Interpretations of the alternative set theory

Summary We show an axiom A such that there is no nontrivial interpretation of the alternative set theory (AST) inAST+A keeping , sets and the class of all standard natural numbers. Furthermore, there is no interpretation ofAST inAST without the prolongation axiom, but there is an interpretation ofAST in the theory having the prolongation axiom and the basic set-theoretical axioms only.     

On the axiom of extensionality in the positive set theory

This is a study of the relative interpretability of the axiom of extensionality in the positive set theory. This work has to be considered in the line of works of R. O. Gandy, D. Scott and R. Hinnion who have studied the relative interpretability of the axiom of extensionality in set theories of Zermelo and Zermelo‐Fraenkel.     

The rigid relation principle,a new weak choice principle

The rigid relation principle, introduced in this article, asserts that every set admits a rigid binary relation. This follows from the axiom of choice, because well‐orders are rigid, but we prove that it is neither equivalent to the axiom of choice nor provable in Zermelo‐Fraenkel set theory without the axiom of choice. Thus, it is a new weak choice principle. Nevertheless, the restriction of the principle to sets of reals (among other general instances) is provable without the axiom of choice.     

A set of axioms for nonstandard extensions

We give an axiomatic characterization for complete elementary extensions, that is, elementary extensions of the first‐order structure consisting of all finitary relations and functions on the underlying set. Such axiom systems have been studied using various types of primitive notions (e.g., [1, 3, 6]). Our system uses the notion of partial functions as primitive. Properties of nonstandard extensions are derived from five axioms in a rather algebraic way, without the use of metamathematical notions such as formulas or satisfaction. For example, when applied to the real number system, it provides a complete framework for developing nonstandard analysis based on hyperreals without having to construct them and without any use of logic. This has possible pedagogical and expository applications as presented in, e.g., [5], [6]. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Biprobability logic with conditional expectation

This paper is devoted to fill the gap in studying logics for biprobability structures. We introduce the logic $L_{\mathbb AE_1E_2}^a$ with two conditional expectation operators and prove the completeness theorem. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Non‐well‐founded extensions of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbf {V}$\end{document}

This paper describes a new and user‐friendly method for constructing models of non‐well‐founded set theory. Given a sufficiently well‐behaved system θ of non‐well‐founded set‐theoretic equations, we describe how to construct a model M_θ for $\mathsf {ZFC}^-$ in which θ has a non‐degenerate solution. We shall prove that this M_θ is the smallest model for $\mathsf {ZFC}^-$ which contains $\mathbf {V}$ and has a non‐degenerate solution of θ.     

Computability in structures representing a Scott set

Continuing work begun in [10], we utilize a notion of forcing for which the generic objects are structures and which allows us to determine whether these “generic” structures compute certain sets and enumerations. The forcing conditions are bounded complexity types which are consistent with a given theory and are elements of a given Scott set. These generic structures will “represent” this given Scott set, in the sense that the structure has a certain weak saturation property with respect to bounded complexity types in the Scott set. For example, if ? is a nonstandard model of PA, then ? represents the Scott set ? = n∈ω | ?⊧“the nth prime divides a” | a∈?. The notion of forcing yields two main results. The first characterizes the sets of natural numbers computable in all models of a given theory representing a given Scott set. We show that the characteristic function of such a set must be enumeration reducible to a complete existential type which is consistent with the given theory and is an element of the given Scott set. The second provides a sufficient condition for the existence of a structure ? such that ? represents a countable jump ideal and ? does not compute an enumeration of a given family of sets ?. This second result is of particular interest when the family of sets which cannot be enumerated is ? = Rep[Th(?)]. Under this additional assumption, the second result generalizes a result on TA [6] and on certain other completions of PA [10]. For example, we show that there also exist models of completions of ZF from which one cannot enumerate the family of sets represented by the theory. Received: 8 October 1997 / Published online: 25 January 2001     

Relative isoperimetric inequalities for minimal submanifolds outside a convex set

Let C be a closed convex set in a complete simply connected Riemannian manifold M with sectional curvature bounded above by a positive constant K. Assume that Σ is a compact minimal surface outside C such that Σ is orthogonal to ?C along ?Σ∩?C and ?Σ ～ ?C is radially connected from a point p ∈ ?Σ∩?C. We introduce a modified volume M_p(Σ) of Σ and obtain a sharp isoperimetric inequality where equality holds if and only if Σ is a geodesic half disk with constant Gaussian curvature K. We also prove higher dimensional isoperimetric inequalities for minimal submanifolds outside a closed convex set in a Riemannian manifold using the modified volume.     

