Search

It is known that there is a close relation between Prikry forcing and the iteration of ultrapowers: If U is a normal ultrafilter on a measurable cardinal κ and 〈Mn, jm,n | m ≤ n ≤ ω〉 is the iteration of ultrapowers of V by U, then the sequence of critical points 〈j0,n(κ) | n ∈ ω〉 is a Prikry generic sequence over Mω. In this paper we generalize this for normal precipitous filters. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]

There are two versions of the Proper Iteration Lemma. The stronger (but less well-known) version can be used to give simpler proofs of iteration theorems (e.g., [7, Lemma 24] versus [9, Theorem IX.4.7]). In this paper we give another demonstration of the fecundity of the stronger version by giving a short proof of Shelah's theorem on the preservation of the ωω-bounding property. (© 2003 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]

In [4] the authors studied the residue field of a model M of IΔ0 + Ω1 for the principal ideal generated by a prime p. One of the main results is that M/ has a unique extension of each finite degree. In this paper we are interested in understanding the structure of any quotient field of M, i.e. we will study the quotient M/I for I a maximal ideal of M. We prove that any quotient field of M satisfies the property of having a unique extension of each finite degree. We will use some of Cherlin's ideas from [3], where he studies the ideal theory of non standard algebraic integers. [ABSTRACT FROM AUTHOR]

In this paper we define the polyadic Pavelka algebras as algebraic structures for Rational Pavelka predicate calculus (RPL∀). We prove two representation theorems which are the algebraic counterpart of the completness theorem for RPL∀. [ABSTRACT FROM AUTHOR]

In [4] the authors studied the residue field of a model M of IΔ0 + Ω1 for the principal ideal generated by a prime p. One of the main results is that M/ has a unique extension of each finite degree. In this paper we are interested in understanding the structure of any quotient field of M, i.e. we will study the quotient M/I for I a maximal ideal of M. We prove that any quotient field of M satisfies the property of having a unique extension of each finite degree. We will use some of Cherlin's ideas from [3], where he studies the ideal theory of non standard algebraic integers. [ABSTRACT FROM AUTHOR]

For any ordinal δ, let λδ be the least inaccessible cardinal above δ. We force and construct a model in which the least supercompact cardinal κ is indestructible under κ-directed closed forcing and in which every measurable cardinal δ < κ is < λδ strongly compact and has its < λδ strong compactness indestructible under δ-directed closed forcing of rank less than λδ. In this model, κ is also the least strongly compact cardinal. We also establish versions of this result in which κ is the least strongly compact cardinal but is not supercompact. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]

A spatial modal logic (SML) is introduced as an extension of the modal logic S4 with the addition of certain spatial operators. A sound and complete Kripke semantics with a natural space (or location) interpretation is obtained for SML. The finite model property with respect to the semantics for SML and the cut-elimination theorem for a modified subsystem of SML are also presented. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]

Many existence propositions in constructive analysis are implied by the lesser limited principle of omniscience LLPO; sometimes one can even show equivalence. It was discovered recently that some existence propositions are equivalent to Bouwer's fan theorem FAN if one additionally assumes that there exists at most one object with the desired property. We are providing a list of conditions being equivalent to FAN, such as a unique version of weak König's lemma. This illuminates the relation between FAN and LLPO. Furthermore, we give a short and elementary proof of the fact that FAN is equivalent to each positive valued function with compact domain having positive infimum. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]

We prove the Banach-Steinhaus theorem for distributions on the space &Dscr;(ℝ) within Bishop's constructive mathematics. To this end, we investigate the constructive sequential completion $ \tilde {\cal D} $(ℝ) of &Dscr;(ℝ). [ABSTRACT FROM AUTHOR]

The spectral mapping theorem in a unital Banach algebra is examined for its constructive content. [ABSTRACT FROM AUTHOR]

We shall investigate certain set-theoretic pigeonhole principles which arise as generalizations of the usual (finitary) pigeonhole principle; and we shall show that many of them are equivalent to full AC. We discuss also several restricted cases and variations of those principles and relate them to restricted choice principles. In this sense the pigeonhole principle is a rich source of weak choice principles. It is shown that certain sequences of restricted pigeonhole principles form implicational hierarchies with respect to ZF. We state also several open problems in order to indicate the extent to which the whole subject of pigeonhole principles requires further exploration. [ABSTRACT FROM AUTHOR]

The article discusses the interpolation by a game, which is a generalisation of the Karchmer-Wigderson game and of real communication complexity. It provides an overview of an exponential lower bound for tree-like monotone protocols of small real communication complexity solving the monotone communication complexity problem associated with the bipartite perfect matching problem. It also obtains an effective interpolation in a form of a protocol of small real communication complexity.

The article discusses the model of subsystems for the theory of New Foundations (NF) that are inspired by and have models in the Zermelo-Fraenkel (ZF) set theory. It relates that NF theory is a finitely axiomatizable concept that was introduced by William Van Oman Quine in 1937. It mentions that the subsystem called K that was used in NF theory will be presented to satisfy typical properties of ZF.

The article focuses the equation that aims to determine the structure of stone spaces and the TD separation axiom. It outlines the useful characteristics of T0 -spaces in defining an order structure in topology and investigates the class of Stone spaces that contain all Boolean and finite structures. It outlines the solution of the equation, which indicates the Stone space of the Boolean algebra of all finite and cofinite subsets of the elements in topological space.

The article offers information on the elementary theory related to real numbers. It offers information on the advantages of real numbers in numerical analysis. It mentions that the axioms are used in the first-order (elementary) theory. It presents information on the Tarski elimination procedure also called the Tarski- Seidenberg results related to algebraic geometry.

The article focuses on the equation that proves the Barwise completeness theorem for the class of biprobability models. It outlines the equation and formula that aim to introduce the weak model of the logic via Henkin type construction to prove the existence of the property and graded model. It also mentions the results, which can be obtained for other kinds of biprobablity models that can also be added with one more relation symbol to the language.