Many inspections are manipulable

A self-proclaimed expert uses past observations of a stochastic process to make probabilistic predictions about the process. An inspector applies a test function to the infinite sequence of predictions provided by the expert and the observed realization of the process in order to check the expert's reliability. If the test function is Borel and the inspection is such that a true expert always passes it, then it is also manipulable by an ignorant expert. The proof uses Martin's theorem about the determinacy of Blackwell games. Under the axiom of choice, there exist non-Borel test functions that are not manipulable.Forecasting, calibration, zero-sum games

Probabilistic modal {\mu}-calculus with independent product

The probabilistic modal {\mu}-calculus is a fixed-point logic designed for expressing properties of probabilistic labeled transition systems (PLTS's). Two equivalent semantics have been studied for this logic, both assigning to each state a value in the interval [0,1] representing the probability that the property expressed by the formula holds at the state. One semantics is denotational and the other is a game semantics, specified in terms of two-player stochastic parity games. A shortcoming of the probabilistic modal {\mu}-calculus is the lack of expressiveness required to encode other important temporal logics for PLTS's such as Probabilistic Computation Tree Logic (PCTL). To address this limitation we extend the logic with a new pair of operators: independent product and coproduct. The resulting logic, called probabilistic modal {\mu}-calculus with independent product, can encode many properties of interest and subsumes the qualitative fragment of PCTL. The main contribution of this paper is the definition of an appropriate game semantics for this extended probabilistic {\mu}-calculus. This relies on the definition of a new class of games which generalize standard two-player stochastic (parity) games by allowing a play to be split into concurrent subplays, each continuing their evolution independently. Our main technical result is the equivalence of the two semantics. The proof is carried out in ZFC set theory extended with Martin's Axiom at an uncountable cardinal

The Wonder of Colors and the Principle of Ariadne

The Principle of Ariadne, formulated in 1988 ago by Walter Carnielli and Carlos Di Prisco and later published in 1993, is an infinitary principle that is independent of the Axiom of Choice in ZF, although it can be consistently added to the remaining ZF axioms. The present paper surveys, and motivates, the foundational importance of the Principle of Ariadne and proposes the Ariadne Game, showing that the Principle of Ariadne, corresponds precisely to a winning strategy for the Ariadne Game. Some relations to other alternative. set-theoretical principles are also briefly discussed

NOTES ON Bl-mathrmADomega1mathrm{AD}_{omega_{1}} (Infinite Combinatorics and Forcing Theory)

In this note, we show that the axiom B1-mathrm{AD}_{$omega$_{1} is inconsistent under ZF + mathrm{AC}_{(v}(mathbb{R}). This answers the question of Löwe in [3, Question 52]

Calibration: Respice, Adspice, Prospice

“Those who claim for themselves to judge the truth are bound to possess a criterion of truth.” JEL Code: C18, C53, D89calibration, prediction

What is Absolute Undecidability?†

It is often alleged that, unlike typical axioms of mathematics, the Continuum Hypothesis (CH) is indeterminate. This position is normally defended on the ground that the CH is undecidable in a way that typical axioms are not. Call this kind of undecidability “absolute undecidability”. In this paper, I seek to understand what absolute undecidability could be such that one might hope to establish that (a) CH is absolutely undecidable, (b) typical axioms are not absolutely undecidable, and (c) if a mathematical hypothesis is absolutely undecidable, then it is indeterminate. I shall argue that on no understanding of absolute undecidability could one hope to establish all of (a)–(c). However, I will identify one understanding of absolute undecidability on which one might hope to establish both (a) and (c) to the exclusion of (b). This suggests that a new style of mathematical antirealism deserves attention—one that does not depend on familiar epistemological or ontological concerns. The key idea behind this view is that typical mathematical hypotheses are indeterminate because they are relevantly similar to CH

The Free Will Theorem

On the basis of three physical axioms, we prove that if the choice of a particular type of spin 1 experiment is not a function of the information accessible to the experimenters, then its outcome is equally not a function of the information accessible to the particles. We show that this result is robust, and deduce that neither hidden variable theories nor mechanisms of the GRW type for wave function collapse can be made relativistic. We also establish the consistency of our axioms and discuss the philosophical implications.Comment: 31 pages, 6figure