In this paper, we continue the study of a left-distributive algebra of elementary embeddings from the collection of sets of rank less than lambda to itself, as well as related finite left-distributive algebras (which can be defined without reference to large cardinals). In particular, we look at the critical points (least ordinals moved) of the elementary embeddings; simple statements about these ordinals can be reformulated as purely algebraic statements concerning the left distributive law. Previously, lower bounds on the number of critical points have been used to show that certain such algebraic statements, known to follow from large cardinals, require more than Primitive Recursive Arithmetic to prove. Here we present the first few steps of a program that, if it can be carried to completion, should give exact computations of the number of critical points, thereby showing that hypotheses only slightly beyond Primitive Recursive Arithmetic would suffice to prove the aforementioned algebraic statements.
Some of the main results here were discovered through examination of bit patterns from extensive calculations in the finite algebras; this may be the first serious example of computer-aided research in the theory of large cardinals. The results in this paper have a large overlap with results of A. Drapal, which were obtained independently at about the same time, but from a quite different point of view.
Let $j:V_λ---> V_λ$ be an elementary embedding, with critical point $κ$, and let $f(n)$ be the number of critical points of embeddings in the algebra generated by $j$ which lie between $j^n(κ)$ and $j^{n+1}(κ)$. It is shown that $f(n)$ is finite for all $n$.
Every transformation monoid comes equipped with a canonical topology-the topology of pointwise convergence. For some structures, the topology of the endomorphism monoid can be reconstructed from its underlying abstract monoid. This phenomenon is called automatic homeomorphicity. In this paper we show that whenever the automorphism group of a countable saturated structure has automatic homeomorphicity and a trivial center, then the monoid of elementary self-embeddings has automatic homeomorphicity, too. As a second result we strengthen a result by Lascar by showing that whenever $\mathbf{A}$ is…
We consider the role of the foundation axiom and various anti-foundation axioms in connection with the nature and existence of elementary self-embeddings of the set-theoretic universe.
We consider embeddings in a torsion-free hyperbolic group which are elementary in the sense of first-order logic. We give a description of these embeddings in terms of Sela's hyperbolic towers. We deduce as a corollary that subgroups elementarily embedded in finitely generated free groups are free factors.
We give a sharper version of a theorem of Rosicky, Trnkova and Adamek, and a new proof of a theorem of Rosicky, both about colimit preservation between categories of structures. Unlike the original proofs, which use category-theoretic methods, we use set-theoretic arguments involving elementary embeddings given by large cardinals such as alpha-strongly compact and C^(n)-extendible cardinals.
One of the numerous characterizations of a Ramsey cardinal kappa involves the existence of certain types of elementary embeddings for transitive sets of size κsatisfying a large fragment of ZFC. We introduce new large cardinal axioms generalizing the Ramsey elementary embeddings characterization and show that they form a natural hierarchy between weakly compact cardinals and measurable cardinals. These new axioms serve to further our knowledge about the elementary embedding properties of smaller large cardinals, in particular those still consistent with V=L.
There is a polymodal provability logic $GLP$. We consider generalizations of this logic: the logics $GLP_α$, where $α$ ranges over linear ordered sets and play the role of the set of indexes of modalities. We consider the varieties of modal algebras that corresponds to the polymodal logics. We prove that the elementary theories of the free $\emptyset$-generated $GLP_{n}$ -algebras are decidable for all finite ordinals $n$.
Given two elementary embeddings from the collection of sets of rank less than $λ$ to itself, one can combine them to obtain another such embedding in two ways: by composition, and by applying one to (initial segments of) the other. Hence, a single such nontrivial embedding $j$ generates an algebra of embeddings via these two operations, which satisfies certain laws (for example, application distributes over both composition and application). Laver has shown, among other things, that this algebra is free on one generator with respect to these laws. The set of critical points of members of this …
In a remarkable series of papers, Zlil Sela classified the first-order theories of free groups and torsion-free hyperbolic groups using geometric structures he called towers. It was later proved by Chloé Perin that if $H$ is an elementarily embedded subgroup (or elementary submodel) of a torsion-free hyperbolic group $G$, then $G$ is a tower over $H$. We prove a generalization of Perin's result to toral relatively hyperbolic groups using JSJ and shortening techniques.
The following pcf results are proved: 1. Assume that kappa > aleph_0 is a weakly compact cardinal. Let mu > 2^kappa be a singular cardinal of cofinality kappa. Then for every regular lambda < pp^+_{Gamma(kappa)} (mu) there is an increasing sequence (lambda_i | i < kappa) of regular cardinals converging to mu such that lambda = tcf(prod_{i < kappa} lambda_i, <_{J^{bd}_kappa}). 2. Let mu be a strong limit cardinal and theta a cardinal above mu. Suppose that at least one of them has an uncountable cofinality. Then there is sigma_* < mu such that for every chi < theta the following holds: theta > …
Large cardinals arising from the existence of arbitrarily long end elementary extension chains over models of set theory are studied here. In particular, we show that the large cardinals obtained that way (`Unfoldable cardinals') behave as a `boundary' between properties consistent with `V=L' and existence of indiscernibles. We also provide an `embedding characterisation' of the unfoldable cardinals and study their preservation and destruction by various different forcings.
