We show, assuming PD, that every complete finitely axiomatized second order theory with a countable model is categorical, but that there is, assuming again PD, a complete recursively axiomatized second order theory with a countable model which is non-categorical. We show that the existence of even very large (e.g. supercompact) cardinals does not imply the categoricity of all finite complete second order theories. More exactly, we show that a non-categorical complete finitely axiomatized second order theory can always be obtained by (set) forcing. We also show that the categoricity of all finite complete second order theories with a model of a certain singular cardinality kappa of uncountable cofinality can be forced over any model of set theory. Previously, Solovay had proved, assuming V=L, that every complete finitely axiomatized second order theory (with or without a countable model) is categorical, and that in a generic extension of L there is a complete finitely axiomatized second order theory with a countable model which is non-categorical.
We continue the work of [KlSh:362] and prove that for lambda successor, a lambda-categorical theory T in L_{kappa^*, omega} is mu-categorical for every mu, mu <= lambda which is above the (2^{LS(T)})^+-beth cardinal.
We deal with the systematic development of stability for the context of approximate elementary submodels of a monster metric space, which is not far, but still very distinct from the first order case. In particular we prove the analogue of Morley's theorem for the classes of complete metric spaces.
We prove Los conjecture = Morley theorem in ZF, with the same characterization (of first order countable theories categorical in aleph_alpha for some (equivalently for every) ordinal alpha>0. Another central result here is, in this context: the number of models of a countable first order T of cardinality aleph_alpha is either >=|alpha| for every aleph_αor it has a small upper bound (close to beth_2).
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…
Let ${\bf K}$ be an $\mathrm{LS}({\bf K})$-short abstract elementary class and assume more than the existence of a monster model (amalgamation over sets and arbitrarily large models). Suppose ${\bf K}$ is categorical in some $μ>\mathrm{LS}({\bf K})$, then it is categorical in all $μ'\geqμ$. Our result removes the successor requirement of $μ$ made by Grossberg-VanDieren, at the cost of using shortness instead of tameness; and of using amalgamation over sets instead of over models. It also removes the primes requirement by Vasey which assumes tameness and amalgamation over models. As a corollary…
Informally speaking, the categoricity of an axiom system means that its non-logical symbols have only one possible interpretation that renders the axioms true. Although non-categoricity has become ubiquitous in the second half of the 20th century whether one looks at number theory, geometry or analysis, the first axiomatizations of such mathematical theories by Dedekind, Hilbert, Huntington, Peano and Veblen were indeed categorical. A common resolution of the difference between the earlier categorical axiomatizations and the more modern non-categorical axiomatizations is that the latter derive…
The categoricity spectrum of a class of structures is the collection of cardinals in which the class has a single model up to isomorphism. Assuming that cardinal exponentiation is injective (a weakening of the generalized continuum hypothesis, GCH), we give a complete list of the possible categoricity spectrums of an abstract elementary class with amalgamation and arbitrarily large models. Specifically, the categoricity spectrum is either empty, an end segment starting below the Hanf number, or a closed interval consisting of finite successors of the Löwenheim-Skolem-Tarski number (there are e…
We investigate a notion called uniqueness in power kappa that is akin to categoricity in power kappa, but is based on the cardinality of the generating sets of models instead of on the cardinality of their universes. The notion is quite useful for formulating categoricity-like questions regarding powers below the cardinality of a theory. We prove, for (uncountable) universal theories T, that if T is kappa-unique for one uncountable kappa, then it is kappa-unique for every uncountable kappa ; in particular, it is categorical in powers greater than the cardinality of T.
We provide a complete classification of all the possible categoricity spectra, in terms of internal size, that can appear in a large accessible category with directed colimits, assuming the Singular Cardinal Hypothesis ($SCH$), and providing as well explicit threshold cardinals for eventual categoricity. This includes as a particular case the first complete classification of categoricity spectra of abstract elementary classes (AEC's) entirely in $ZFC$. More specifically, we have the following theorem: Let $\mathcal{K}$ be a large $κ$-accessible category with directed colimits. Assume the Singu…
We give a model-theoretic characterization of the class of geometric theories classified by an atomic topos having enough points; in particular, we show that every complete geometric theory classified by an atomic topos is countably categorical. Some applications are also discussed.
A countable semigroup is $\aleph_0$-categorical if it can be characterised, up to isomorphism, by its first-order properties. In this paper we continue our investigation into the $\aleph_0$-categoricity of semigroups. Our main results are a complete classification of $\aleph_0$-categorical orthodox completely 0-simple semigroups, and descriptions of the $\aleph_0$-categorical members of certain classes of strong semilattices of semigroups.
