to show has a model theory, in particular classification theory. At this point having found significant dissimilarities to the first order case on the one hand, and solving the parallel of serious theorems on the other hand, there is no reason to abandon this direction.
We may wonder
Question 0.3.
Characterize such that is not saturated whenever is a model of a regular complete ultrafilter on .
Question 0.4.
Can we prove nice things on the following logics:

let be : for every large enough we have and if is increasing, a directed partial order then iff . How close is to a.e.c. when is a compact cardinal?

As above but is linearly ordered.
A work in preparation of BoneyShelah intends to deal with it.
0(b). Preliminaries
Hypothesis 0.5.
is a compact uncountable cardinal (of course, we use only restricted versions of this).
Notation 0.6.
1) Let mean: is a formula of is a sequence of variables with no repetitions including the variables occuring freely in and if not said otherwise. We use to denote formulas and or or is if is true or 1 and if is false or 0.
2) For a set , usually of ordinals, let , now may be an ordinal but, e.g. if we may write ; similarly for ; let .
3) denotes a vocabulary, i.e. a set of predicates and function symbols each with places (but in §3 the number of places is finite).
4) denotes a theory in ; usually complete in the vocabulary and with a model of cardinality if not said otherwise.
5) Let be the class of models of .
6) For a model let its vocabulary be .
Notation 0.7.
1) are ordinals .
2) For a linear order let be its completion.
Definition 0.8.
1) Let be the set of complete ultrafilters on , nonprincipal if not said otherwise. Let be the set of complete filters on ; mainly we use regular ones (see below).
2) is called regular when there is a witness which means: for and .
3) For with and which is not complete let and for some we have and let : for some the cardinality of is .
4) Let be the set of regular ; let be the set of regular ; when we may omit .
Note that
Observation 0.9.
If and and is the cardinal then is and is or ; moreover, if is regular then hence .
Notation 0.10.
1) A vocabulary means with if not said otherwise, where is a predicate (or function symbol) from , of course, where is the number of places of .
2) If and then and .
3) : for some and .
4) If then we may omit .
4A) If is the set of quantifier free formulas from , we may write instead of .
Definition 0.11.
1) is the set of formulas of in the vocabulary .
2) For models let means: if and then .
Definition 0.12.
For a set of ordinals, a sequence and models of the same vocabulary and a set of formulas we define a game but when we may write :

a play lasts some finite number of moves not known in advance

in the th move the antagonist chooses

such that

sequence where




in the th move (after the antagonist’s move) the protagonist chooses for

the play ends when the antagonist cannot choose

the protagonist wins a play when :

the set and the th move was done
is a function and even

is a partial onetoone function from into and moreover

it preserves satisfaction of formulas and their negations.
We know (see, e.g. [Dic85])
Fact 0.13.
The models are equivalent iff for every set of atomic formulas and the protagonist wins in the game .
And, of course
Fact 0.14.
For a complete .
1) has amalgamation and the joint embedding property (JEP), that is:

amalgamation: if for then there are such that


for is a embedding of into over , that is, for some model we have and is an isomorphism from onto over ;


JEP: if are equivalent models then there is a model and embedding of into for .
2) Types are well defined, see [Sh:300b], i.e. the orbital type tp and the types as a set of formula are essentially equivalent, that is:
The well known generalization of Łos theorem is:
Theorem 0.15.
If and is a model for and for then iff the set belongs to .
Recall
Fact 0.16.
Assume is not complete and .
1) If and for then there is such that is a sequence of length with the th element being ” for^{1}^{1}1We are identifying elements of with ones of naturally, see 0.22(2). Alternatively, expand by having , so is an individual constant for , so is an expansion of and is the th element of the sequence ”. every .
2) If and is regular and for then there is such that and , (in fact, also the inverse holds).
3) For some function from onto is a normal ultrafilter on .
Proof..
1) Let where . Let be such that , exists by the assumption on . We define by:

.
Now is as required, check.
Definition 0.17.
Assume and is a linear order and and .
1) We say is a middle convergent or strongly convergent in when for every and there is of cardinality or respectively, such that:

if and then .
So in the case of middle convergent ; omitting means for some .
2) We say add strictly “middle/strongly convergent” when we can demand “J.
Definition 0.18.
For a linear order .
1) is its inverse.
2) A cut is a pair such that is an initial segment of and .
3) The cofinality of the cut is the pair of regular cardinals (or 0 or 1) such that .
4) We say is a precut of [of cofinality ] when is a cut of [of cofinality ].
Definition 0.19.
0) We say respects when for some set is an equivalence relation on and and .
1) We say is a (limitultrafilteriteration triple) when :

is a filter on the set

is a family of equivalence relations on

is directed, i.e. if and for then there is refining for every

if then is a complete ultrafilter on where and respects .
1A) Let be a mean that above we weaken (d) to

if then is a complete filter.
2) Omitting “” means , recalling is our fixed compact cardinal.
3) Let mean that:

is a function from onto

if then where and

if and then .
Remark 0.20.
Note that in 0.19(3), if then .
Definition 0.21.
Assume is a l.u.f.t.
1) For a function let . If and then .
2) For a set let is refined by some .
3) For a model let and is refined by some , pedantically (as may be ), ; stands for limit reduced power.
4) If is we may in (3) write .
We now give the generalization of Keisler [Kei63]; HodgeShelah [HoSh:109, Lemma 1,pg.80] is the case .
Theorem 0.22.
1) If is so for then iff .
2) Moreover , pedantically is a elementary embedding of into where .
3) We define similarly when is refined by some , may use this more in end of the proof of 3.2.
Convention 0.23.
1) Abusing a notation in we allow for when .
2) For we can find such that and which means: if then and .
Remark 0.24.
1) Why the “pedantically” in 0.21(3)? Otherwise if is a is not directed, then defining , we have freedom: if , i.e. on and no refines so we have no restrictions.
2) So, e.g. for categoricity we better restrict ourselves to vocabularies such that .
Definition 0.25.
We say is a complete model when for every and there are such that .
Observation 0.26.
1) If is a model of cardinality then there is a complete expansion of so and has cardinality .
2) For models and as above the following conditions are equivalent:

identifying with , for some

there is such that and is isomorphic to over .
3) For a model , if is a directed partial order and and then for some , the model satisfies has a cofinal increasing sequence of length and .
Proof..
Easy, e.g.
3) Let be as in part (1). Note that has Skolem functions and let is a term so and .
Clearly

is satisfiable in .
[Why? Because if has cardinality then the set appears in has cardinality and let ; now for each the set and for has cardinality . Now we choose by induction on such that the assignment for in satisfies , possible because and is directed. So the with the assignment for is a model of , so is (satisfiable indeed.]
Recalling that an individual constant, is realized in some elementary extension of by the assignment . Without loss of generality is the Skolem hull of , so is as required. Now is as required exists by part (3). ∎
Observation 0.27.
1) If is a nontrivial