Models of expansions of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathbb N}$\end{document} with no end extensions

We deal with models of Peano arithmetic (specifically with a question of Ali Enayat). The methods are from creature forcing. We find an expansion of ${\mathbb N}$ such that its theory has models with no (elementary) end extensions. In fact there is a Borel uncountable set of subsets of ${\mathbb N}$ such that expanding ${\mathbb N}$ by any uncountably many of them suffice. Also we find arithmetically closed ${\mathcal A}$ with no ultrafilter on it with suitable definability demand (related to being Ramsey). © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Coding into HOD via normal measures with some applications

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     

Sequence encoding without induction

We show that the universally axiomatized, induction‐free theory $\mathsf {PA}^-$ is a sequential theory in the sense of Pudlák's 5 , in contrast to the closely related Robinson's arithmetic.     

A note on parameter free Π1‐induction and restricted exponentiation

We characterize the sets of all Π₂ and all $\mathcal {B}(\Sigma _{1})We characterize the sets of all Π₂ and all $\mathcal {B}(\Sigma _{1})$ (= Boolean combinations of Σ₁) 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 $\mathcal {B}(\Sigma _{n+1})$ sentences cannot be extended to Π_{n + 2} sentences. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Turing cones and set theory of the reals

We investigate Turing cones as sets of reals, and look at the relationship between Turing cones, measures, Baire category and special sets of reals, using these methods to show that Martin's proof of Turing Determinacy (every determined Turing closed set contains a Turing cone or is disjoint from one) does not work when you replace “determined” with “Blackwell determined”. This answers a question of Tony Martin. Received: 6 December 1999 / Revised version: 28 June 2000 Published online: 3 October 2001     

The approachability ideal without a maximal set

We develop a forcing poset with finite conditions which adds a partial square sequence on a given stationary set, with adequate sets of models as side conditions. We then develop a kind of side condition product forcing for simultaneously adding partial square sequences on multiple stationary sets. We show that certain quotients of such forcings have the ${\omega }_1$-approximation property. We apply these ideas to prove, assuming the consistency of a greatly Mahlo cardinal, that it is consistent that the approachability ideal $I[{\omega }_2]$ does not have a maximal set modulo clubs.     

On quasi‐varieties of multiple valued logic models

We extend the concept of quasi‐variety of first‐order models from classical logic to multiple valued logic (MVL) and study the relationship between quasi‐varieties and existence of initial models in MVL. We define a concept of ‘Horn sentence’ in MVL and based upon our study of quasi‐varieties of MVL models we derive the existence of initial models for MVL ‘Horn theories’. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Indestructibility,HOD, and the Ground Axiom

Let φ₁ stand for the statement V = HOD and φ₂ stand for the Ground Axiom. Suppose T_i 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, A_i = _df{δ < κ∣δ is an inaccessible cardinal which is not a limit of inaccessible cardinals and V_δ?T_i} must be unbounded in κ. The large cardinal hypothesis on λ is necessary, as we further demonstrate by constructing via forcing four models in which A_i = ?? for i = 1, …, 4. In each of these models, there is an indestructibly supercompact cardinal κ, and no cardinal δ > κ is inaccessible. We show it is also the case that if κ is indestructibly supercompact, then V_κ?T₁, so by reflection, B₁ = _df{δ < κ∣δ is an inaccessible limit of inaccessible cardinals and V_δ?T₁} is unbounded in κ. Consequently, it is not possible to construct a model in which κ is indestructibly supercompact and B₁ = ??. On the other hand, assuming κ is supercompact and no cardinal 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, V_δ?T₁. It is thus not possible to prove in ZFC that B_i = _df{δ < κ∣δ is an inaccessible limit of inaccessible cardinals and V_δ?T_i} for i = 2, …, 4 is unbounded in κ if κ is indestructibly supercompact. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Weak saturation of ideals on Pκ(λ)

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 (proper, fine, κ‐complete) ideal on P_κ(λ) is weakly λ⁺‐saturated. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Sturmian theory for linear Hamiltonian systems without controllability

In this paper we present Sturmian separation and comparison theorems for linear Hamiltonian systems when no controllability assumption is imposed. This generalizes the traditional results of W. T. Reid for controllable (or normal) linear Hamiltonian systems to the possibly abnormal case. Our new theory is based on several recent results on linear Hamiltonian systems without controllability by W. Kratz, M. Wahrheit, V. Zeidan, and the author regarding the piecewise constant kernel for conjoined bases, the oscillation and eigenvalue theorems, and the Rayleigh principle. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

The Kripke schema in metric topology

A form of Kripke's schema turns out to be equivalent to each of the following two statements from metric topology: every open subspace of a separable metric space is separable; every open subset of a separable metric space is a countable union of open balls. Thus Kripke's schema serves as a point of reference for classifying theorems of classical mathematics within Bishop‐style constructive reverse mathematics.