Suppose L is a relational language and P in L is a unary predicate. If M is an L-structure then P(M) is the L-structure formed as the substructure of M with domain {a: M models P(a)}. Now suppose T is a complete first order theory in L with infinite models. Following Hodges, we say that T is relatively lambda-categorical if whenever M, N models T, P(M)=P(N), |P(M)|= lambda then there is an isomorphism i:M-> N which is the identity on P(M). T is relatively categorical if it is relatively lambda-categorical for every lambda. The question arises whether the relative lambda-categoricity of T for s…
We examine categoricity issues for computable algebraic fields. We give a structural criterion for relative computable categoricity of these fields, and use it to construct a field that is computably categorical, but not relatively computably categorical. Finally, we show that computable categoricity for this class of fields is $Π^0_4$-complete.
We provide a proof, in $ZFC$, of Shelah's eventual categoricity conjecture for abstract elementary classes (AEC's). Moreover, assuming in addition the Singular Cardinal Hypothesis ($SCH$), we prove a direct generalization to the more general context of accessible categories with directed colimits. If $\mathcal{K}$ is such a category, we show that there is a cardinal $μ$ such that if $\mathcal{K}$ is $λ$-categorical for some $λ\geq μ$ (i.e., it has only one object of internal size $λ$ up to isomorphism), then $\mathcal{K}$ is eventually categorical (i.e., it is $λ'$-categorical for every $λ' \g…
University of Helsinki（赫尔辛基大学）；University College London（伦敦大学学院）；University of Torino（都灵大学）；Istituto Nazionale di Alta Matematica “Francesco Severi”（弗朗切斯科·塞韦里国家高等数学研究所）
We prove that if $T$ is an $ω$-categorical supersimple theory with nontrivial dependence (given by forking), then there is a nontrivial regular 1-type over a finite set of reals which is realized by real elements; hence forking induces a nontrivial pregeometry on the solution set of this type and the pregeometry is definable (using only finitely many parameters). The assumption about $ω$-categoricity is necessary. This result is used to prove the following: If $V$ is a finite relational vocabulary with maximal arity 3 and $T$ is a supersimple $V$-theory with elimination of quantifiers, then $T…
We study approximate $\aleph_0$-categoricity of theories of beautiful pairs of randomizations, in the sense of continuous logic. This leads us to disprove a conjecture of Ben Yaacov, Berenstein and Henson, by exhibiting $\aleph_0$-categorical, $\aleph_0$-stable metric theories $Q$ for which the corresponding theory $Q_P$ of beautiful pairs is not approximately $\aleph_0$-categorical, i.e., has separable models that are not isomorphic even up to small perturbations of the smaller model of the pair. The theory $Q$ of randomized infinite vector spaces over a finite field is such an example. On th…
The fundamental aim of the paper is to correct an harmful way to interpret a Goedel's erroneous remark at the Congress of Koenigsberg in 1930. Despite the Goedel's fault is rather venial, its misreading has produced and continues to produce dangerous fruits, as to apply the incompleteness Theorems to the full second-order Arithmetic and to deduce the semantic incompleteness of its language by these same Theorems. The first three paragraphs are introductory and serve to define the languages inherently semantic and its properties, to discuss the consequences of the expression order used in a lan…
We prove that many seemingly simple theories have Borel complete reducts. Specifically, if a countable theory has uncountably many complete 1-types, then it has a Borel complete reduct. Similarly, if $Th(M)$ is not small, then $M^{eq}$ has a Borel complete reduct, and if a theory $T$ is not $ω$-stable, then the elementary diagram of some countable model of $T$ has a Borel complete reduct.
We show that degrees containing a complete extensions of arithmetic have the random join property: they are the supremum of any random real they compute, with another random real. The same is true for the truth-table and weak truth-table reducibilities.
We classify the computability-theoretic complexity of two index sets of classes of first-order theories: We show that the property of being an $\aleph_0$-categorical theory is $Π^0_3$-complete; and the property of being an Ehrenfeucht theory $Π^1_1$-complete. We also show that the property of having continuum many models is $Σ^1_1$-hard. Finally, as a corollary, we note that the properties of having only decidable models, and of having only computable models, are both $Π^1_1$-complete.