For any pair of ordinals $α<β$, $\sf CA_α$ denotes the class of cylindric algebras of dimension $α$, $\sf RCA_α$ denote the class of representable $\sf CA_α$s and $\sf Nr_αCA_β$ ($\sf Ra CA_β)$ denotes the class of $α$-neat reducts (relation algebra reducts) of $\sf CA_β$. We show that any class $\sf K$ such that $\sf RaCA_ω\subseteq \sf K\subseteq RaCA_5$, $\sf K$ is not elementary, i.e not definable in first order logic. Let $2<n<ω$. It is also shown that any class $\sf K$ such that $\sf Nr_nCA_ω\cap {\sf CRCA}_n\subseteq {\sf K}\subseteq \mathbf{S}_c\sf Nr_nCA_{n+3}$, where $\sf CRCA_n$ is …
Let K be an abstract elementary classes which has arbitrarily large models and satisfies the amalgamation and joint embedding properties. Theorem 1. Suppose K is χ-tame. If K is categorical in some λ^+ >LS(K) then it is categorical in all μ\geq (λ+χ)^+. Theorem 2. If K is LS(K)-tame and is categorical both in LS(K) and in LS(K)^+ then K is categorical in all μ\geq LS(K).
We introduce the notion of $λ$-equivalence and $λ$-embeddings of objects in suitable categories. This notion specializes to $L_{\inftyλ}$-equivalence and $L_{\inftyλ}$-elementary embedding for categories of structures in a language of arity less than $λ$, and interacts well with functors and $λ$-directed colimits. We recover and extend results of Feferman and Eklof on "local functors" without fixing a language in advance. This is convenient for formalizing Lefschetz's principle in algebraic geometry, which was one of the main applications of the work of Eklof.
The ultrapower theorem of Keisler-Shelah allows such model-theoretic notions as elementary equivalence, elementary embedding and existential embedding to be couched in the language of categories (limits, morphism diagrams). This in turn allows analogs of these (and related) notions to be transported into unusual settings, chiefly those of Banach spaces and of compacta. Our interest here is the enrichment of the theory of compacta, especially the theory of continua, brought about by the immigration of model-theoretic ideas and techniques.
We show that for 1<n<m, the class Nr_nCA_m known to be non-elementary is pseudo elementary. When n and m are finite we use a two sorted theory, when n is finite and m infinite we use a three sorted one, and finally when both are infinite we use a four sorted defining theory. Our non finite axiomatizability result, follows from the fact that for 2<n<m, and any r\in ωthere exists a finite (Monk like) algebra C(m,n,r), such that C(m,n,r)\in Nr_nCA_m C(m,n,r)\notin SNr_nCA_{m+1}, and any non trivial ultraproduct on r of such algebras in in ElNr_nCA_m. Finally we use such algebras, to show that for…
Let $A$ be an infinite set. Let $Ω(A)$ be the algebra over $A$ where every constant is a fundamental constant and every finitary function is a fundamental operation. We shall give a method of representing any algebra $\mathcal{L}$ in the variety generated by $Ω(A)$ as limit reduced powers and even direct limits of limit reduced powers of $\mathcal{L}$. If the algebra $\mathcal{L}$ is elementarily equivalent to $Ω(A)$, then this construction represents $Ω(A)$ as a limit ultrapower and also as direct limits of limit ultrapowers of $Ω(A)$. This method therefore gives a method of representing Bool…
We provide a thorough algebraic analysis of three known completions having a central role in the exact completions of Lawvere's doctrines: the one adding comprehensive diagonals (i.e. forcing equality on terms to coincide with the equality predicate), the one adding full comprehensions and the one adding quotients. We show that all these 2-adjunctions are 2-monadic and that the 2-monads arising from these adjunctions are all property-like. This entails that comprehensive diagonals, full comprehensions and quotients are algebraic properties of an elementary doctrine. Finally, we discuss and pre…
We characterize when the elementary diagram of a mutually algebraic structure has a model complete theory, and give an explicit description of a set of existential formulas to which every formula is equivalent. This characterization yields a new, more constructive proof that the elementary diagram of any model of a strongly minimal, trivial theory is model complete.
We explore the possibilities for elementary embeddings $j : M \to N$, where $M$ and $N$ are models of ZFC with the same ordinals, $M \subseteq N$, and $N$ has access to large pieces of $j$. We construct commuting systems of such maps between countable transitive models that are isomorphic to various canonical linear and partial orders, including the real line $\mathbb R$.
We study the saturation properties of several classes of $C^*$-algebras. Saturation has been shown by Farah and Hart to unify the proofs of several properties of coronas of $σ$-unital $C^*$-algebras; we extend their results by showing that some coronas of non-$σ$-unital $C^*$-algebras are countably degree-$1$ saturated. We then relate saturation of the abelian $C^*$-algebra $C(X)$, where $X$ is $0$-dimensional, to topological properties of $X$, particularly the saturation of $CL(X)$.
We use continuous model theory to obtain several results concerning isomorphisms and embeddings between II_1 factors and their ultrapowers. Among other things, we show that for any II_1 factor M, there are continuum many nonisomorphic separable II_1 factors that have an ultrapower isomorphic to an ultrapower of M. We also give a poor man's resolution of the Connes Embedding Problem: there exists a separable II_1 factor such that all II_1 factors embed into one of its ultrapowers.
In this paper we provide a complete algebraic characterization of elementary equivalence of rings with a finitely generated additive group in the language of pure rings. The rings considered are arbitrary otherwise.
We prove that there is a lattice embedded from every countable distributive lattice into the Boolean algebra of computable subsets of $\mathbb{N}$. Along the way, we discuss all relevant results about lattices, Boolean algebras and embeddings between them.
