Continuous Restricted Radial Motion of a Gas under the Action of a Piston

The possibility of continuous conjugation of the straightline radial motion of a gas sphere toward the center and away from the center with the motion where the gas in the entire sphere stops simultaneously is shown. The motion is described by an invariant submodel of rank 1. Time reflections allow one to construct a solution that describes a periodic continuous restricted motion of the gas sphere under the action of a piston.     

Kripke submodels and universal sentences

We define two notions for intuitionistic predicate logic: that of a submodel of a Kripke model, and that of a universal sentence. We then prove a corresponding preservation theorem. If a Kripke model is viewed as a functor from a small category to the category of all classical models with (homo)morphisms between them, then we define a submodel of a Kripke model to be a restriction of the original Kripke model to a subcategory of its domain, where every node in the subcategory is mapped to a classical submodel of the corresponding classical model in the range of the original Kripke model. We call a sentence universal if it is built inductively from atoms (including ? and ⊥) using ∧, ∨, ?, and →, with the restriction that antecedents of → must be atomic. We prove that an intuitionistic theory is axiomatized by universal sentences if and only if it is preserved under Kripke submodels. We also prove the following analogue of a classical model‐consistency theorem: The universal fragment of a theory Γ is contained in the universal fragment of a theory Δ if and only if every rooted Kripke model of Δ is strongly equivalent to a submodel of a rooted Kripke model of Γ. Our notions of Kripke submodel and universal sentence are natural in the sense that in the presence of the rule of excluded middle, they collapse to the classical notions of submodel and universal sentence. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Strong zero-dimensionality of hyperspaces

For a space X, X² denotes the collection of all non-empty closed sets of X with the Vietoris topology, and K(X) denotes the collection of all non-empty compact sets of X with the subspace topology of X². The following are known:

•

ω² is not normal, where ω denotes the discrete space of countably infinite cardinality.

•

For every non-zero ordinal γ with the usual order topology, K(γ) is normal iff whenever cf γ is uncountable.

In this paper, we will prove:

(1)

ω² is strongly zero-dimensional.

(2)

K(γ) is strongly zero-dimensional, for every non-zero ordinal γ.

In (2), we use the technique of elementary submodels.     

Erratum to “Normality and countable paracompactness of hyperspaces of ordinals” [Topology Appl. 154 (2007) 358-362]

We correct the proof of Theorem 8 in “Normality and countable paracompactness of hyperspaces of ordinals” [Topology Appl. 154 (2007) 358-362].     

Preservation theorems for Kripke models

There are several ways for defining the notion submodel for Kripke models of intuitionistic first‐order logic. In our approach a Kripke model A is a submodel of a Kripke model B if they have the same frame and for each two corresponding worlds A^α and B^α of them, A^α is a subset of B^α and forcing of atomic formulas with parameters in the smaller one, in A and B, are the same. In this case, B is called an extension of A. We characterize theories that are preserved under taking submodels and also those that are preserved under taking extensions as universal and existential theories, respectively. We also study the notion elementary submodel defined in the same style and give some results concerning this notion. In particular, we prove that the relation between each two corresponding worlds of finite Kripke models A ≤ B is elementary extension (in the classical sense) (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Model-Theoretic Property of Sharply Bounded Formulae,with some Applications

We define a property of substructures of models of arithmetic, that of being length-initial, and show that sharply bounded formulae are absolute between a model and its length-initial submodels. We use this to prove independence results for some weak fragments of bounded arithmetic by constructing appropriate models as length-initial submodels of some given model.     

On the Number of Elementary Submodels of an Unsuperstable Homogeneous Structure

We show that if M is a stable unsuperstable homogeneous structure, then for most κ ? |M|, the number of elementary submodels of M of power κ is 2^κ.     

Homogeneity in powers of zero-dimensional first-countable spaces

A construction of L. Brian Lawrence is extended to show that the -power of every subset of the Cantor set is homogeneous via a continuous translation modulo a dense set. It follows that every zero-dimensional first-countable space has a homogeneous -power.

     

On Nonstructure of Elementary Submodels of an Unsuperstable Homogeneous Structure

In the first part of this paper we let M be a stable homogeneous model and we prove a nonstructure theorem for the class of all elementary submodels of M, assuming that M is ‘unsuperstable’ and has Skolem functions. In the second part we assume that M is an unstable homogeneous model of large cardinality and we prove a nonstructure theorem for the class of all elementary submodels of M.     

A Remark on Independence Results for Sharply Bounded Arithmetic

The purpose of this note is to show that the independence results for sharply bounded arithmetic of Takeuti [4] and Tada and Tatsuta [3] can be obtained and, in case of the latter, improved by the model-theoretic method developed by the author in [2].