Strongly dependent theories
Saharon Shelah
我们进一步研究了强相依（一阶完全）理论T的模型类，继续数学。LO/0406440。如果|A| + |T| <= μ，I子集C，|I| >= beth_{|T|^+}（μ），则基数μ ^+的某个J子集I是A上的不可分辨序列。
We further investigate the class of models of a strongly dependent (first order complete) theory T, continuing math.LO/0406440. If |A|+|T|<= mu, I subseteq C, |I| >=beth_{|T|^+}(mu) then some J subseteq I of cardinality mu^+ is an indiscernible sequence over A .
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We apply a general approach for distributions of binary isolating and semi-isolating formulas to the class of strongly minimal theories.
Constructing strongly equivalent nonisomorphic models for unsuperstable theories. Part B
我们研究如何等价的非同构模型的非超稳定理论可以。我们衡量的等价性Eschericht-Fraisse游戏。本文继续[HySh：474]。
Constructing strongly equivalent nonisomorphic models for unsuperstable theories, Part A
我们研究如何等价的非同构模型的非超稳定理论可以有。我们衡量的等价性Eschericht-Fraisse游戏。
Constructing strongly equivalent nonisomorphic models for unsuperstable theories, Part C
本文证明了具有dop的稳定理论T的kappa（T）-饱和模型的一个强非结构定理。
On generically stable types in dependent theories
我们发展了一般稳定类型理论、基于非分叉的独立关系理论和依赖理论背景下的稳定权理论。
We show that for a model complete strongly minimal theory whose pregeometry is flat, the recursive spectrum (SRM($T$)) is either of the form $[0,α)$ for $α\in ω+2$ or $[0,n]\cup\{ω\}$ for $n\in ω$, or $\{ω\}$, or contained in $\{0,1,2\}$. Combined with previous results, this leaves precisely 4 sets for which it is not yet determined whether each is the spectrum of a model complete strongly minimal theory with a flat pregeometry.
We conjecture that for a strongly minimal theory T in a finite signature satisfying the Zilber Trichotomy, there are only three possibilities for the recursive spectrum of T: all countable models of T are recursively presentable; none of them are recursively presentable; or only the zero-dimensional model of T is recursively presentable. We prove this conjecture for disintegrated (formerly, trivial) theories and for modular groups. The conjecture also holds via known results for fields. The conjecture remains open for finite covers of groups and fields.
Let $T$ be a (first order complete) dependent theory, ${\mathfrak{C}}$ a $\barκ$-saturated model of $T$ and $G$ a definable subgroup which is abelian. Among subgroups of bounded index which are the union of $<\barκ$ type definable subsets there is a minimal one, i.e. their intersection has bounded index. In fact, the bound is $\leq 2^{|T|}$. We then deal with 2-dependent theories, a wider class of first order theories.
There is much more known about the family of superstable theories when compared to stable theories. This calls for a search of an analogous "super-dependent" characterization in the context of dependent theories. This problem has been treated in \cite{Sh:783,Sh:863}, where the candidates "Strongly dependent", "Strongly dependent^2" and others were considered. These families generated new families when we are considering intersections with the stable family. Here, continuing \cite[§2, §5E,F,G]{Sh:863}, we deal with several candidates, defined using dividing properties and related ranks of types…
Minimal bounded index subgroup for dependent theories
对于依赖理论T，在C_T中，对每个型可定义群G，指数有界的型可定义子群的交是指数有界的型可定义子群。
Morley sequences in dependent theories
我们的特点nonforking（Morley）序列在依赖理论的Poizat的特殊序列的推广，并表明平均类型的Morley序列是平稳的。我们的特征一般稳定类型的“最终”类型的结构。然后，我们研究的基本性质的“严格的Morley序列”，基于Shelah的严格nonforking的概念。特别是，我们证明了“金引理”这样的序列，和一个弱版本的本地字符。
Dependent measures in independent theories
本文提出Keisler测度的依赖性概念，将NIP理论中一般稳定测度的结果推广到任意理论，并证明依赖测度刻画Glivenko-Cantelli函数类。
Minimal types in super-dependent theories
本文给出了一个可定义在o-极小结构中的理论解释实封闭的场的充要几何条件。证明通过分析有限秩超玫瑰相关理论中的刺极小型。我们证明了这样的理论是协调的荆棘极小型，这样的类型是不稳定的，当且仅当每个非代数扩张的。我们的结论是，一个类型是稳定的，当且仅当它允许在刺最小稳定类型中进行协调。我们还证明了非平凡的刺极小稳定类型扩展了稳定集。
In this short note we show that if we add predicate for a dense complete indiscernible sequence in a dependent theory then the result is still dependent. This answers a question of Baldwin and Benedikt and implies that every unstable dependent theory has a dependent expansion interpreting linear order.
The classical Baldwin-Lachlan characterization of uncountably categorical theories is known to fail in continuous logic in that not every inseparably categorical theory has a strongly minimal set. Here we investigate these issues by developing the theory of strongly minimal sets in continuous logic and by examining inseparably categorical expansions of Banach space. To this end we introduce and characterize 'dictionaric theories,' theories in which definable sets are prevalent enough that many constructions familiar in discrete logic can be carried out. We also introduce, in the context of Ban…
A dependent theory is a (first order complete theory) T which does not have the independence property. A main result here is: if we expand a model of T by the traces on it of sets definable in a bigger model then we preserve its being dependent. Another one justifies the cofinality restriction in the theorem (from a previous work) saying that pairwise perpendicular indiscernible sequences, can have arbitrary dual-cofinalities in some models containing them.
n-dependent continuous theories and hyperdefinable sets
本文定义了一阶结构的连续建模性质并证明其等价于age的嵌入Ramsey性质，同时利用连续逻辑中的广义不可辨序列，基于不可辨序列的坍缩刻画了连续理论与一阶超可定义集的n-依赖性。
We try to understand complete types over a somewhat saturated model of a complete first order theory which is dependent (previously called NIP), by "decomposition theorems for such types". Our thesis is that the picture of dependent theory is the combination of the one for stable theories and the one for the theory of dense linear order or trees (and first we should try to understand the quite saturated case). As a measure of our progress, we give several applications considering some test questions; in particular we try to prove the generic pair conjecture and do it for measurable cardinals.
