We show that Shoenfield's functional interpretation of Peano arithmetic can be factorized as a negative translation due to J. L. Krivine followed by Gödel's Dialectica interpretation. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim).

We show that if κ is an infinite successor cardinal, and λ > κ a cardinal of cofinality less than κ satisfying certain conditions, then no ideal on Pκ is weakly λ+-saturated. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

In this paper we study the logical strength of the determinacy of infinite binary games in terms of second order arithmetic. We define new determinacy schemata inspired by the Wadge classes of Polish spaces and show the following equivalences over the system RCA0*, which consists of the axioms of discrete ordered semi‐rings with exponentiation, Δ10 comprehension and Π00 induction, and which is known as a weaker system than the popularbase theory RCA0: 1. Bisep(Δ10, Σ10)‐Det* ↔ WKL0, 2. Bisep(Δ10, Σ20)‐Det* ↔ (...) ATR0 + Σ11 induction, 3. Bisep(Σ10, Σ20)‐Det* ↔ Sep(Σ10, Σ20)‐Det* ↔ Π11‐CA0, 4. Bisep(Δ20, Σ20)‐Det* ↔ Π11‐TR0, where Det* stands for the determinacy of infinite games in the Cantor space (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim). (shrink)

The Necessary Maximality Principle for c. c. c. forcing with real parameters is equiconsistent with the existence of a weakly compact cardinal. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim).

We study sincere-strategy preference-based approval voting (SP-AV), a system proposed by Brams and Sanver [1] and here adjusted so as to coerce admissibility of the votes (rather than excluding inadmissible votes a priori), with respect to procedural control. In such control scenarios, an external agent seeks to change the outcome of an election via actions such as adding/deleting/partitioning either candidates or voters. SP-AV combines the voters' preference rankings with their approvals of candidates, where in elections with at least two candidates (...) the voters' approval strategies are adjusted – if needed – to approve of their most-preferred candidate and to disapprove of their least-preferred candidate. This rule coerces admissibility of the votes even in the presence of control actions, and hybridizes, in effect, approval with pluralitiy voting. We prove that this system is computationally resistant (i.e., the corresponding control problems are NP-hard) to 19 out of 22 types of constructive and destructive control. Thus, SP-AV has more resistances to control than is currently known for any other natural voting system with a polynomial-time winner problem. In particular, SP-AV is (after Copeland voting, see Faliszewski et al. [2, 3]) the second natural voting system with an easy winner-determination procedure that is known to have full resistance to constructive control, and unlike Copeland voting it in addition displays broad resistance to destructive control (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim). (shrink)

We characterize the sets of all Π2 and all equation image theorems of IΠ−1 in terms of restricted exponentiation, and use these characterizations to prove that both sets are not deductively equivalent. We also discuss how these results generalize to n > 0. As an application, we prove that a conservation theorem of Beklemishev stating that IΠ−n + 1 is conservative over IΣ−n with respect to equation image sentences cannot be extended to Πn + 2 sentences. © 2011 WILEY-VCH (...) class='Hi'>Verlag GmbH & Co. KGaA, Weinheim. (shrink)

We present a way of topologizing sets of Galois types over structures in abstract elementary classes with amalgamation. In the elementary case, the topologies thus produced refine the syntactic topologies familiar from first order logic. We exhibit a number of natural correspondences between the model-theoretic properties of classes and their constituent models and the topological properties of the associated spaces. Tameness of Galois types, in particular, emerges as a topological separation principle. © 2011 WILEY-VCH Verlag GmbH & Co. (...) KGaA, Weinheim. (shrink)

Schnorr randomness and computable randomness are natural concepts of random sequences. However van Lambalgen’s Theorem fails for both randomnesses. In this paper we define truth-table Schnorr randomness and truth-table reducible randomness, for which we prove that van Lambalgen's Theorem holds. We also show that the classes of truth-table Schnorr random reals relative to a high set contain reals Turing equivalent to the high set. It follows that each high Schnorr random real is half of a real for which van Lambalgen's (...) Theorem fails. Moreover we establish the coincidence between triviality and lowness notions for truth-table Schnorr randomness. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. (shrink)

