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Unfolding feasible arithmetic and weak truth
, 2012
"... In this paper we continue Feferman’s unfolding program initiated in [11] which uses the concept of the unfolding U(S) of a schematic system S in order to describe those operations, predicates and principles concerning them, which are implicit in the acceptance of S. The program has been carried thro ..."
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In this paper we continue Feferman’s unfolding program initiated in [11] which uses the concept of the unfolding U(S) of a schematic system S in order to describe those operations, predicates and principles concerning them, which are implicit in the acceptance of S. The program has been carried through for a schematic system of nonfinitist arithmetic NFA in Feferman and Strahm [13] and for a system FA (with and without Bar rule) in Feferman and Strahm [14]. The present contribution elucidates the concept of unfolding for a basic schematic system FEA of feasible arithmetic. Apart from the operational unfolding U0(FEA) of FEA, we study two full unfolding notions, namely the predicate unfolding U(FEA) and a more general truth unfolding UT(FEA) of FEA, the latter making use of a truth predicate added to the language of the operational unfolding. The main results obtained are that the provably convergent functions on binary words for all three unfolding systems are precisely those being computable in polynomial time. The upper bound computations make essential use of a specific theory of truth TPT over combinatory logic, which has recently been introduced in Eberhard and Strahm [7] and Eberhard [6] and whose involved prooftheoretic analysis is due to Eberhard [6]. The results of this paper were first announced in [8].
Weak theories of truth and explicit mathematics. Submitted for publication. 19
"... We study weak theories of truth over combinatory logic and their relationship to weak systems of explicit mathematics. In particular, we consider two truth theories TPR and TPT of primitive recursive and feasible strength. The latter theory is a novel abstract truththeoretic setting which is able t ..."
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We study weak theories of truth over combinatory logic and their relationship to weak systems of explicit mathematics. In particular, we consider two truth theories TPR and TPT of primitive recursive and feasible strength. The latter theory is a novel abstract truththeoretic setting which is able to interpret expressive feasible subsystems of explicit mathematics. 1
1 Logic, Mathematics and Conceptual Structuralism
"... Abstract. Conceptual structuralism is a nonrealist philosophy of mathematics according to which the objects of mathematical thought are humanly conceived “idealworld” structures. Basic conceptions of structures, such as those of the natural numbers, the continuum, and sets in the cumulative hierar ..."
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Abstract. Conceptual structuralism is a nonrealist philosophy of mathematics according to which the objects of mathematical thought are humanly conceived “idealworld” structures. Basic conceptions of structures, such as those of the natural numbers, the continuum, and sets in the cumulative hierarchy, differ in their degree of clarity. One may speak of what is true in a given conception, but that notion of truth may be partial. Mathematics proceeds from such basic conceptions by reflective expansion and carefully reasoned argument, the last of which is analyzed in logical terms. The main questions for the role of logic here is whether there are principled demarcations on its use. It is claimed that in the case of a completely clear conception, such as that of the natural numbers, the logical notions are just those of firstorder classical logic and hence that that is the appropriate vehicle for reasoning. At the other extreme, in the case of set theory, where each set is conceived of as a definite totality but the universe of “all ” sets is an indefinite totality, it is proposed that the appropriate logic is semiintuitionistic in which classical logic applies only to (set) bounded formulas. Certain subsystems of classical set theory in which extensive parts of mathematics can be formalized are reducible to
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"... A symbolic approach to the state graph based analysis of highlevel Markov reward models Ein symbolischer Ansatz für die Zustandsgraphbasierte Analyse von ..."
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A symbolic approach to the state graph based analysis of highlevel Markov reward models Ein symbolischer Ansatz für die Zustandsgraphbasierte Analyse von
1 The Operational Perspective: Three Routes
"... Let me begin with a few personal words of appreciation, since Gerhard Jäger is one of my most valued friends and long time collaborators. It’s my pleasure to add my tribute to him for his outstanding achievements and leadership over the years, and most of all for having such a wonderful open spirit ..."
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Let me begin with a few personal words of appreciation, since Gerhard Jäger is one of my most valued friends and long time collaborators. It’s my pleasure to add my tribute to him for his outstanding achievements and leadership over the years, and most of all for having such a wonderful open spirit and being such a fine person. I first met Gerhard at the 1978 logic colloquium meeting in Mons, Belgium. He was attending that with Wolfram Pohlers and Wilfried Buchholz, with both of whom I had long enjoyed a stimulating working relationship on theories of iterated inductive definitions. From our casual conversations there, it was clear that Gerhard was already someone with great promise in proof theory. But things really took off between us a year later when we both visited Oxford University for the academic year 19791980. Gerhard had just finished his doctoral dissertation with Kurt Schütte and Wolfram Pohlers. I remember that we did a lot of walking and talking together, though I had to walk twice as fast to keep up with him. We talked a lot about proof theory and in particular about my explicit mathematics program that I had introduced in 1975 and had expanded on in my Mons lectures; Gerhard was quick to take up all my questions and to deal with them