Sticks and clubs

We study combinatorial principles known as stick and club. Several variants of these principles and cardinal invariants connected to them are also considered. We introduce a new kind of side by-side product of partial orderings which we call pseudo-product. Using such products, we give several generic extensions where some of these principles hold together with ¬CH and Martin's axiom for countable p.o.-sets. An iterative version of the pseudo-product is used under an inaccessible cardinal to show the consistency of the club principle for every stationary subset of limits of ω₁ together with ¬CH and Martin's axiom for countable p.o.-sets.     

Elementary inductive dichotomy: Separation of open and clopen determinacies with infinite alternatives

We introduce a new axiom called inductive dichotomy, a weak variant of the axiom of inductive definition, and analyze the relationships with other variants of inductive definition and with related axioms, in the general second order framework, including second order arithmetic, second order set theory and higher order arithmetic. By applying these results to the investigations on the determinacy axioms, we show the following. (i) Clopen determinacy is consistency-wise strictly weaker than open determinacy in these frameworks, except second order arithmetic; this is an enhancement of Schweber–Hachtman separation of open and clopen determinacy into the consistency-wise separation. (ii) Hausdorff–Kuratowski hierarchy of differences of opens is faithfully reflected by the hierarchy of consistency strengths of corresponding parameter-free determinacies in the aforementioned frameworks; this result is valid also in second order arithmetic only except clopen determinacy.     

The Axiom of Choice in Second-Order Predicate Logic

The present article deals with the power of the axiom of choice (AC) within the second-order predicate logic. We investigate the relationship between several variants of AC and some other statements, known as equivalent to AC within the set theory of Zermelo and Fraenkel with atoms, in Henkin models of the one-sorted second-order predicate logic with identity without operation variables. The construction of models follows the ideas of Fraenkel and Mostowski. It is e. g. shown that the well-ordering theorem for unary predicates is independent from AC for binary predicates and from the trichotomy law for unary predicates. Moreover, we show that the AC for binary predicates follows neither from the trichotomy law for unary predicates nor from Zorn's lemma for unary predicates nor from the formalization of the axiom of choice for disjoint families of sets for binary predicates, and that the trichotomy law for unary predicates does not follow from AC for binary predicates. Mathematics Subject Classification: 03B15, 03E25, 04A25.     

L-拓扑空间的新U0公理

在L-拓扑空间引入了一种新的U0公理,它被叫做次U0公理。次U0公理蕴涵次T0公理。次U0公理是一般拓扑中U0公理的好的扩张。另外,L-实直线和L-单位区间满足次U0公理。     

A general axiomatic system for image resolution quantification

In this paper, we generalize our previously published axiom system for quantification of image resolution and prove that any resolution measure consistent with the new axiom system must be a homogeneous symmetric function of order 1/2 of the eigenvalues of the covariance matrix of the PSF. We demonstrate that the previous axiom system is not consistent with the affine transformation axiom. We propose a weak combination axiom to replace the previous strong combination axiom and use it to solve this conflict. It is remarkable that the original finding in one-dimension by Wang and Li can be easily rediscovered with aid of the weak combination axiom, instead of using the previous strong combination axiom. If the previous axiom system is modified with the weak combination axiom and augmented with the affine transformation axiom, the resolution measure is shown to be proportional to the squared root of the geometric mean of the eigenvalues of the covariance matrix of the PSF. Relevant discussions and possible extensions are also provided.     

The equivalence of Side-Angle-Side and Side-Angle-Angle in the absolute plane

Following the approach of G.D. Birkhoff, we develop a new axiom system for a continuous absolute plane. In this new axiom system we remove Side-Angle-Side as an axiom and replace it with Side-Angle-Angle as a new axiom. We prove that the new axiom system is also a continuous absolute plane, and in particular, that Side-Angle-Side holds in the new axiom system. In addition, we give new proofs of well known results. These new proofs do not depend on Side-Angle-Side, but instead use Side-Angle-Angle.     

Jump Theorems for REA Operators

In [2], Jockusch and Shore have introduced a new hierarchy of sets and operators called the REA hierarchy. In this note we prove analogues of the Friedberg Jump Theorem and the Sacks Jump Theorem for many REA operators. MSC: 03D25, 03D55.     

Aristotle's axiom in the foundations of geometry

In the foundations of non-Euclidean geometry without Dedekind's axiom, Archimedes' axiom suffices to insure that the geometry is hyperbolic, but this axiom is not necessary. The weaker axiom of Aristotle is necessary and sufficient; it uses only geometric variables, not integer variables.     

The Shapley value without efficiency and additivity

We provide a new characterization of the Shapley value neither using the efficiency axiom nor the additivity axiom. In this characterization, efficiency is replaced by the gain-loss axiom (Einy and Haimanko, 2011), i.e., whenever the total worth generated does not change, a player can only gain at the expense of another one. Additivity and the equal treatment axiom are substituted by fairness (van den Brink, 2001) or differential marginality (Casajus, 2011), where the latter requires equal productivity differentials of two players to translate into equal payoff differentials. The third axiom of our characterization is the standard dummy player axiom.     

