On bar recursion of types 0 and 1

For general information on bar recursion the reader should consult the papers of Spector [8], where it was introduced, Howard [2] and Tait [11]. In this note we shall prove that the terms of Godel's theory T (in its extensional version of Spector [8]) are closed under the rule BRo•1 of bar recursion of types 0 and 1. Our method of proof is based on the notion of an infinite term introduced by Tait [9]. The main tools of the proof are (i) the normalization theorem for (notations for) infinite terms and (ii) valuation functionals. Both are elaborated in [6]; for brevity some familiarity with this paper is assumed here. Using (i) and (ii) we reduce BRo.1 to ';-recursion with'; < co. From this the result follows by work of Tait [10], who gave a reduction of 2E-recursion to ';-recursion at a higher type. At the end of the paper we discuss a perhaps more natural variant of bar recursion introduced by Kreisel in [4]. Related results are due to rKeisel (in his appendix to [8]), who obtains results which imply, using the reduction given by Howard [2] of the constant of bar recursion of type '0 to the rule of bar recursion of type (0 ~ '0) ~ '0, that T is not closed under the rule of bar recursion of a type of level ~ 2, to Diller [1], who gave a reduction of BRo.1 to ';-recursion with'; bounded by the least (V-critical number, and to Howard [3], who gave an ordinal analysis of the constant of bar recursion of type O. I am grateful to H. Barendregt, W. Howard and G. Kreisel for many useful comments and discussions. Recall that a functional F of type 0 ~ (0 ~ '0) ~ (J is said to be defined by (the rule of) bar recursion of type '0 from Yand functionals G, H of the proper types i

Catastrophic Risks with Finite or Infinite States

Catastrophic risks are rare events with major consequences, such as market crashes, catastrophic climate change, asteroids or the extinction of a species. We show that classic expected utility theory based on Von Neumann axioms is insensitive to rare events no matter how catastrophic. Its insensitivity emerges from a requirement of continuity (e.g. Arrow's Monotone Continuity Axiom, and its relatives as defined by De Groot, Hernstein and Milnor) that anticipate average responses to extreme events. This leads to countably additive measures and `expected utility' that are insensitive to extreme risks. In a new axiomatic extension, the author (Chichilnisky 1996, 2000, 2002) requires equal treatment of rare and frequent events, deriving the new decision criterion the axioms imply. These are expected utility combined with purely finitely additive measures that focus on catastrophes, and explain the presistent observations of distributions with "fat tails" in earth sciences and financial markets. Continuity is based on the `topology of fear' introduced in Chichilnisky (2009), and is linked to Debreu's 1953 work on Adam Smith's Invisible Hand. The balance between the classic and the new axioms tests the limits of non- parametric estimation in Hilbert spaces, Chichilnisky (2008).. extending the foundations of probability & statistics (Chichilnisky 2009 and 2010) to include "black swans" or rare events, and finite as well as infinite state spaces

Catastrophic Risks with Finite or Infinite States

Catastrophic risks are rare events with major consequences, such as market crashes, catastrophic climate change, asteroids or the extinction of a species. We show that classic expected utility theory based on Von Neumann axioms is insensitive to rare events no matter how catastrophic. Its insensitivity emerges from a requirement of continuity (e.g. Arrow's Monotone Continuity Axiom, and its relatives as defined by De Groot, Hernstein and Milnor) that anticipate average responses to extreme events. This leads to countably additive measures and `expected utility' that are insensitive to extreme risks. In a new axiomatic extension, the author (Chichilnisky 1996, 2000, 2002) requires equal treatment of rare and frequent events, deriving the new decision criterion the axioms imply. These are expected utility combined with purely finitely additive measures that focus on catastrophes, and explain the presistent observations of distributions with "fat tails" in earth sciences and financial markets. Continuity is based on the `topology of fear' introduced in Chichilnisky (2009), and is linked to Debreu's 1953 work on Adam Smith's Invisible Hand. The balance between the classic and the new axioms tests the limits of non- parametric estimation in Hilbert spaces, Chichilnisky (2008).. extending the foundations of probability & statistics (Chichilnisky 2009 and 2010) to include "black swans" or rare events, and finite as well as infinite state spaces

ON THE FOUNDATIONS OF COMPUTABILITY THEORY

The principal motivation for this work is the observation that there are significant deficiencies in the foundations of conventional computability theory. This thesis examines the problems with conventional computability theory, including its failure to address discrepancies between theory and practice in computer science, semantic confusion in terminology, and limitations in the scope of conventional computing models. In light of these difficulties, fundamental notions are re-examined and revised definitions of key concepts such as “computer,” “computable,” and “computing power” are provided. A detailed analysis is conducted to determine desirable semantics and scope of applicability of foundational notions. The credibility of the revised definitions is ascertained by demonstrating by their ability to address identified problems with conventional definitions. Their practical utility is established through application to examples. Other related issues, including hidden complexity in computations, subtleties related to encodings, and the cardinalities of sets involved in computing, are examined. A resource-based meta-model for characterizing computing model properties is introduced. The proposed definitions are presented as a starting point for an alternate foundation for computability theory. However, formulation of the particular concepts under discussion is not the sole purpose of the thesis. The underlying objective of this research is to open discourse on alternate foundations of computability theory and to inspire re-examination of fundamental notions