An axiom system for the weak monadic second order theory of two successors

A compelte axiom system for the weak monadic second order theory of two successor functions, W2S, is presented. The axiom system consists, roughly, of the generalized Peano axioms and of an inductive definition of the finite sets. For the proof, methods of J. R. Buchi and J. Doner are used to obtain a new decision procedure for W2S, whose proofs are easily formalized. Different finiteness axioms are discussed. This paper was written while the author was visiting at Purdue University, and appeared first as Report CSD TR-56, Purdue University, 1971.     

Players indifferent to cooperate and characterizations of the Shapley value

In this paper we provide new axiomatizations of the Shapley value for TU-games using axioms that are based on relational aspects in the interactions among players. Some of these relational aspects, in particular the economic or social interest of each player in cooperating with each other, can be found embedded in the characteristic function. We define a particular relation among the players that it is based on mutual indifference. The first new axiom expresses that the payoffs of two players who are not indifferent to each other are affected in the same way if they become enemies and do not cooperate with each other anymore. The second new axiom expresses that the payoff of a player is not affected if players to whom it is indifferent leave the game. We show that the Shapley value is characterized by these two axioms together with the well-known efficiency axiom. Further, we show that another axiomatization of the Shapley value is obtained if we replace the second axiom and efficiency by the axiom which applies the efficiency condition to every class of indifferent players. Finally, we extend the previous results to the case of weighted Shapley values.     

On almost clopen continuity

Recently the class of clopen continuous functions between topological spaces has been generalized by the definition of the class of almost clopen continuous functions. The aim of this paper is to reconsider this second class of functions from the perspective of change of topology. Indeed, we show that the concept of almost clopen continuity coincides with the classical notion of continuity provided that suitable changes are made to the topologies of the domain and codomain of the function. We investigate some of the consequences of this situation.     

Discounting axioms imply risk neutrality

Although most applications of discounting occur in risky settings, the best-known axiomatic justifications are deterministic. This paper provides an axiomatic rationale for discounting in a stochastic framework. Consider a representation of time and risk preferences with a binary relation on a real vector space of vector-valued discrete-time stochastic processes on a probability space. Four axioms imply that there are unique discount factors such that preferences among stochastic processes correspond to preferences among present value random vectors. The familiar axioms are weak ordering, continuity and nontriviality. The fourth axiom, decomposition, is non-standard and key. These axioms and the converse of decomposition are assumed in previous axiomatic justifications for discounting with nonlinear intraperiod utility functions in deterministic frameworks. Thus, the results here provide the weakest known sufficient conditions for discounting in deterministic or stochastic settings. In addition to the four axioms, if there exists a von Neumann-Morgenstern utility function corresponding to the binary relation, then that function is risk neutral (i.e., affine). In this sense, discounting axioms imply risk neutrality.     

Weak values,the core,and new axioms for the Shapley value

An explicit formula for all linear symmetric values is established and is then used to show that, in the definition of the Shapley value, the efficiency and dummy axioms may be replaced by a projection axiom, a monotonicity axiom, and Roth's (1977) strategic risk neutrality axiom. It is also demonstrated that, for the set of super-additive games, there is no efficient linear symmetric value which is always in the core.     

Starkes und überstarkes Auswahlaxiom

It is well known that equivalence holds between the weak axiom of choice (AC) and the well ordering principle (WOP) for sets, resp. between strong AC and WOP for classes. It will be shown that in a theory PC^* with inpredicative classes (i. e. with no restriction of quantification in the defining formula) the super-strong AC used by the informally working mathematician is equivalent to a superstrong WOP. The equivalence between strong AC and super-strong AC is implied by a conditionC refutable in PC^* but provable in PC which is PC^* with predicative classes only and with the general ordered pair axiom. PC^* [super-strong AC] is inconsistent because the super-strong AC impliesC. Therefore the application of choice functions to non-empty classes generally makes a predicative definition of these classes necessary. Connected with these problems is a statement equivalent to the conjunction of the axioms of power set and foundation based on a function which coincides with the von Neumann-function under the assumption of one of the mentioned axioms.     

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.     

Sur l'existence de modèles du second ordre non dénombrables avec domaine du premier ordre dénombrable

We generalize to second order logic a result of Keisler concerning second order arithmetic. We prove that for any countable second order model, verifying certain axioms, there exists an elementary extension having the same domain of intrepretation for individuals and whose domain of interpretation for relations is uncountable. The axioms we ask for are the comprehension scheme, a choice scheme and a pairing scheme that allow us not to have explicitly a pairing function in the language.     

Complex analysis in subsystems of second order arithmetic

This research is motivated by the program of Reverse Mathematics. We investigate basic part of complex analysis within some weak subsystems of second order arithmetic, in order to determine what kind of set existence axioms are needed to prove theorems of basic analysis. We are especially concerned with Cauchy’s integral theorem. We show that a weak version of Cauchy’s integral theorem is proved in RCAo. Using this, we can prove that holomorphic functions are analytic in RCAo. On the other hand, we show that a full version of Cauchy’s integral theorem cannot be proved in RCAo but is equivalent to weak König’s lemma over RCAo.     