Attempts to extend the classical Hausdorff difference hierarchy to the case of partitions of a space to k > 2 subsets lead to non‐equivalent notions. In a hope to identify the “right” extension we consider the extensions appeared in the literature so far: the limit‐, level‐, Boolean and Wadge hierarchies of k ‐partitions. The advantages and disadvantages of the four hierarchies are discussed. The main technical contribution of this paper is a complete characterization of the Wadge degrees of Δ02‐measurable k (...) ‐partitions of the Baire space. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim). (shrink)

The aim of this paper is to determine the logical and computational strength of instances of the Bolzano-Weierstraß principle and a weak variant of it.We show that BW is instance-wise equivalent to the weak König’s lemma for Σ01-trees . This means that from every bounded sequence of reals one can compute an infinite Σ01-0/1-tree, such that each infinite branch of it yields an accumulation point and vice versa. Especially, this shows that the degrees d ≫ 0′ are exactly those containing (...) an accumulation point for all bounded computable sequences.Let BWweak be the principle stating that every bounded sequence of real numbers contains a Cauchy subsequence . We show that BWweak is instance-wise equivalent to the cohesive principle and—using this—obtain a classification of the computational and logical strength of BWweak. Especially we show that BWweak does not solve the halting problem and does not lead to more than primitive recursive growth. Therefore it is strictly weaker than BW. We also discuss possible uses of BWweak. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. (shrink)

This article provides a detailed comparison between two systems of collapsing functions. These functions play a crucial role in proof theory, in the analysis of patterns of resemblance, and the analysis of maximal order types of well partial orders. The exact correspondence given here serves as a starting point for far reaching extensions of current results on patterns and well partial orders. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

We compute here the Borel complexity of the relation of isometry between separable Banach spaces, using results of Gao, Kechris [2], Mayer-Wolf [5], and Weaver [8]. We show that this relation is Borel bireducible to the universal relation for Borel actions of Polish groups. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim).

This research is motivated by the program of reverse mathematics and non-standard arguments in second-order arithmetic. Within a weak subsystem of second-order arithmetic ACA0, we investigate some aspects of non-standard analysis related to sequential compactness. Then, using arguments of non-standard analysis, we show the equivalence of the Riemann mapping theorem and ACA0 over WKL0. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim).

We develop a new method for coding sets while preserving GCH in the presence of large cardinals, particularly supercompact cardinals. We will use the number of normal measures carried by a measurable cardinal as an oracle, and therefore, in order to code a subset A of κ, we require that our model contain κ many measurable cardinals above κ. Additionally we will describe some of the applications of this result. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

Let φ1 stand for the statement V = HOD and φ2 stand for the Ground Axiom. Suppose Ti for i = 1, …, 4 are the theories “ZFC + φ1 + φ2,” “ZFC + ¬φ1 + φ2,” “ZFC + φ1 + ¬φ2,” and “ZFC + ¬φ1 + ¬φ2” respectively. We show that if κ is indestructibly supercompact and λ > κ is inaccessible, then for i = 1, …, 4, Ai = df{δ κ is inaccessible. We show it is also (...) the case that if κ is indestructibly supercompact, then Vκ⊨T1, so by reflection, B1 = df{δ κ is inaccessible, we demonstrate that it is possible to construct a model in which κ is indestructibly supercompact and for every inaccessible cardinal δ < κ, Vδ⊨T1. It is thus not possible to prove in ZFC that Bi = df{δ < κ∣δ is an inaccessible limit of inaccessible cardinals and Vδ⊨Ti} for i = 2, …, 4 is unbounded in κ if κ is indestructibly supercompact. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. (shrink)

We study constructive set theories, which deal with operations applying both to sets and operations themselves. Our starting point is a fully explicit, finitely axiomatized system ESTE of constructive sets and operations, which was shown in 10 to be as strong as PA. In this paper we consider extensions with operations, which internally represent description operators, unbounded set quantifiers and local fixed point operators. We investigate the proof theoretic strength of the resulting systems, which turn out to be impredicative . (...) © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. (shrink)

