Arrow’s theorem has fostered a lot of work on the problem to aggregate individual preferences into a collective preference or more generally into a collective choice function. Still more generally the ”russian school” (see e.g. Aizerman and Aleskerov 1995, Aizerman and Malishevski 1981 or Aleskerov 1999) has considered the problem to aggregate individual choice functions into a collective choic...

This survey is divided into three major sections. The first concerns mathematical results about the choice axiom and the choice models that devoIve from it. For example, its relationship to Thurstonian theory is satisfyingly understood; much is known about how choice and ranking probabilities may relate, although little of this knowledge seems empirically useful; and there are certain interesti...

Finitely Supported Mathematics represents a part of mathematics for experimental sciences which has a continuous evolution in the last century. It is developed according to the Fraenkel-Mostowski axioms of set theory. The axiom of choice is inconsistent in the Finitely Supported Mathematics. We prove that several weaker forms of the axiom of choice are also inconsistent in the Finitely Supporte...

We investigate the connection between choice functions and lexicographic probabilities, by means of the convexity axiom considered by [7] but without imposing any Archimedean condition. We show that lexicographic probabilities are related to a particular type of sets of desirable gambles, and investigate the properties of the coherent choice function this induces via maximality. Finally, we sho...

We work in set-theory without choice ZF. Given a commutative field K, we consider the statement D(K): “On every non null K-vector space there exists a non-null linear form.” We investigate various statements which are equivalent to D(K) in ZF. Denoting by Z2 the two-element field, we deduce that D(Z2) implies the axiom of choice for pairs. We also deduce that D(Q) implies the axiom of choice fo...

We present a cut-elimination proof for simple type theory with axiom of choice modeled after Takahashi’s proof of cut-elimination for simple type theory with extensionality. The same proof works when types are restricted, for example for second-order classsical logic with axiom of choice.