Related papers: Coherence of countably many bets
We use a generalization of a construction by Ziegler to show that for any field $F$ and any countable collection of countable subsets $A_i \subseteq F, i \in \calI \subset \Z_{>0}$ there exist infinitely many fields $K$ of arbitrary…
We recall some of the history of the information-theoretic approach to deriving core results in probability theory and indicate parts of the recent resurgence of interest in this area with current progress along several interesting…
We present a version of arithmetic in all finite types which allows for a definition of equality at higher types for which all congruence are derivable, for which the soundness of the Dialectica interpretation is provable inside the system…
We provide some examples showing how game-theoretic arguments can be used in computability theory and algorithmic information theory: unique numbering theorem (Friedberg), the gap between conditional complexity and total conditional…
We introduce a general framework for de Finetti reduction results, applicable to various notions of partially exchangeable probability distributions. Explicit statements are derived for the cases of exchangeability, Markov exchangeability,…
We recently introduced a bounded rationality approach for the theory of desirable gambles. It is based on the unique requirement that being non-negative for a gamble has to be defined so that it can be provable in polynomial-time. In this…
We provide game-theoretic proofs of some well-known existence theorems of Friedberg numberings for the class of all partial computable functions, including (1) the existence of two incomparable Friedberg numberings; (2) the existence of a…
Infinity, in various guises, has been invoked recently in order to `explain' a number of important questions regarding observable phenomena in science, and in particular in cosmology. Such explanations are by their nature speculative. Here…
Picture countably many logicians all wearing a hat in one of $\kappa$-many colours. They each get to look at finitely many other hats and afterwards make finitely many guesses for their own hat's colour. For which $\kappa$ can the logicians…
The premises of an argument give evidence or other reasons to support a conclusion. However, the amount of support required depends on the generality of a conclusion, the nature of the individual premises, and similar. An argument whose…
In the modern Bayesian view classical probability theory is simply an extension of conventional logic, i.e., a quantitative tool that allows for consistent reasoning in the presence of uncertainty. Classical theory presupposes, however,…
The interrelations between various classes of convergence spaces defined by countability conditions are studied. Remarkably, they all find characterizations in the usual space of ultrafilters in terms of classical topological properties.…
We develop an approach to solve Barberis (2012)'s casino gambling model in which a gambler whose preferences are specified by the cumulative prospect theory (CPT) must decide when to stop gambling by a prescribed deadline. We assume that…
In this paper we consider propositional calculi, which are finitely axiomatizable extensions of intuitionistic implicational propositional calculus together with the rules of modus ponens and substitution. We give a proof of undecidability…
In this paper we show examples for applications of the Bombieri-Lang conjecture in additive combinatorics, giving bounds on the cardinality of sumsets of squares and higher powers of integers. Using similar methods we give bounds on the…
A classical Borel Cantelli Lemma gives conditions for deciding whether an infinite number of rare events will almost surely happen. In this article, we propose an extension of Borel Cantelli Lemma to characterize the multiple occurrence of…
This note is concerned with a formal analysis of the problem of non-monotonic reasoning in intelligent systems, especially when the uncertainty is taken into account in a quantitative way. A firm connection between logic and probability is…
The motivation for extending secret sharing schemes to cases when either the set of players is infinite or the domain from which the secret and/or the shares are drawn is infinite or both, is similar to the case when switching to abstract…
In this paper we discuss contrastive explanations for formal argumentation - the question why a certain argument (the fact) can be accepted, whilst another argument (the foil) cannot be accepted under various extension-based semantics. The…
We consider how to make probability forecasts of binary labels. Our main mathematical result is that for any continuous gambling strategy used for detecting disagreement between the forecasts and the actual labels, there exists a…