Algebraic Approach to Colombeau Theory

We present a differential algebra of generalized functions over a field of generalized scalars by means of several axioms in terms of general algebra and topology. Our differential algebra is of Colombeau type in the sense that it contains a copy of the space of Schwartz distributions, and the set of regular distributions with $\mathcal C^\infty$-kernels forms a differential subalgebra. We discuss the uniqueness of the field of scalars as well as the consistency and independence of our axioms. This article is written mostly to satisfy the interest of mathematicians and scientists who do not necessarily belong to the \emph{Colombeau community}; that is to say, those who do not necessarily work in the \emph{non-linear theory of generalized functions}.Comment: 16 page

Logical Dreams

We discuss the past and future of set theory, axiom systems and independence results. We deal in particular with cardinal arithmetic

When Allais meets Ulysses: Dynamic Consistency and the Certainty Effect

We report experimental findings about subjects’ behavior in dynamic decision problems involving multistage lotteries with different timings of resolution of uncertainty. Our within subject design allows us to study violations of the independence axiom in the light of the dynamic axioms' ones : dynamic consistency, consequentialism and reduction of compound lotteries.

G\"odel's Notre Dame Course

This is a companion to a paper by the authors entitled "G\"odel's natural deduction", which presented and made comments about the natural deduction system in G\"odel's unpublished notes for the elementary logic course he gave at the University of Notre Dame in 1939. In that earlier paper, which was itself a companion to a paper that examined the links between some philosophical views ascribed to G\"odel and general proof theory, one can find a brief summary of G\"odel's notes for the Notre Dame course. In order to put the earlier paper in proper perspective, a more complete summary of these interesting notes, with comments concerning them, is given here.Comment: 18 pages. minor additions, arXiv admin note: text overlap with arXiv:1604.0307

Intertemporal substitution and recursive smooth ambiguity preferences

In this paper, we establish an axiomatically founded generalized recursive smooth ambiguity model that allows for a separation among intertemporal substitution, risk aversion, and ambiguity aversion. We axiomatize this model using two approaches: the second-order act approach à la Klibanoff, Marinacci, and Mukerji (2005) and the two-stage randomization approach à la Seo (2009). We characterize risk attitude and ambiguity attitude within these two approaches. We then discuss our model's application in asset pricing. Our recursive preference model nests some popular models in the literature as special cases.Ambiguity, ambiguity aversion, risk aversion, intertemporal substitution, model uncertainty, recursive utility, dynamic consistency

Rationality and dynamic consistency under risk and uncertainty

For choice with deterministic consequences, the standard rationality hypothesis is ordinality - i.e., maximization of a weak preference ordering. For choice under risk (resp. uncertainty), preferences are assumed to be represented by the objectively (resp. subjectively) expected value of a von Neumann{Morgenstern utility function. For choice under risk, this implies a key independence axiom; under uncertainty, it implies some version of Savage's sure thing principle. This chapter investigates the extent to which ordinality, independence, and the sure thing principle can be derived from more fundamental axioms concerning behaviour in decision trees. Following Cubitt (1996), these principles include dynamic consistency, separability, and reduction of sequential choice, which can be derived in turn from one consequentialist hypothesis applied to continuation subtrees as well as entire decision trees. Examples of behavior violating these principles are also reviewed, as are possible explanations of why such violations are often observed in experiments

Logic Programming as Constructivism

The features of logic programming that seem unconventional from the viewpoint of classical logic can be explained in terms of constructivistic logic. We motivate and propose a constructivistic proof theory of non-Horn logic programming. Then, we apply this formalization for establishing results of practical interest. First, we show that 'stratification can be motivated in a simple and intuitive way. Relying on similar motivations, we introduce the larger classes of 'loosely stratified' and 'constructively consistent' programs. Second, we give a formal basis for introducing quantifiers into queries and logic programs by defining 'constructively domain independent* formulas. Third, we extend the Generalized Magic Sets procedure to loosely stratified and constructively consistent programs, by relying on a 'conditional fixpoini procedure