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Reverse Mathematics: The Playground of Logic
, 2010
"... The general enterprise of calibrating the strength of classical mathematical theorems in terms of the axioms (typically of set existence) needed to prove them was begun by Harvey Friedman in [1971] (see also [1967]). His goals were both philosophical and foundational. What existence assumptions are ..."
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The general enterprise of calibrating the strength of classical mathematical theorems in terms of the axioms (typically of set existence) needed to prove them was begun by Harvey Friedman in [1971] (see also [1967]). His goals were both philosophical and foundational. What existence assumptions are really needed to develop classical mathematics
The extent of constructive game labellings
 Proceedings of the 5th Panhellenic Logic Symposium
, 2005
"... Abstract. We develop a theory of labellings for infinite trees, define the notion of a combinatorial labelling, and show that ∆ 0 2 is the largest boldface pointclass in which every set admits a combinatorial labelling. 1 ..."
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Abstract. We develop a theory of labellings for infinite trees, define the notion of a combinatorial labelling, and show that ∆ 0 2 is the largest boldface pointclass in which every set admits a combinatorial labelling. 1
Why sets?
 PILLARS OF COMPUTER SCIENCE: ESSAYS DEDICATED TO BORIS (BOAZ) TRAKHTENBROT ON THE OCCASION OF HIS 85TH BIRTHDAY, VOLUME 4800 OF LECTURE NOTES IN COMPUTER SCIENCE
, 2008
"... Sets play a key role in foundations of mathematics. Why? To what extent is it an accident of history? Imagine that you have a chance to talk to mathematicians from a faraway planet. Would their mathematics be setbased? What are the alternatives to the settheoretic foundation of mathematics? Besi ..."
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Sets play a key role in foundations of mathematics. Why? To what extent is it an accident of history? Imagine that you have a chance to talk to mathematicians from a faraway planet. Would their mathematics be setbased? What are the alternatives to the settheoretic foundation of mathematics? Besides, set theory seems to play a significant role in computer science; is there a good justification for that? We discuss these and some related issues.
Probability measures and effective randomness
, 2007
"... Abstract. We study the question, “For which reals x does there exist a measure µ such that x is random relative to µ? ” We show that for every nonrecursive x, there is a measure which makes x random without concentrating on x. We give several conditions on x equivalent to there being continuous meas ..."
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Abstract. We study the question, “For which reals x does there exist a measure µ such that x is random relative to µ? ” We show that for every nonrecursive x, there is a measure which makes x random without concentrating on x. We give several conditions on x equivalent to there being continuous measure which makes x random. We show that for all but countably many reals x these conditions apply, so there is a continuous measure which makes x random. There is a metamathematical aspect of this investigation. As one requires higher arithmetic levels in the degree of randomness, one must make use of more iterates of the power set of the continuum to show that for all but countably many x’s there is a continuous µ which makes x random to that degree. 1.
Games for Truth
, 2008
"... Die Wahrheit liegt weder in der unendlichen Annährung an einer objektiv Gegebenes noch in der Mitte, sondern rundherum wie ein Sack, der mit jeder neuen Meinung, die man hineinstopft, seine Form ändert, aber immer fester wird. R. Musil We represent truth sets for a variety of the well known semantic ..."
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Die Wahrheit liegt weder in der unendlichen Annährung an einer objektiv Gegebenes noch in der Mitte, sondern rundherum wie ein Sack, der mit jeder neuen Meinung, die man hineinstopft, seine Form ändert, aber immer fester wird. R. Musil We represent truth sets for a variety of the well known semantic theories of truth as those sets consisting of those sentences for which a player has a winning strategy in an infinite two person game. The classifications of the games are simple, those over the natural model of arithmetic being all within the arithmetical class of Σ 0 3. 1
RANDOMNESS – BEYOND LEBESGUE MEASURE
"... Much of the recent research on algorithmic randomness has focused on randomness for Lebesgue measure. While, from a computability theoretic point of view, the picture remains unchanged if one passes to arbitrary computable measures, interesting phenomena occur if one studies the the set of reals wh ..."
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Much of the recent research on algorithmic randomness has focused on randomness for Lebesgue measure. While, from a computability theoretic point of view, the picture remains unchanged if one passes to arbitrary computable measures, interesting phenomena occur if one studies the the set of reals which are random for an arbitrary (continuous) probability measure or a generalized Hausdorff measure on Cantor space. This paper tries to give a survey of some of the research that has been done on randomness for nonLebesgue measures.
What Is Logic?
"... It is far from clear what is meant by logic or what should be meant by it. It is nevertheless reasonable to identify logic as the study of inferences and inferential relations. The obvious practical use of logic is in any case to help us to reason well, to draw good inferences. And the typical form ..."
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It is far from clear what is meant by logic or what should be meant by it. It is nevertheless reasonable to identify logic as the study of inferences and inferential relations. The obvious practical use of logic is in any case to help us to reason well, to draw good inferences. And the typical form the theory of any part of logic seems to be a set of rules of inference. This answer already introduces some structure into a discussion of the nature of logic, for in an inference we can distinguish the input called a premise or premises from the output known as the conclusion. The transition from a premise or a number of premises to the conclusion is governed by a rule of inference. If the inference is in accordance with the appropriate rule, it is called valid. Rules of inference are often thought of as the alpha and omega of logic. Conceiving of logic as the study of inference is nevertheless only the first approximation to the title question, in that it prompts more questions than it answers. It is not clear what counts as an inference or what a theory of such inferences might look like. What are the rules of inference based on? Where do we find them? The ultimate end