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Randomness in Computability Theory
, 2000
"... We discuss some aspects of algorithmic randomness and state some open problems in this area. The first part is devoted to the question "What is a computably random sequence?" Here we survey some of the approaches to algorithmic randomness and address some questions on these concepts. In the seco ..."
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We discuss some aspects of algorithmic randomness and state some open problems in this area. The first part is devoted to the question "What is a computably random sequence?" Here we survey some of the approaches to algorithmic randomness and address some questions on these concepts. In the second part we look at the Turing degrees of MartinLof random sets. Finally, in the third part we deal with relativized randomness. Here we look at oracles which do not change randomness. 1980 Mathematics Subject Classification. Primary 03D80; Secondary 03D28. 1 Introduction Formalizations of the intuitive notions of computability and randomness are among the major achievements in the foundations of mathematics in the 20th century. It is commonly accepted that various equivalent formal computability notions  like Turing computability or recursiveness  which were introduced in the 1930s and 1940s adequately capture computability in the intuitive sense. This belief is expressed in the w...
Hilbert's Programs: 19171922
, 1999
"... . Hilbert's finitist programwas not created at the beginning of the twenties solely to counteract Brouwer's intuitionism, but rather emerged out of broad philosophical reflections on the foundations of mathematics and out of detailed logical work; that is evident from notes of lecture courses tha ..."
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. Hilbert's finitist programwas not created at the beginning of the twenties solely to counteract Brouwer's intuitionism, but rather emerged out of broad philosophical reflections on the foundations of mathematics and out of detailed logical work; that is evident from notes of lecture courses that were given by Hilbert and prepared in collaboration with Bernays during the period from 1917 to 1922. These notes reveal a dialectic progression from a critical logicism through a radical constructivism toward finitism; the progression has to be seen against the background of the stunning presentation of mathematical logic in the lectures given during the winter term 1917/18. In this paper, I sketch the connection of Hilbert's considerations to issues in the foundations of mathematics during the second half of the 19th century, describe the work that laid the basis of modern mathematical logic, and analyze the first steps in the new subject of proof theory. A revision of the standar...
From Cournot’s principle to market efficiency
 Cournot: Modelling Economics
, 2007
"... The efficientmarkets hypothesis was formulated in the early 1960s, when Cournot’s principle was no longer widely understood and accepted as a philosophical foundation for probability theory. A revival of Cournot’s principle can help us distinguish clearly among different aspects of market efficienc ..."
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The efficientmarkets hypothesis was formulated in the early 1960s, when Cournot’s principle was no longer widely understood and accepted as a philosophical foundation for probability theory. A revival of Cournot’s principle can help us distinguish clearly among different aspects of market efficiency.
The Borel hierarchy and the projective hierarchy in intuitionistic mathematics
 University of Nijmegen Department of Mathematics
, 2001
"... this paper. Once we agree to accept and use it we enter a new world and discover many facts for which there does not exist a classical counterpart. The principle entails for instance that the union of the two closed sets [0, 1] and [1, 2] is not a countable intersection of open subsets of R. One ma ..."
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this paper. Once we agree to accept and use it we enter a new world and discover many facts for which there does not exist a classical counterpart. The principle entails for instance that the union of the two closed sets [0, 1] and [1, 2] is not a countable intersection of open subsets of R. One may also infer that there are unions of three closed sets di#erent from every union of two closed sets. These observations are the tip of an iceberg. The intuitionistic Borel Hierarchy shows o# an exquisite fine structure
Understanding And Using Brouwer's Continuity Principle
, 2000
"... Brouwer's Continuity Principle distinguishes intuitionistic mathematics from other varieties of constructive mathematics, giving it its own avour. We discuss the plausibility of this assumption and show how it is used. We explain how one may understand its consequences even if one hesitates to ac ..."
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Brouwer's Continuity Principle distinguishes intuitionistic mathematics from other varieties of constructive mathematics, giving it its own avour. We discuss the plausibility of this assumption and show how it is used. We explain how one may understand its consequences even if one hesitates to accept it as an axiom. 1 Brouwer's Continuity Principle We let N be the set of all natural numbers. Its elements 0; 1; 2; : : : are produced one by one. N is a never nished project that is executed stepbystep. We let N be the set of all innite sequences of natural numbers. The acceptance of N as a totality has been a major step in the history of mathematical thinking, and led to the development of set theory. With Cantor's diagonal argument in mind, Brouwer probed the meaning of the words: \every possible innite sequence of natural numbers" and found a way to sensibly use them. An element of N is a function from N to N, = (0); (1); (2); : : : Every such element is produced st...
Arguments for the Continuity Principle
, 2000
"... Contents 1 The continuity principle 1 2 A phenomenological consideration 8 2.1 An argument for G(raph)WCN . . . . . . . . . . . . . . . . . 8 2.2 Two arguments against WCN . . . . . . . . . . . . . . . . . . 13 3 Other arguments for continuity 15 3.1 Undecidability of equality of choice sequences ..."
