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Foundational and mathematical uses of higher types
 REFLECTIONS ON THE FOUNDATIONS OF MATHEMATICS: ESSAY IN HONOR OF SOLOMON FEFERMAN
, 1999
"... In this paper we develop mathematically strong systems of analysis in higher types which, nevertheless, are prooftheoretically weak, i.e. conservative over elementary resp. primitive recursive arithmetic. These systems are based on noncollapsing hierarchies ( n WKL+ ; n WKL+ ) of principles ..."
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In this paper we develop mathematically strong systems of analysis in higher types which, nevertheless, are prooftheoretically weak, i.e. conservative over elementary resp. primitive recursive arithmetic. These systems are based on noncollapsing hierarchies ( n WKL+ ; n WKL+ ) of principles which generalize (and for n = 0 coincide with) the socalled `weak' König's lemma WKL (which has been studied extensively in the context of second order arithmetic) to logically more complex tree predicates. Whereas the second order context used in the program of reverse mathematics requires an encoding of higher analytical concepts like continuous functions F : X ! Y between Polish spaces X;Y , the more exible language of our systems allows to treat such objects directly. This is of relevance as the encoding of F used in reverse mathematics tacitly yields a constructively enriched notion of continuous functions which e.g. for F : IN ! IN can be seen (in our higher order context)
A New Approach to Predicative Set Theory
"... We suggest a new basic framework for the WeylFeferman predicativist program by constructing a formal predicative set theory PZF which resembles ZF. The basic idea is that the predicatively acceptable instances of the comprehension schema are those which determine the collections they define in an a ..."
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We suggest a new basic framework for the WeylFeferman predicativist program by constructing a formal predicative set theory PZF which resembles ZF. The basic idea is that the predicatively acceptable instances of the comprehension schema are those which determine the collections they define in an absolute way, independent of the extension of the “surrounding universe”. This idea is implemented using syntactic safety relations between formulas and sets of variables. These safety relations generalize both the notion of domainindependence from database theory, and Godel notion of absoluteness from set theory. The language of PZF is typefree, and it reflects real mathematical practice in making an extensive use of statically defined abstract set terms. Another important feature of PZF is that its underlying logic is ancestral logic (i.e. the extension of FOL with a transitive closure operation). 1
Amending Frege’s Grundgesetze der Arithmetik
, 2002
"... Frege’s Grundgesetze der Arithmetik is formally inconsistent. This system is, except for minor differences, secondorder logic together with an abstraction operator governed by Frege’s Axiom V. A few years ago, Richard Heck showed that the ramified predicative secondorder fragment of the Grundgeset ..."
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Frege’s Grundgesetze der Arithmetik is formally inconsistent. This system is, except for minor differences, secondorder logic together with an abstraction operator governed by Frege’s Axiom V. A few years ago, Richard Heck showed that the ramified predicative secondorder fragment of the Grundgesetze is consistent. In this paper, we show that the above fragment augmented with the axiom of reducibility for concepts true of only finitely many individuals is still consistent, and that elementary Peano arithmetic (and more) is interpretable in this extended system.
On the calculating power of Laplace’s demon (Part I)
, 2006
"... We discuss several ways of making precise the informal concept of physical determinism, drawing on ideas from mathematical logic and computability theory. We outline a programme of investigating these notions of determinism in detail for specific, precisely articulated physical theories. We make a s ..."
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We discuss several ways of making precise the informal concept of physical determinism, drawing on ideas from mathematical logic and computability theory. We outline a programme of investigating these notions of determinism in detail for specific, precisely articulated physical theories. We make a start on our programme by proposing a general logical framework for describing physical theories, and analysing several possible formulations of a simple Newtonian theory from the point of view of determinism. Our emphasis throughout is on clarifying the precise physical and metaphysical assumptions that typically underlie a claim that some physical theory is ‘deterministic’. A sequel paper is planned, in which we shall apply similar methods to the analysis of other physical theories. Along the way, we discuss some possible repercussions of this kind of investigation for both physics and logic. 1
The Hilbert problems and Hilbert’s Program
, 2008
"... In 1900 the great mathematician David Hilbert laid down a list of 23 mathematical problems [32] which exercised a great influence on subsequent mathematical research. From the perspective of foundational studies, it is noteworthy that Hilbert’s Problems 1 and 2 are squarely in the area of foundation ..."
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In 1900 the great mathematician David Hilbert laid down a list of 23 mathematical problems [32] which exercised a great influence on subsequent mathematical research. From the perspective of foundational studies, it is noteworthy that Hilbert’s Problems 1 and 2 are squarely in the area of foundations of mathematics, while Problems 10 and 17 turned out to be closely related to mathematical logic.
TO THE BEAT OF A DIFFERENT DRUMMER♣....FREEDOM, ANARCHY AND
, 2010
"... ♣ As in Thoreau's enchanting elegy to the independent thinker, in Walden. He reminded the `conformist ' that `If a man does not keep pace with his companions', it may well be because `he hears the beat of a different drummer.' ♦ As almost always in my idle speculations, I have b ..."
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♣ As in Thoreau's enchanting elegy to the independent thinker, in Walden. He reminded the `conformist ' that `If a man does not keep pace with his companions', it may well be because `he hears the beat of a different drummer.' ♦ As almost always in my idle speculations, I have been fortified and sustained in these endeavours by the enlightened wisdom of my friend and colleague, Stefano Zambelli. He is more wedded to the validity of the ChurchTuring Thesis than I. I suspect this has to do with age and the increasing awareness of the fragility and fallibility of my mind. Unfortunately, I cannot blame him for any of the many remaining infelicities in this paper. Abstract1 In this paper I attempt to make a case for promoting the courage of rebels within the citadels of orthodoxy in academic research environments. Wicksell in Macroeconomics, Brouwer in the Foundations of Mathematics,Turing in Computability Theory, Sra¤a in the Theories of Value and Distribution are, in my own
elds of research, paradigmatic examples of rebels, adventurers and nonconformists of the highest calibre in scienti
c research within University
Grundlagenstreit in the Theory of Programming
, 2003
"... The explication of (declarative) computer programming by the theories of of mathematical logic has become quite important with the advent of practically usable intelligent proof assistants targeted to the veri cation of computer programs. We will argue that the present day situation is reminisc ..."
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The explication of (declarative) computer programming by the theories of of mathematical logic has become quite important with the advent of practically usable intelligent proof assistants targeted to the veri cation of computer programs. We will argue that the present day situation is reminiscent of the situation in the foundations of mathematics during the socalled Foundation Wars of 1920's. The platonists advocating the full use of Cantor's set theory battled with the intuitionists led by Brouwer. The rst camp is represented in the theory of programming by the adherents of Higher Order Logic (HOL) and the second by the Type Theory. Ironically, the two camps are not as much inimical to each other as they jointly take on the middle ground explicating computer programming in Peano Arithmetic (PA), or in its secondorder conservative extension ACA0 . In this paper the author argues the middle ground.