7 research outputs found
Lexicographic choice functions without archimedeanicity
We investigate the connection between choice functions and lexicographic probabilities, by means of the convexity axiom considered by Seidenfeld, Schervisch and Kadane (2010) 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 show that the convexity axiom is necessary but not sufficient for a coherent choice function to be the infimum of a class of lexicographic ones
A local-global principle for preordered semirings and abstract Positivstellens\"atze
Motivated by trying to find a new proof of Artin's theorem on positive
polynomials, we state and prove a Positivstellensatz for preordered semirings
in the form of a local-global principle. It relates the given algebraic order
on a suitably well-behaved semiring to the geometrical order defined in terms
of a probing by homomorphisms to "test algebras". We introduce and study the
latter as structures intended to capture the behaviour of a semiring element in
the infinitesimal neighbourhoods of a real point of the real spectrum. As first
applications of our local-global principle, we prove two abstract
non-Archimedean Positivstellens\"atze. The first one is a non-Archimedean
generalization of the classical Positivstellensatz of Krivine-Kadison-Dubois,
while the second one is deeper. A companion paper will use our second
Positivstellensatz to derive an asymptotic classification of random walks on
locally compact abelian groups.
As an important intermediate result, we develop an abstract
Positivstellensatz for preordered semifields which states that a semifield
preorder is always the intersection of its total extensions. We also introduce
quasiordered rings and develop some of their theory. While these are related to
Marshall's -modules, we argue that quasiordered rings offer an improved
definition which puts them among the basic objects of study for real algebra.Comment: 111 pages. v2: added relevant assumptions to main results
(zerosumfreeness, no zero divisors), other minor fixe
Bernoulli Without Bayes: A Theory of Utility-Sophisticated Preferences under Ambiguity
A decision-maker is utility-sophisticated if he ranks acts according to their expected utility whenever such comparisons are meaningful. We characterize utility sophistication in cases in which probabilistic beliefs are not too imprecise, and show that in these cases utility-sophisticated preferences are completely determined by consequence utilities and event attitudes captured by preferences over bets. The Anscombe-Aumann framework as employed in the classical contributions of Schmeidler (1989) and Gilboa-Schmeidler (1989) can be viewed as an important special case. For the class of utility sophisticated preferences with sufficiently precise beliefs, we also propose a definition of revealed probabilistic beliefs that overcomes the limitations of existing definitions.Expected Utility, Ambiguity, Probalistic Sophistication, Revealed Probabilistic Beliefs
Views from a peak:Generalisations and descriptive set theory
This dissertation has two major threads, one is mathematical, namely descriptive set theory, the other is philosophical, namely generalisation in mathematics. Descriptive set theory is the study of the behaviour of definable subsets of a given structure such as the real numbers. In the core mathematical chapters, we provide mathematical results connecting descriptive set theory and generalised descriptive set theory. Using these, we give a philosophical account of the motivations for, and the nature of, generalisation in mathematics.In Chapter 3, we stratify set theories based on this descriptive complexity. The axiom of countable choice for reals is one of the most basic fragments of the axiom of choice needed in many parts of mathematics. Descriptive choice principles are a further stratification of this fragment by the descriptive complexity of the sets. We provide a separation technique for descriptive choice principles based on Jensen forcing. Our results generalise a theorem by Kanovei.Chapter 4 gives the essentials of a generalised real analysis, that is a real analysis on generalisations of the real numbers to higher infinities. This builds on work by Galeotti and his coauthors. We generalise classical theorems of real analysis to certain sets of functions, strengthening continuity, and disprove other classical theorems. We also show that a certain cardinal property, the tree property, is equivalent to the Extreme Value Theorem for a set of functions which generalize the continuous functions.The question of Chapter 5 is whether a robust notion of infinite sums can be developed on generalisations of the real numbers to higher infinities. We state some incompatibility results, which suggest not. We analyse several candidate notions of infinite sum, both from the literature and more novel, and show which of the expected properties of a notion of sum they fail.In Chapter 6, we study the descriptive set theory arising from a generalization of topology, κ-topology, which is used in the previous two chapters. We show that the theory is quite different from that of the standard (full) topology. Differences include a collapsing Borel hierarchy, a lack of universal or complete sets, Lebesgue’s ‘great mistake’ holds (projections do not increase complexity), a strict hierarchy of notions of analyticity, and a failure of Suslin’s theorem.Lastly, in Chapter 7, we give a philosophical account of the nature of generalisation in mathematics, and describe the methodological reasons that mathematicians generalise. In so doing, we distinguish generalisation from other processes of change in mathematics, such as abstraction and domain expansion. We suggest a semantic account of generalisation, where two pieces of mathematics constitute a generalisation if they have a certain relation of content, along with an increased level of generality