Chanoch Havlin, Saharon Shelah
We prove the existence of pairs of models of the same cardinality lambda which are very equivalent according to EF games, but not isomorphic. We continue the paper math.LO/0404222, but we don't rely on it.
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We construct non-isomorphic models M, N, e.g. of cardinality aleph_1 such that in the Ehrenfeucht-Fraisse game of length zeta < omega_1 the isomorphism player wins
It is well-known that the first order Peano axioms PA have a continuum of non-isomorphic countable models. The question, how close to being isomorphic such countable models can be, seems to be less investigated. A measure of closeness to isomorphism of countable models is the length of back-and-forth sequences that can be established between them. We show that for every countable ordinal alpha there are countable non-isomorphic models of PA with a back-and-forth sequence of length alpha between them. This implies that the Scott height (or rank) of such models is bigger than $α$. We also prove …
We prove that if ZF is consistent then ZFC+GCH is consistent with the following statement: There is for every k<omega a model of cardinality aleph_1 which is L_{infty,omega_1}-equivalent to exactly k non-isomorphic models of cardinality aleph_1. In order to get this result we introduce ladder systems and colourings different from the ``standard'' counterparts, and prove the following purely combinatorial result: For each prime number p and positive integer m it is consistent with ZFC+GCH that there is a ``good'' ladder system having exactly p^m pairwise nonequivalent colourings.
Our "long term and large scale" aim is to characterize the first order theories T (at least the countable ones) such that: for every ordinal alpha there lambda,M_1,M_2 such that M_1,M_2 are non-isomorphic models of T of cardinality lambda which are EF_{alpha, lambda}-equivalent. We expect that as in the main gap we get a strong dichotomy, so in the non-structure side we have stronger, better examples, and in the structure side we have a parallel of [Sh:c,XIII]. We presently prove the consistency of the non-structure side for T which is aleph_0-independent (= not strongly dependent), even for P…
For a cardinal kappa and a model M of cardinality kappa let No(M) denote the number of non-isomorphic models of cardinality kappa which are L_{infty,kappa}--equivalent to M. In [Sh:133] Shelah established that when kappa is a weakly compact cardinal and mu<=kappa is a nonzero cardinal, there exists a model M of cardinality kappa with No(M)=mu. We prove here that if kappa is a weakly compact cardinal, the question of the possible values of No(M) for models M of cardinality kappa is equivalent to the question of the possible numbers of equivalence classes of equivalence relations which are Sigma…
There has been much interest on constructing models which are not isomorphic of cardinality lambda but are equivalent under the Ehrenfeucht-Fraisse game of length alpha even for every alpha<lambda. So under G.C.H. we know much. We deal here with constructions of such pairs of models proven in ZFC and get the existence under mild conditions.
Suppose lambda is a singular cardinal of uncountable cofinality kappa. For a model M of cardinality lambda, let No(M) denote the number of isomorphism types of models N of cardinality lambda which are L_{infty lambda}-equivalent to M. In [Sh:189] inverse kappa-systems A of abelian groups and their certain kind of quotient limits Gr(A)/Fact(A) were considered. It was proved that for every cardinal mu there exists an inverse kappa-system A such that A consists of abelian groups having cardinality at most mu^kappa and card(Gr(A)/Fact(A))= mu. In [Sh:228] a strict connection between inverse kappa-…
We study a notion of potential isomorphism, where two structures are said to be potentially isomorphic if they are isomorphic in some generic extension that preserves stationary sets and does not add new sets of cardinality less than the cardinality of the models. We introduce the notions of semi-proper and weakly semi-proper trees, and note that there is a strong connection between the existence of potentially isomorphic models for a given complete theory and the existence of weakly semi-proper trees. We prove the existence of semi-proper trees under certain cardinal arithmetic assumptions. W…
The logic L^1_θintroduced in [Sh:797]; it is the maximal logic below L_theta theta in which a well ordering is not definable. We investigate it for theta a compact cardinal. We prove 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 L^1_theta-equivalent iff for some omega-sequence of theta-complete ultrafilters, the iterated ultra-powers by it of those two models are isomorphic.
We show that graph products of finite abelian groups are elementarily equivalent if and only if they are $\exists\forall$-equivalent if and only if they are isomorphic. In particular, two right-angled Coxeter groups are elementarily equivalent if and only if they are isomorphic.
We prove that there exist uncountably many separable II$_1$ factors whose ultrapowers (with respect to arbitrary ultrafilters) are non-isomorphic. In fact, we prove that the families of non-isomorphic II$_1$ factors originally introduced by McDuff \cite{MD69a,MD69b} are such examples. This entails the existence of a continuum of non-elementarily equivalent II$_1$ factors, thus settling a well-known open problem in the continuous model theory of operator algebras.
We prove that, consistently with ZFC, no ultraproduct of countably infinite (or separable metric, non-compact) structures is isomorphic to a reduced product of countable (or separable metric) structures associated to the Fréchet filter. Since such structures are countably saturated, the Continuum Hypothesis implies that they are isomorphic when elementarily equivalent.
It has been a longstanding problem whether every amenable operator algebra is isomorphic to a (necessarily nuclear) C*-algebra. In this note, we give a nonseparable counterexample. The existence of a separable counterexample remains an open problem. We also initiate a general study of unitarizability of representations of amenable groups in C*-algebras and show that our method cannot produce a separable counterexample.
We extend the classical Feferman-Vaught theorem to logic for metric structures. This implies that the reduced powers of elementarily equivalent structures are elementarily equivalent, and therefore they are isomorphic under the Continuum Hypothesis. We also prove the existence of two separable C*-algebras of the form $\bigoplus_i M_{k(i)}(\mathbb{C})$ such that the assertion that their coronas are isomorphic is independent from ZFC, which gives the first example of genuinely non-commutative coronas of separable C*-algebras with this property.
Irreducible Artin groups of finite type can be parametrized via their associated Coxeter diagrams into six sporadic examples and four infinite families, each of which is further parametrized by the natural numbers. Within each of these four infinite families, we investigate the relationship between elementary equivalence and isomorphism. For three out of the four families, we show that two groups in the same family are equivalent if and only if they are isomorphic; a positive, but weaker, result is also attained for the fourth family. In particular, we show that two braid groups are elementari…
Given two structures $\mathcal{M}$ and $\mathcal{N}$ on the same domain, we say that $\mathcal{N}$ is a reduct of $\mathcal{M}$ if all $\emptyset$-definable relations of $\mathcal{N}$ are $\emptyset$-definable in $\mathcal{M}$. In this article the reducts of the Henson digraphs are classified. Henson digraphs are homogeneous countable digraphs that omit some set of finite tournaments. As the Henson digraphs are $\aleph_0$-categorical, determining their reducts is equivalent to determining all closed supergroups $G<$ Sym$(\mathbb{N})$ of their automorphism groups. A consequence of the classific…
As a consequence of identifying the principle described in the title, we prove that for any uncountable cardinal lambda, if there is a lambda-free Whitehead group of cardinality lambda which is not free, then there are many ``nice'' Whitehead groups of cardinality lambda which are not free.
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