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.     

Taylor's theorem for a function of several variablesClassroom notes

In this note the measure problem for the Lebesgue measure is discussed in terms of metric space theory. It is illuminated that under the axiom of choice most of the subsets of [0, 1) with positive outer measure are non‐Lebesgue measurable. This fact is adequate to emphasize the significance of Lebesgue measurability as well as the essentiality of the axiom of choice.     

On the existence of product stochastic measures

In this paper, we prove that the existence of product stochastic measures depends on the axiom-system of set theory: If one accepts the axiom of choice, the answer is negative, and we give a counter-example where the product stochastic measure doesn't exist; but in the Solovay model (one kind of set theory which refuses the axiom of choice), the answer is positive, and we give a proof.     

Injective power objects and the axiom of choice

“The axiom of choice states that any set X of non-empty sets has a choice function—i.e. a function satisfying f(x)∈x for all x∈X. When we want to generalise this to a topos, we have to choose what we mean by non-empty, since in , the three concepts non-empty, inhabited, and injective are equivalent, so the axiom of choice can be thought of as any of the three statements made by replacing “non-empty” by one of these notions.It seems unnatural to use non-empty in an intuitionistic context, so the first interpretation to be used in topos theory was the notion based on inhabited objects. However, Diaconescu (1975) [1] showed that this interpretation implied the law of the excluded middle, and that without the law of the excluded middle, even the finite version of the axiom of choice does not hold! Nevertheless some people still view this as the most appropriate formulation of the axiom of choice in a topos.In this paper, we study the formulation based upon injective objects. We argue that it can be considered a more natural formulation of the axiom of choice in a topos, and that it does not have the undesirable consequences of the inhabited formulation. We show that if it holds for , then it holds in a wide variety of topoi, including all localic topoi. It also has some of the classical consequences of the axiom of choice, although a lot of classical results rely on both the axiom of choice and the law of the excluded middle. An additional advantage of this formulation is that it can be defined for a slightly more general class of categories than just topoi.We also examine the corresponding injective formulations of Zorn’s lemma and the well-order principle. The injective form of Zorn’s lemma is equivalent to the axiom of injective choice, and the injective well-order principle implies the axiom of injective choice.     

A Reconciliation Among Discrete Compromise Solutions

The application of compromise solutions to discrete multi-objective problems brings about some technical flexibilities, such as the selection of distance function for computing both normalized attribute ratings and distances between two alternatives, and the choice between the ideal and negative-ideal alternatives for implementing the axiom of choice. These flexibilities are undesirable, since the method may yield conflicting preference-alternative rankings, depending on parameter choice. This paper introduces a credibility measurement of distance function and takes a broader concept of the axiom of choice in order to reconcile disagreement among compromise solutions.     

Group preference aggregation in the multiplicative AHP The model of the group decision process and Pareto optimality

A recent paper has focused awareness on group aggregation procedures in the AHP, showing that geometric mean aggregation violates the desirable social choice axiom of Pareto optimality. We show that this violation can be attributed to the representation used to model the group decision process, thereby questioning the legitimacy of the Pareto optimality axiom. We furthermore propose a geometric mean group aggregation procedure which satisfies all the social choice axioms suggested.     

Some equivalents of AC in algebra II

We show that the axiom of choice AC is equivalent to the statement Any quotient group of any abelian group has a selector. We also show that the multiple choice axiom MC is equivalent to the assertion: Any filter in any Boolean ring has a well ordered filterbase. Received October 3, 1996; accepted in final form May 1, 1998.     

σ-kompakte Räume

The theorem, that -compact spaces are Lindelöf, is equivalent to the countable axiom of choice. Variants of this theorem are compared with weak versions of the axiom of choice.     

Sequential topological conditions in ℝ in the absence of the axiom of choice

It is known that – assuming the axiom of choice – for subsets A of ? the following hold: (a) A is compact iff it is sequentially compact, (b) A is complete iff it is closed in ?, (c) ? is a sequential space. We will show that these assertions are not provable in the absence of the axiom of choice, and that they are equivalent to each     

On the Consistency of a Positive Theory

In positive theories, we have an axiom scheme of comprehension for positive formulas. We study here the “generalized positive” theory GPK_∞⁺. Natural models of this theory are hyperuniverses. The author has shown in [2] that GPK_∞⁺ interprets the Kelley Morse class theory. Here we prove that GPK_∞⁺ + AC_WF (AC_WF being a form of the axiom of choice allowing to choose elements in well-founded sets) and the Kelley-Morse class theory with the axiom of global choice and the axiom “On is ramifiable” are mutually interpretable. This shows that GPK_∞⁺ + AC_WF is a “strong” theory since “On is ramifiable” implies the existence of a proper class of inaccessible cardinals.     

The Baire Category Property and Some Notions of Compactness

We work in set theory without the axiom of choice: ZF. We showthat the axiom BC: Compact Hausdorff spaces are Baire, is equivalentto the following axiom: Every tree has a subtree whose levelsare finite, which was introduced by Blass (cf. [4]). This settlesa question raised by Brunner (cf. [9, p. 438]). We also showthat the axiom of Dependent Choices is equivalent to the axiom:In a Hausdorff locally convex topological vector space, convex-compactconvex sets are Baire. Here convex-compact is the notion whichwas introduced by Luxemburg (cf. [16]).     

Some weak forms of the Baire category theorem

We show that the statement (K12) “separable, countably compact, regular spaces are Baire” is deducible from a strictly weaker form than AC, namely, CAC(?) (the axiom of choice for countable families of non‐empty subsets of the real line ?). We also find some characterizations of the axiom of dependent choices.     

Unions and the axiom of choice

We study statements about countable and well‐ordered unions and their relation to each other and to countable and well‐ordered forms of the axiom of choice. Using WO as an abbreviation for “well‐orderable”, here are two typical results: The assertion that every WO family of countable sets has a WO union does not imply that every countable family of WO sets has a WO union; the axiom of choice for WO families of WO sets does not imply that the countable union of countable sets is WO. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A refinement of prudent choices

We characterize choice correspondences that can be rationalized by a procedure that is a refinement of the prudent choices exposed in (Houy, 2010). Our characterization is made by means of the usual expansion axiom γ and a weakening of the usual contraction axiom α. We also make a link with traditional rationality.     

Choice principles for special subsets of the real line

We study the role the axiom of choice plays in the existence of some special subsets of ? and its power set ?(?).     

The axiom of choice and the law of excluded middle in weak set theories

A weak form of intuitionistic set theory WST lacking the axiom of extensionality is introduced. While WST is too weak to support the derivation of the law of excluded middle from the axiom of choice, we show that bee.ng up WST with moderate extensionality principles or quotient sets enables the derivation to go through. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Rigid Unary Functions and the Axiom of Choice

We shall investigate certain statements concerning the rigidity of unary functions which have connections with (weak) forms of the axiom of choice.     

The axiom of choice holds iff maximal closed filters exist

It is shown that in ZF set theory the axiom of choice holds iff every non empty topological space has a maximal closed filter.     

Sequential rationalization of multivalued choice

This paper contributes to the theory of rational choice under sequential criteria. Following the approach initiated by Manzini and Mariotti (2007) for single-valued choice functions, we characterize choice correspondences that are rational by two sequential criteria under a mild consistency axiom. Rationales ensuring the sequential rationalization are explicitly constructed and a uniquely determined, canonical solution is provided.     

Metric spaces and the axiom of choice

We study conditions for a topological space to be metrizable, properties of metrizable spaces, and the role the axiom of choice plays in these matters.