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Contents 1 The continuity principle 1 2 A phenomenological consideration 8 2.1 An argument for G(raph)WCN . . . . . . . . . . . . . . . . . 8 2.2 Two arguments against WCN . . . . . . . . . . . . . . . . . . 13 3 Other arguments for continuity 15 3.1 Undecidability of equality of choice sequences . . . . . . . . . 15 3.2 Kripke's Schema and full PEM . . . . . . . . . . . . . . . . . 15 3.3 The KLST theorem . . . . . . . . . . . . . . . . . . . . . . . . 16 4 Conclusion 19 1 The continuity principle There are two principles that lend Brouwer's mathematics the extra power beyond arithmetic. Both are presented in Brouwer's writings with little or no argument. One, the principle of bar induction, will not concern us here. The other, the continuity principle for numbers, occurs for the rst time in print in [Brouwer 1918]. It is formulated and immediately applied to show that the set of numerical choice sequences is not enumerable. In fa
Randomness And Foundations Of Probability: Von Mises' Axiomatisation Of Random Sequences
 Institute for Mathematical Statistics
, 1996
"... This paper appeared in T. Ferguson et al (eds.): Probability, statistics and game theory, papers in honor of David Blackwell, Institute for Mathematical Statistics 1996 ..."
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This paper appeared in T. Ferguson et al (eds.): Probability, statistics and game theory, papers in honor of David Blackwell, Institute for Mathematical Statistics 1996
Hermann Weyl’s Intuitionistic Mathematics. Dirk
"... It is common knowledge that for a short while Hermann Weyl joined Brouwer in his pursuit of a revision of mathematics according to intuitionistic principles. There is, however, little in the literature that sheds light on Weyl’s role, and in particular on Brouwer’s reaction to Weyl’s allegiance to t ..."
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It is common knowledge that for a short while Hermann Weyl joined Brouwer in his pursuit of a revision of mathematics according to intuitionistic principles. There is, however, little in the literature that sheds light on Weyl’s role, and in particular on Brouwer’s reaction to Weyl’s allegiance to the cause of intuitionism. This short episode certainly raises a number of questions: what made Weyl give up his own program, spelled out in “Das Kontinuum”, how come Weyl was so wellinformed about Brouwer’s new intuitionism, in what respect did Weyl’s intuitionism differ from Brouwer’s intuitionism, what did Brouwer think of Weyl’s views,........? To some of these questions at least partial answers can be put forward on the basis of some of the available correspondence and notes. The present paper will concentrate mostly on the historical issues of the intuitionistic episode in Weyl’s career. Weyl entered the foundational controversy with a bang in 1920 with his sensational paper “On the new foundational crisis in mathematics ” 1. He had already made a name for himself in the foundations of mathematics in 1918 with his monograph “The Continuum” [Weyl 1918] ; this contained in addition to a technical logical – mathematical construction of the continuum, a fairly extensive discussion of the shortcomings of the traditional construction of the continuum on the basis of arbitrary — and hence also impredicative — Dedekind cuts. This book did not cause much of a stir in mathematics, that is to say, it was ritually quoted in the literature but, probably, little understood. It had to wait for a proper appreciation until the phenomenon of impredicativity was better understood 2. The paper “On the new foundational crisis in mathematics ” had a totally different effect, it was the proverbial stone thrown into the quiet pond of mathematics. Weyl characterised it in retrospect with the somewhat apologetic words: Only with some hesitation I acknowledge these lectures, which reflect in their style, which was here and there really bombastic, the mood of excited times — the times immediately following the First World War. 3 Indeed, Weyl’s “New crisis ” reads as a manifesto to the mathematical community, it uses an evocative language with a good many explicit references to the political
History of Constructivism in the 20th Century
"... notions, such as `constructive proof', `arbitrary numbertheoretic function ' are rejected. Statements involving quantifiers are finitistically interpreted in terms of quantifierfree statements. Thus an existential statement 9xAx is regarded as a partial communication, to be supplemented by providi ..."
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notions, such as `constructive proof', `arbitrary numbertheoretic function ' are rejected. Statements involving quantifiers are finitistically interpreted in terms of quantifierfree statements. Thus an existential statement 9xAx is regarded as a partial communication, to be supplemented by providing an x which satisfies A. Establishing :8xAx finitistically means: providing a particular x such that Ax is false. In this century, T. Skolem 4 was the first to contribute substantially to finitist 4 Thoralf Skolem 18871963 History of constructivism in the 20th century 3 mathematics; he showed that a fair part of arithmetic could be developed in a calculus without bound variables, and with induction over quantifierfree expressions only. Introduction of functions by primitive recursion is freely allowed (Skolem 1923). Skolem does not present his results in a formal context, nor does he try to delimit precisely the extent of finitist reasoning. Since the idea of finitist reasoning ...
On the Quantitative Structure of ...
, 2000
"... We analyze the quantitative structure of 0 2 . Among other things, we prove that a set is Turing complete if and only if its lower cone is nonnegligible, and that the sets of r.e.degree form a small subset of 0 2 . Mathematical Subject Classification: 03D15, 03D30, 28E15 Keywords: Comput ..."
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We analyze the quantitative structure of 0 2 . Among other things, we prove that a set is Turing complete if and only if its lower cone is nonnegligible, and that the sets of r.e.degree form a small subset of 0 2 . Mathematical Subject Classification: 03D15, 03D30, 28E15 Keywords: Computable measure theory, Turing degrees, completeness. 1 Introduction We study an eective measure theory suited for the study of 0 2 , the second level of the arithmetical hierarchy (alternatively, the sets computable relative to the halting problem K). This work may be seen as part of the constructivist tradition in mathematics as documented in [6]. The framework for eectivizing measure theory that we employ uses martingales. Martingales were rst applied to the study of random sequences by J. Ville [22]. Recursive martingales were studied in Schnorr [19], and became popular in complexity theory in more recent years through the work of Lutz [14, 15]. Lutz Research supported by a Ma...