The continuum hypothesis : independence and truth-value

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Philosophy, 1974.MIT Humanities Library copy: issued in two vols.Leaf number 84 used twice. Also issued as a two-volume set.Includes bibliographical references (leaves 217-258).by Thomas S. Weston.Ph.D

Borel Determinacy and Metamathematics

Borel determinacy states that if G(T;X) is a game and X is Borel, then G(T;X) is determined. Proved by Martin in 1975, Borel determinacy is a theorem of ZFC set theory, and is, in fact, the best determinacy result in ZFC. However, the proof uses sets of high set theoretic type (N1 many power sets of ω). Friedman proved in 1971 that these sets are necessary by showing that the Axiom of Replacement is necessary for any proof of Borel Determinacy. To prove this, Friedman produces a model of ZC and a Borel set of Turing degrees that neither contains nor omits a cone; so by another theorem of Martin, Borel Determinacy is not a theorem of ZC. This paper contains three main sections: Martin's proof of Borel Determinacy; a simpler example of Friedman's result, namely, (in ZFC) a coanalytic set of Turing degrees that neither contains nor omits a cone; and finally, the Friedman result

On the computational content of the axiom of choice

We present a possible computational content of the negative translation of classical analysis with the Axiom of (countable) Choice. Interestingly, this interpretation uses a refinement of the realizability semantics of the absurdity proposition, which is not interpreted as the empty type here. We also show how to compute witnesses from proofs in classical analysis of ∃-statements and how to extract algorithms from proofs of ∀∃-statements. Our interpretation seems computationally more direct than the one based on Gödel's Dialectica interpretatio

New Perspectives on Games and Interaction

This volume is a collection of papers presented at the 2007 colloquium on new perspectives on games and interaction at the Royal Dutch Academy of Sciences in Amsterdam. The purpose of the colloquium was to clarify the uses of the concepts of game theory, and to identify promising new directions. This important collection testifies to the growing importance of game theory as a tool to capture the concepts of strategy, interaction, argumentation, communication, cooperation and competition. Also, it provides evidence for the richness of game theory and for its impressive and growing application

The Poetry of Logical Ideas: Towards a Mathematical Genealogy of Media Art

In this dissertation I chart a mathematical genealogy of media art, demonstrating that mathematical thought has had a significant influence on contemporary experimental moving image production. Rather than looking for direct cause and effect relationships between mathematics and the arts, I will instead examine how mathematical developments have acted as a cultural zeitgeist, an indirect, but significant, influence on the humanities and the arts. In particular, I will be narrowing the focus of this study to the influence mathematical thought has had on cinema (and by extension media art), given that mathematics lies comfortably between the humanities and sciences, and that cinema is the object par excellence of such a study, since cinema and media studies arrived at a time when the humanities and sciences were held by many to be mutually exclusive disciplines. It is also shown that many media scholars have been implicitly engaging with mathematical concepts without necessarily recognizing them as such. To demonstrate this, I examine many concepts from media studies that demonstrate or derive from mathematical concepts. For instance, Claude Shannon's mathematical model of communication is used to expand on Stuart Hall's cultural model, and the mathematical concept of the fractal is used to expand on Rosalind Krauss' argument that video is a medium that lends itself to narcissism. Given that the influence of mathematics on the humanities and the arts often occurs through a misuse or misinterpretation of mathematics, I mobilize the concept of a productive misinterpretation and argue that this type of misreading has the potential to lead to novel innovations within the humanities and the arts. In this dissertation, it is also established that there are many mathematical concepts that can be utilized by media scholars to better analyze experimental moving images. In particular, I explore the mathematical concepts of symmetry, infinity, fractals, permutations, the Axiom of Choice, and the algorithmic to moving images works by Hollis Frampton, Barbara Lattanzi, Dana Plays, T. Marie, and Isiah Medina, among others. It is my desire that this study appeal to scientists with an interest in cinema and media art, and to media theorists with an interest in experimental cinema and other contemporary moving image practices

Ancient Logic and its Modern Interpretations: Proceedings of the Buffalo Symposium on Modernist Interpretations of Ancient Logic, 21 and 22 April, 1972

Articles by Ian Mueller, Ronald Zirin, Norman Kretzmann, John Corcoran, John Mulhern, Mary Mulhern,Josiah Gould, and others. Topics: Aristotle's Syllogistic, Stoic Logic, Modern Research in Ancient Logic

Limited foresight

This thesis is about multi-period problems in which the decision-maker or players cannot see far enough ahead to solve the problem completely. The thesis considers why it might be that players reason forwards at all, let alone reasoning forwards only finitely far. It shows, using finite automata, that there is a class of problems for which forwards reasoning is more efficient than backwards reasoning. It goes on to use these finite automata to solve for an optimal foresight length. It then discusses solution concepts, and applies its preferred solution concept to two problems - one macro problem involving a central banker, and one micro problem concerning the decision whether to smoke

Le problème de la valeur dans les jeux stochastiques

La théorie des jeux est un outils standard quand il s'agit de l'étude des systèmes réactifs. Ceci est une conséquence de la variété des modèle de jeux tant au niveau de l'interaction des joueurs qu'au niveau de l'information que chaque joueur possède.Dans cette thèse, on étudie le problème de la valeur pour des jeux où les joueurs possèdent une information parfaite, information partiel et aucune information. Dans le cas où les joueurs possèdent une information parfaite sur l'état du jeu,on étudie le problème de la valeur pour des jeux dont les objectifs sont des combinaisons booléennes d'objectifs qualitatifs et quantitatifs.Pour les jeux stochastiques à un joueur, on montre que les valeurs sont calculables en temps polynomiale et on montre que les stratégies optimalespeuvent être implementées avec une mémoire finie.On montre aussi que notre construction pour la conjonction de parité et de la moyenne positivepeut être étendue au cadre des jeux stochastiques à deux joueurs. Dans le cas où les joueurs ont une information partielle,on étudie le problème de la valeur pour la condition d'accessibilité.On montre que le calcul de l'ensemble des états à valeur 1 est un problème indécidable,on introduit une sous classe pour laquelle ce problème est décidable.Le problème de la valeur 1 pour cette sous classe est PSPACE-complet dansle cas de joueur aveugle et dans EXPTIME dans le cas de joueur avec observations partielles.Game theory proved to be very useful in the fieldof verification of open reactive systems. This is due to the widevariety of games' model that differ in the way players interactand the amount of information players have.In this thesis, we study the value problem forgames where players have full knowledge on their current configurationof the game, partial knowledge, and no knowledge.\\In the case where players have perfect information,we study the value problem for objectives that consist in combinationof qualitative and quantitative conditions.In the case of one player stochastic games, we show thatthe values are computable in polynomial time and show thatthe optimal strategies exist and can be implemented with finite memory.We also showed that our construction for parity and positive-average Markov decisionprocesses extends to the case of two-player stochastic games.\\In the case where the players have partial information,we study the value problem for reachability objectives.We show that computing the set of states with value 1 is an undecidableproblem and introduce a decidable subclass for the value 1 problem.This sub class is PSPACE-complete in the case of blind controllersand EXPTIME is the setting of games with partial observations.BORDEAUX1-Bib.electronique (335229901) / SudocSudocFranceF