H(λ)-completely Hausdorff axiom on L-topological spaces

This paper defines the new concept of completely Hausdorff axiom of an L-topological space by means of L-continuous mappings from an L-topological space to the refined Hutton's unit L-interval by Wang. Some characterizations of the completely Hausdorff axiom, defined in this paper, are given, and many nice properties of this kind of completely Hausdorff axiom are proved. For example, it is hereditary and product invariant; the refined Hutton's unit L-interval satisfy this kind of completely Hausdorff axiom, and when an L-topological space satisfy this kind of completely Hausdorff axiom, every f-convergent ideal does not have f-limit points with different supports etc. The relation between the completely Hausdorff axiom defined in the paper and other separation axioms is discussed also.     

The Generalized Axiom of Spheres

Some generalization is proposed for the axiom of spheres. A collection of Riemannian spaces is constructed which satisfy the generalized axiom of spheres, but do not satisfy the earlier-known axioms of submanifolds. The structure is found of the curvature tensor of manifolds satisfying the generalized axiom of spheres. This structure mostly resembles the structure of the curvature tensor of manifolds with the generalized axiom of planes.     

Dependent Choices and Anti‐Foundation

In Zermelo‐Fraenkel set theory without the Axiom of Foundation we study the schema version of the principle of dependent choices in connection with Aczel's antifoundation axiom (a part of it), Boffa's anti‐foundation axiom, and axiom of collection.     

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.     

基于半范数的风险测度

由于一致性风险测度公理中的平移不变性公理存在不合理性,故可将该该公理从一致性风险测度公理中去掉.将半范数的概念加以扩展,增加单调性要求,则去掉平移不变性公理之后一致性风险测度公理与半范数的要求就完全相同,这样风险度量从本质上讲就是定义在某空间上的半范数.本文发现F ishburn的风险测度是满足正齐次性、次可加性、单调性要求的.从这个意义讲,F ishburn的风险测度是一个比较科学的风险度量方法.     

关于三角范畴的八面体公理

三角范畴是一个带有自同构且满足四条公理的加法范畴,其中的一条重要公理就是八面体公理,该公理形式复杂不易理解难以应用.在本文中,作者讨论了三角范畴定义中八面体公理的几个等价命题,给出了新的八面体公理的等价命题,证明了各个公理间的相互等价关系,同时简化了八面体的表达形式,并且给出了该定理的一个具体应用.     

Weakly ordered plane geometry

A plane geometry is introduced, which although ordered such that both every line is densely linearly ordered without first or last element and such that every line partitions the points of the plane not incident with it into two convex half-planes, does not need to satisfy the Pasch axiom. An alternate axiomatization of ordered planes is obtained by adding several axioms, including the inner form of the Pasch axiom. The missing link between the inner form of Pasch’s axiom and the full Pasch axiom is herewith determined.     

公理信息论

本文阐述了公理信息论的产生 ,并且给出公理信息论的公理系统     

Ternary Operations as Primitive Notions for Constructive Plane Geometry V

In this paper we provide a quantifier-free, constructive axiomatization of metric-Euclidean and of rectangular planes (generalizations of Euclidean planes). The languages in which the axiom systems are expressed contain three individual constants and two ternary operations. We also provide an axiom system in algorithmic logic for finite Euclidean planes, and for several minimal metric-Euclidean planes. The axiom systems proposed will be used in a sequel to this paper to provide ‘the simplest possible’ axiom systems for several fragments of plane Euclidean geometry. Mathematics Subject Classification: 51M05, 51M15, 03F65.     

Parameterized fairness axioms on cycle-free graph games

We study cooperative transferable utility games with a communication structure represented by an undirected graph, i.e., a group of players can cooperate only if they are connected on the graph. This type of games is called graph games and the best-known solution for them is the Myerson value, which is characterized by the component efficiency axiom and the fairness axiom. Recently the average tree solution has been proposed on cycle-free graph games, and shown to be characterized by the component efficiency axiom and the component fairness axiom. We propose e{\epsilon} -parameterized fairness axiom on cycle-free graph games that incorporates the preceding fairness axioms, and show the existence and the uniqueness of the solution. We then discuss a relationship between the existing and our proposed solutions by a numerical example.     

The power of some forms of the induction axiom in the multiplicative arithmetic

In this paper, the sequential variants of the multiplicative arithmetic with the below-defined formsAIO,I ⁺, andI ⁻ of the induction axiom are investigated, and some relations between the classes of theorems in these systems are determined. In addition, the multiplicative systems with these induction axioms, for which the classes of derivable formulas are equivalent, are presented. Published in Lietuvos Matematikos Rinkinys, Vol. 40, No. 1, pp. 36–47, January–March, 2000.