We characterize all ordered Abelian groups whose first order theory in the language {+,<} is strongly dependent. The main result of this note was obtained independently by Halevi-Hasson and Farré.
We prove elimination of field quantifiers for strongly dependent henselian fields in the Denef-Pas language. This is achieved by proving the result for a class of fields generalizing algebraically maximal Kaplansky fields. We deduce that if $(K,v)$ is strongly dependent then so is its henselization.
In the first part we show a counterexample to a conjecture by Shelah regarding the existence of indiscernible sequences in dependent theories (up to the first inaccessible cardinal). In the second part we discuss generic pairs, and give an example where the pair is not dependent. Then we define the notion of directionality which deals with counting the number of coheirs of a type and we give examples of the different possibilities. Then we discuss non-splintering, an interesting notion that appears in the work of Rami Grossberg, Andrés Villaveces and Monica VanDieren, and we show that it is no…
In this note we prove and disprove some chain conditions in type definable and definable groups in dependent, strongly dependent and strongly^{2} dependent theories.
In this paper, we propose an abstract definition of dependent type theories as essentially algebraic theories. One of the main advantages of this definition is its composability: simple theories can be combined into more complex ones, and different properties of the resulting theory may be deduced from properties of the basic ones. We define a category of algebraic dependent type theories which allows us not only to combine theories but also to consider equivalences between them. We also study models of such theories and show that one can think of them as contextual categories with additional …
We define a general class of dependent type theories, encompassing Martin-Löf's intuitionistic type theories and variants and extensions. The primary aim is pragmatic: to unify and organise their study, allowing results and constructions to be given in reasonable generality, rather than just for specific theories. Compared to other approaches, our definition stays closer to the direct or naive reading of syntax, yielding the traditional presentations of specific theories as closely as possible. Specifically, we give three main definitions: raw type theories, a minimal setup for discussing depe…
We survey the history of Shelah's conjecture on strongly dependent fields, give an equivalent formulation in terms of a classification of strongly dependent fields and prove that the conjecture implies that every strongly dependent field has finite dp-rank.
We define a notion of equivalence between algebraic dependent type theories which we call Morita equivalence. This notion has a simple syntactic description and an equivalent description in terms of models of the theories. The category of models of a type theory often carries a natural structure of a model category. If this holds for the categories of models of two theories, then a map between them is a Morita equivalence if and only if the adjunction generated by it is a Quillen equivalence.
Relating homotopy equivalences to conservativity in dependent type theories with computation axioms
本文借助同伦类型理论与带属性范畴，通过上下文间典范同伦等价证明外延类型理论对带命题计算公理的依赖类型理论具有保守性。
Strongly Dependent Ordered Abelian Groups and Henselian Fields
本文刻画有限dp-秩的强依赖有序阿贝尔群，并据此证明强依赖域上任意Henselian赋值所形成的赋值域仍为强依赖。
The nonexistence of unicorns and many-sorted Löwenheim-Skolem theorems
本文证明了可满足性模理论中“独角兽理论”不存在，即稳定无穷且强有限可见证的理论必是强礼貌的，并证明了多类逻辑的 Löwenheim-Skolem 定理与 Łoś-Vaught 测试版本。
We study the coherence and conservativity of extensions of dependent type theories by additional strict equalities. By considering notions of congruences and quotients of models of type theory, we reconstruct Hofmann's proof of the conservativity of Extensional Type Theory over Intensional Type Theory. We generalize these methods to type theories without the Uniqueness of Identity Proofs principle, such as variants of Homotopy Type Theory, by introducing a notion of higher congruence over models of type theory. Our definition of higher congruence is inspired by Brunerie's type-theoretic defini…
For an infinite cardinal $κ$, let $dedκ$ denote the supremum of the number of Dedekind cuts in linear orders of size $κ$. It is known that $κ<dedκ\leq 2^κ$ for all $κ$ and that $dedκ<2^κ$ is consistent for any $κ$ of uncountable cofinality. We prove however that $2^κ\leq ded ( ded ( ded ( dedκ)))$ always holds. Using this result we calculate the Hanf numbers for the existence of two-cardinal models with arbitrarily large gaps and for the existence of arbitrarily large models omitting a type in the class of countable dependent first-order theories. Specifically, we show that these bounds are as…
Can a problem undecidable with classical resources be decidable with quantum ones? The answer expected is no; as both being Turing theories, they should not solve the Halting problem - a problem unsolvable by any Turing machine. Yet, we provide an affirmative answer to the aforesaid question. We come up with a novel logical structure to formulate infinitely many such problems for any pair of distinct Turing theories, including but not limited to the classical and quantum theories. Importantly, a class of other decision problems, such as the Halting one, remains unsolvable in all those theories…
Shininess, strong politeness, and unicorns
本文细化光泽性与强礼貌性理论组合性质的关系，证明光泽理论必可判定且强礼貌，并构造不可判定但强礼貌的非光泽理论，最终完成理论组合性质关系的完整分类。
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