In this article, we introduce the notion of weakly measurable cardinal, a new large cardinal concept obtained by weakening the familiar concept of a measurable cardinal. Specifically, a cardinal κ is weakly measurable if for any collection equation image containing at most κ+ many subsets of κ, there exists a nonprincipal κ-complete filter on κ measuring all sets in equation image. Every measurable cardinal is weakly measurable, but a weakly measurable cardinal need not be measurable. Moreover, while the GCH cannot (...) fail first at a measurable cardinal, I will show that it can fail first at a weakly measurable cardinal. More generally, if κ is measurable, then we can make its weak measurability indestructible by the forcing Add for any η while forcing the GCH to hold below κ. Nevertheless, I shall prove that weakly measurable cardinals and measurable cardinals are equiconsistent. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. (shrink)

Built on the foundations laid by Peirce, Schröder, and others in the 19th century, the modern development of relation algebras started with the work of Tarski and his colleagues [21, 22]. They showed that relation algebras can capture strong first‐order theories like ZFC, and so their equational theory is undecidable. The less expressive class WA of weakly associative relation algebras was introduced by Maddux [7]. Németi [16] showed that WA's have a decidable universal theory. There has been extensive research on (...) increasing the expressive power of WA by adding new operations [1, 4, 11, 13, 20]. Extensions of this kind usually also have decidable universal theories. Here we give an example – extending WA's with set‐theoretic projection elements – where this is not the case. These “logical” connectives are set‐theoretic counterparts of the axiomatic quasi‐projections that have been investigated in the representation theory of relation algebras [22, 6, 19]. We prove that the quasi‐equational theory of the extended class PWA is not recursively enumerable. By adding the difference operator D one can turn WA and PWA to discriminator classes where each universal formula is equivalent to some equation. Hence our result implies that the projections turn the decidable equational theory of “WA + D ” to non‐recursively enumerable (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim). (shrink)

In this paper we prove that the equational class generated by bounded BCK-algebras is the variety generated by the class of finite simple bounded BCK-algebras. To obtain these results we prove that every simple algebra in the equational class generated by bounded BCK-algebras is also a relatively simple bounded BCK-algebra. Moreover, we show that every simple bounded BCK-algebra can be embedded into a simple integral commutative bounded residuated lattice. We extend our main results to some richer subreducts of the class (...) of integral commutative bounded residuated lattices and to the involutive case. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim). (shrink)

The Craig interpolation theorem is shown for an extended LJ with strong negation. A new simple proof of this theorem is obtained. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

Orthogonality of all families of pairwise weakly orthogonal 1-types for ℵ0-categorical weakly o-minimal theories of finite convexity rank has been proved in 6. Here we prove orthogonality of all such families for binary 1-types in an arbitrary ℵ0-categorical weakly o-minimal theory and give an extended criterion for binarity of ℵ0-categorical weakly o-minimal theories . © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

Walter Reese-Schäfer, Karl-Otto Apel, Zur Einführung (with an Afterword by Jürgen Habermas), Junis Verlag GmbH, Hamburg 1990, 176pp. DM 17.80 -/- The author, presently a freelance writer published in the newspaper “Die Zeit” and the magazine “Stern,” pro­vides in this small book a clear and concise introduction to sources, themes and conclusions in the philosophy of Karl-Otto Apel. Apel, Emeritus Pro­fessor at Frank­furt, and close colleague of Habermas, characterizes his viewpoint as a “transcen­dental pragmatism” in which a Kantian (...) concern for question regarding “the conditions for the possi­bility of some­thing,” (p.10) mixes with deontological discourse-ethics, semeiotic themes from Peirce, an ap­proach to fallibilism, the demand for “final justifica­tions” (Letzt­begründung) and German her­meneutics. In view of the “den­sity and concentration” of Apel’s texts, which often have a “deterrent effect” upon those not already at home with “the phil­osophical language game,” it is the announced aim of this book to provide a work of translation and clarification of Apel’s special­ized efforts --in effect an orientation to Apel’s work. The book divides into an Introduction, 8 chapters and a summary conclusion --with Habermas’ appreciation, “A Master Builder with Hermeneutic Feeling --The Way of Philosopher Karl-Otto Apel,” bringing up the rear. There is also a bibliography of primary and secondary sources and a short table outlining highlights of Apel’s career. The “Introduction” provides a good overview of the aims of the book and gets one started on central themes. (shrink)