Given a complete theory $T$ and a subset $Y \subseteq X^k$, we precisely determine the {\em worst case complexity}, with respect to further monadic expansions, of an expansion $(M,Y)$ by $Y$ of a model $M$ of $T$ with universe $X$. In particular, although by definition monadically stable/NIP theories are robust under arbitrary monadic expansions, we show that monadically NFCP (equivalently, mutually algebraic) theories are the largest class that is robust under anything beyond monadic expansions. We also exhibit a paradigmatic structure for the failure of each of monadic NFCP/stable/NIP and pr…
We show that if A is a linear order then Th(A) is either $\aleph_0$-categorical or Borel complete (in the sense of Friedman and Stanley). We generalize this; if A has countably many unary predicates attached, then Th(A) is $\aleph_0$-categorical, has finitely many countable models (at least three), is Borel equivalent to equality on the reals, is Borel equivalent to "countable sets of reals," or is Borel complete. Furthermore, each of these cases corresponds in a natural way to a count of models of all sizes, up to back-and-forth equivalence. All these cases are possible and we compute precise…
We give an almost entirely model-theoretic account of both Ramsey classes of finite structures and of generalized indiscernibles as studied in special cases in (for example) [7], [9]. We understand "theories of indiscernibles" to be special kinds of companionable theories of finite structures, and much of the work in our arguments is carried in the context of the model-companion. Among other things, this approach allows us to prove that the companion of a theory of indiscernibles whose "base" consists of the quantifier-free formulas is necessarily the theory of the Fraisse limit of a Fraisse c…
For a theory $T$ in $L, T_σ$ is the theory of the models of $T$ with an automorphism $σ$. If $T$ is an unstable model complete theory without the independence property, then $T_σ$ has no model companion. If $T$ is an unstable model complete theory and $T_σ$ has the amalgamation property, then $T_σ$ has no model companion. If $T$ is model complete and has the fcp, then $T_σ$ has no model completion.
We generalise the correspondence between $\aleph 0$-categorical theories and their automorphism groups to arbitrary complete theories in classical logic, and to some theories (including, in particular, all $\aleph 0$-categorical ones) in continuous logic.
A complete first order theory of a relational signature is called monomorphic iff all its models are monomorphic (i.e. have all the $n$-element substructures isomorphic, for each positive integer $n$). We show that a complete theory ${\mathcal T}$ having infinite models is monomorphic iff it has a countable monomorphic model and confirm the Vaught conjecture for monomorphic theories. More precisely, we prove that if ${\mathcal T}$ is a complete monomorphic theory having infinite models, then the number of its non-isomorphic countable models, $I({\mathcal T},ω)$, is either equal to $1$ or to ${…
We show there exists a complete theory in a language of size continuum possessing a unique atomic model which is not constructible. We also show it is consistent with $ZFC + \aleph_1 < 2^{\aleph_0}$ that there is a complete theory in a language of size $\aleph_1$ possessing a unique atomic model which is not constructible. Finally we show it is consistent with $ZFC + \aleph_1 < 2^{\aleph_0}$ that for every complete theory $T$ in a language of size $\aleph_1$, if $T$ has uncountable atomic models but no constructible models, then $T$ has $2^{\aleph_1}$ atomic models of size $\aleph_1$.
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 prove that every many-sorted $ω$-categorical theory is completely interpretable in a one-sorted $ω$-categorical theory. As an application, we give a short proof of the existence of non $G$--compact $ω$-categorical theories.
We show that any countable model of a model complete theory has an elementary extension with a "pseudofinite-like" quasidimension that detects dividing.
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.
A relevant thesis is that for the family of complete first order theories with NIP (i.e. without the independence property) there is a substantial theory, like the family of stable (and the family of simple) first order theories. We examine some properties.
We present distributions of countable models and correspondent structural characteristics of complete theories with continuum many types: for prime models over finite sets relative to Rudin-Keisler preorders, for limit models over types and over sequences of types, and for other countable models of theory.
We present the notions of positively complete theory and general forms of amalgamation in the framework of positive logic. We explore the fundamental properties of positively complete theories and study the behaviour of companion theories by a change of constants in the language. Moreover, we present a general form of amalgamation and discuss some forms of strong amalgamation.
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 .
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.
We introduce a complete many-valued semantics for two normal lattice-based modal logics. This semantics is based on reflexive many-valued graphs. We discuss an interpretation and possible applications of this logical framework in the context of the formal analysis of the interaction between (competing) scientific theories.
There exist two conjectures for constraint satisfaction problems (CSPs) of reducts of finitely bounded homogeneous structures: the first one states that tractability of the CSP of such a structure is, when the structure is a model-complete core, equivalent to its polymorphism clone satisfying a certain non-trivial linear identity modulo outer embeddings. The second conjecture, challenging the approach via model-complete cores by reflections, states that tractability is equivalent to the linear identities (without outer embeddings) satisfied by its polymorphisms clone, together with the natural…
Let $\mathcal{O}$ be the ring of integers of a number field, and let $n\geq 3$. This paper studies bi-interpretability of the ring of integers $\mathbb{Z}$ with the special linear group $\text{SL}_n(\mathcal{O})$, the general linear group $\text{GL}_n(\mathcal{O})$ and solvable group of all invertible uppertriangular matrices over $\mathcal{O}$, $\text{T}_n(\mathcal{O})$. For each of these groups we provide a complete characterization of arbitrary models of their complete elementary theories.