Nonsymmetric Nash solutions and replications of 2-person bargaining

A 2-person fixed threat bargaining problem is considered. A full characterization of the solutions which satisfy all of Nash's axioms except for the axiom of symmetry is given. It is also shown that these nonsymmetric Nash solutions are precisely the solutions that arise from symmetric Nash solutions through replications.     

Axiomatic characterizations of the weighted solidarity values

We define and characterize the class of all weighted solidarity values. Our first characterization employs the classical axioms determining the solidarity value (except symmetry), that is, efficiency, additivity and the A-null player axiom, and two new axioms called proportionality and strong individual rationality. In our second axiomatization, the additivity and the A-null player axioms are replaced by a new axiom called average marginality.     

命題演算的公理系統

<正> §1. 問題的提出 對於傳統的二值邏輯系統(以後叫做系統M)所作的公理系統,優點最多的可說是Hilbert-Bernays[1]Ⅰ册66頁上所载的(一名Munster派公理,以後即用此名).這個公理系統共有兩個模式(又名原則)及五组公理,模式即代入原則     

拓扑分子格的分离公理

在[1]中我们建立了拓扑分子格的理论,它既是古典的点集拓扑学的推广,又是晚近发展起来的Fuzzy拓扑学的推广,对于某些Fuzzy格L(如L是线性序集或L是分子格等),它也是L—Fuzzy拓扑学的推广。因此,凡在拓扑分子格中得到的结果自然都是上述各种拓扑学中相应定理的一般化形式。在本文中我们将讨论拓扑分子格的分离公理。 我们熟知点集拓扑学中的分离公理有多种不同的等价形式。以正则性为例,设X是拓扑空间,X叫正则的,当且仅当对每个点a∈X以及a的每个开邻域U,a有开邻域V满足条件V~-U。这一分离公理又可表述为:设a∈X,F是X中不包含a的闭集,则有开集P     

A new axiomatization of the Shapley–solidarity value for games with a coalition structure

In this paper, we propose a new kind of players as a compromise between the null player and the A-null player. It turns out that the axiom requiring this kind of players to get zero-payoff together with the well-known axioms of efficiency, additivity, coalitional symmetry, and intra-coalitional symmetry characterize the Shapley–solidarity value. This way, the difference between the Shapely–solidarity value and the Owen value is pinpointed to just one axiom.     

Finite axiomatizability of local set theory

The need for modifying axiomatic set theories was caused, in particular, by the development of category theory. The ZF and NBG axiomatic theories turned out to be unsuitable for defining the notion of a model of category theory. The point is that there are constructions such as the category of categories in naïve category theory, while constructions like the set of sets are strongly restricted in the ZF and NBG axiomatic theories. Thus, it was required, on the one hand, to restrict constructions similar to the category of categories and, on the other hand, adapt axiomatic set theory in order to give a definition of a category which survives restricted construction similar to the category of categories. This task was accomplished by promptly inventing the axiom of universality (AU) asserting that each set is an element of a universal set closed under all NBG constructions. Unfortunately, in the theories ZF + AU and NBG + AU, there are toomany universal sets (as many as the number of all ordinals), whereas to solve the problem stated above, a countable collection of universal sets would suffice. For this reason, in 2005, the first-named author introduced local-minimal set theory, which preserves the axiom AU of universality and has an at most countable collection of universal sets. This was achieved at the expense of rejecting the global replacement axiom and using the local replacement axiom for each universal class instead. Local-minimal set theory has 14 axioms and one axiom scheme (of comprehension). It is shown that this axiom scheme can be replaced by finitely many axioms that are special cases of the comprehension scheme. The proof follows Bernays’ scheme with significant modifications required by the presence of the restricted predicativity condition on the formula in the comprehension axiom scheme.     

Separation axioms and frame representation of some topological facts

Similarly as the sobriety is essential for representing continuous maps as frame homo-morphisms, also other separation axioms play a basic role in expressing topological phenomena in frame language. In particular,T _D is equivalent with the correctness of viewing subspaces as sublocates, or with representability of open or closed maps as open or closed homomorphisms. A weaker separation axiom is equivalent with an algebraic recognizability whether the intersection of a system of open sets remains open or not. The role of sobriety is also being analyzed in some detail.In honour of Nico Pumplün on the occasion of his 60th birthdayThe support of the Italian C.N.R. is gratefully acknowledged.Partial financial support of the Italian M.U.R.S.T. is gratefully acknowledged.     

Upper bounds on complexity of Frege proofs with limited use of certain schemata

The paper considers a commonly used axiomatization of the classical propositional logic and studies how different axiom schemata in this system contribute to proof complexity of the logic. The existence of a polynomial bound on proof complexity of every statement provable in this logic is a well-known open question. The axiomatization consists of three schemata. We show that any statement provable using unrestricted number of axioms from the first of the three schemata and polynomially-bounded in size set of axioms from the other schemata, has a polynomially-bounded proof complexity. In addition, it is also established, that any statement, provable using unrestricted number of axioms from the remaining two schemata and polynomially-bounded in size set of axioms from the first scheme, also has a polynomially-bounded proof complexity.