The purpose of this paper is to survey the correspondence between bounded arithmetic and propositional proof systems. In addition, it also contains some new results which have appeared as an extended abstract in the proceedings of the conference TAMC 2008 [11].Bounded arithmetic is closely related to propositional proof systems; this relation has found many fruitful applications. The aim of this paper is to explain and develop the general correspondence between propositional proof systems and arithmetic theories, as introduced by Krajíček and (...) Pudlák [46]. Instead of focusing on the relation between particular proof systems and theories, we favour a general axiomatic approach to this correspondence. In the course of the development we particularly highlight the role played by logical closure properties of propositional proof systems, thereby obtaining a characterization of extensions of EF in terms of a simple combination of these closure properties (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim). (shrink)

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 Aα and Bα of them, Aα is a subset of Bα 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). (shrink)

We prove two embedding and extension theorems in the context of the constructive theory of metric spaces. The first states that Cantor space embeds in any inhabited complete separable metric space (CSM) without isolated points, X, in such a way that every sequentially continuous function from Cantor space to ℤ extends to a sequentially continuous function from X to ℝ. The second asserts an analogous property for Baire space relative to any inhabited locally non‐compact CSM. Both results rely on having (...) careful constructive formulations of the concepts involved.As a first application, we derive new relationships between “continuity principles” asserting that all functions between specified metric spaces are pointwise continuous. In particular, we give conditions that imply the failure of the continuity principle “all functions from X to ℝ are continuous”, when X is an inhabited CSM without isolated points, and when X is an inhabited locally non‐compact CSM. One situation in which the latter case applies is in models based on “domain realizability”, in which the failure of the continuity principle for any inhabited locally non‐compact CSM, X, generalizes a result previously obtained by Escardó and Streicher in the special case X = C[0, 1].As a second application, we show that, when the notion of inhabited complete separable metric space without isolated points is interpreted in a recursion‐theoretic setting, then, for any such space X, there exists a Banach‐Mazur computable function from X to the computable real numbers that is not Markov computable. This generalizes a result obtained by Hertling in the special case that X is the space of computable real numbers. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim). (shrink)

A notion of resource‐bounded Baire category is developed for the classPC[0,1]of all polynomial‐time computable real‐valued functions on the unit interval. The meager subsets ofPC[0,1]are characterized in terms of resource‐bounded Banach‐Mazur games. This characterization is used to prove that, in the sense of Baire category, almost every function inPC[0,1]is nowhere differentiable. This is a complexity‐theoretic extension of the analogous classical result that Banach proved for the classC[0, 1] in 1931. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim).

We present some characterizations of the members of Δta2, that class of the topological arithmetical hierarchy which is just large enough to include several fundamental types of sets of points in Euclidean spaces ℝk. The limit characterization serves as a basic tool in further investigations. The characterization by effective difference chains of effectively exhaustible sets yields only a hierarchy within a subfield of Δta2. Effective difference chains of transfinite (but constructive) order types, consisting of complements of effectively exhaustible sets, as (...) well as another closely related concept, yield a rich hierarchy within the whole class Δta2. The presentation always first reports analogies between Hausdorff's difference hierarchy within the Borel class ΔB2 and Ershov's hierarchy within the class Δ02 of the arithmetical hierarchy; after that the counterparts for Δta2 are developed. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim). (shrink)

We prove that the finite‐model version of arithmetic with the divisibility relation is undecidable (more precisely, it has Π01‐complete set of theorems). Additionally we prove FM‐representability theorem for this class of finite models. This means that a relation R on natural numbers can be described correctly on each input on almost all finite divisibility models if and only if R is of degree ≤0′. We obtain these results by interpreting addition and multiplication on initial segments of finite models with divisibility (...) only. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim). (shrink)