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A ModelTheoretic Approach to Ordinal Analysis
 Bulletin of Symbolic Logic
, 1997
"... . We describe a modeltheoretic approach to ordinal analysis via the finite combinatorial notion of an #large set of natural numbers. In contrast to syntactic approaches that use cut elimination, this approach involves constructing finite sets of numbers with combinatorial properties that, in no ..."
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. We describe a modeltheoretic approach to ordinal analysis via the finite combinatorial notion of an #large set of natural numbers. In contrast to syntactic approaches that use cut elimination, this approach involves constructing finite sets of numbers with combinatorial properties that, in nonstandard instances, give rise to models of the theory being analyzed. This method is applied to obtain ordinal analyses of a number of interesting subsystems of first and secondorder arithmetic. x1. Introduction. Two of proof theory's defining goals are the justification of classical theories on constructive grounds, and the extraction of constructive information from classical proofs. Since Gentzen, ordinal analysis has been a major component in these pursuits, and the assignment of recursive ordinals to theories has proven to be an illuminating way of measuring their constructive strength. The traditional approach to ordinal analysis, which uses cutelimination procedures to transfor...
Realization of Constructive Set Theory into Explicit Mathematics: a lower bound for impredicative Mahlo universe
 Transactions American Math. Soc
, 2000
"... We define a realizability interpretation of Aczel's Constructive Set Theory CZF into Explicit Mathematics. The final results are that CZF extended by Mahlo principles is realizable in corresponding extensions of T0 , thus providing relative lower bounds for the prooftheoretic strength of the l ..."
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We define a realizability interpretation of Aczel's Constructive Set Theory CZF into Explicit Mathematics. The final results are that CZF extended by Mahlo principles is realizable in corresponding extensions of T0 , thus providing relative lower bounds for the prooftheoretic strength of the latter. Introduction Several di#erent frameworks have been founded in the 70es aiming to give a foundation for constructive mathematics. The most welldeveloped of them nowadays are MartinLof type theory, Aczel's constructive set theory, and Feferman's explicit mathematics. While constructive set theory was built to have an immediate type interpretation, no theory stronger than # 1 2 CA, which prooftheoretically is still far below the basic system T 0 of Explicit Mathematics, have been shown up to now to be directly embeddable into explicit systems. It also yielded that the only method for establishing lower bounds for T 0 and its extensions remained to be wellordering proofs. This omissi...
Does Reductive Proof Theory Have A Viable Rationale?
 Erkenntnis
, 2000
"... The goals of reduction and reductionism in the natural sciences are mainly explanatory in character, while those in mathematics are primarily foundational. In contrast to global reductionist programs which aim to reduce all of mathematics to one supposedly "universal " system or founda ..."
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The goals of reduction and reductionism in the natural sciences are mainly explanatory in character, while those in mathematics are primarily foundational. In contrast to global reductionist programs which aim to reduce all of mathematics to one supposedly "universal " system or foundational scheme, reductive proof theory pursues local reductions of one formal system to another which is more justified in some sense. In this direction, two specific rationales have been proposed as aims for reductive proof theory, the constructive consistencyproof rationale and the foundational reduction rationale. However, recent advances in proof theory force one to consider the viability of these rationales. Despite the genuine problems of foundational significance raised by that work, the paper concludes with a defense of reductive proof theory at a minimum as one of the principal means to lay out what rests on what in mathematics. In an extensive appendix to the paper, various reducti...
Universes in Type Theory Part II – Autonomous Mahlo
, 2009
"... We introduce the autonomous Mahlo universe which is an extension of MartinLöf type theory which we consider as predicatively justified and which has a strength which goes substantially beyond that of the Mahlo universe, which is before writing this paper the strongest predicatively justified publis ..."
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We introduce the autonomous Mahlo universe which is an extension of MartinLöf type theory which we consider as predicatively justified and which has a strength which goes substantially beyond that of the Mahlo universe, which is before writing this paper the strongest predicatively justified published extension of MartinLöf type theory. We conjecture it to have the same proof theoretic strength as KripkePlatek set theory extended by one recursively autonomous Mahlo ordinal and finitely many admissibles above it. Here a recursively autonomous Mahlo universe ordinal is an ordinal κ which is recursively hyper αMahlo for all α < κ. We introduce as well as intermediate steps the hyperMahlo and hyper αMahlo universes, and give meaning explanations for these theories as well as for the super and the Mahlo universe. We introduce a model for the autonomous Mahlo universe, and determine an upper bound for its proof theoretic strength, therefore establishing one half of the conjecture mentioned before. The autonomous Mahlo universe is the crucial intermediate step for understanding the Π3reflecting universe, which will be published in a successor of this article and which is even stronger and will slightly exceed the strength of KripkePlatek set theory plus the principle of Π3reflection. 1
Universes in Type Theory Part I
, 2005
"... Abstract We give an overview over universes in MartinL"of type theory and consider the following universe constructions: a simple universe, E. Palmgren's super universe and the Mahlo universe. We then introduce models for these theories in extensions of KripkePlatek set theory having ..."
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Abstract We give an overview over universes in MartinL&quot;of type theory and consider the following universe constructions: a simple universe, E. Palmgren's super universe and the Mahlo universe. We then introduce models for these theories in extensions of KripkePlatek set theory having the same proof theoretic strength. The extensions of KripkePlatek set theory used formalise the existence of a recursively inaccessible ordinal, a recursively hyperinaccessible ordinal, and a recursively Mahlo ordinal. Using these models we determine upper bounds for the proof theoretic strength of the theories in questions. In case of simple universes and the Mahlo universe, these bounds have been shown by the author to be sharp. This article is an overview over the main techniques in developing these models, full details will be presented in a series of future articles. 1 Introduction This article presents some results of a research program with the goal of determining as strong as possible predicatively justified extensions of MartinL&quot;of type theory (MLTT) and to determine their precise proof theoretic strength.We see three main reasons for following this research program:
POSSIBLEWORLDS SEMANTICS FOR MODAL NOTIONS CONCEIVED AS PREDICATES
"... Abstract. If � is conceived as an operator, i.e., an expression that gives applied to a formula another formula, the expressive power of the language is severely restricted when compared to a language where � is conceived as a predicate, i.e., an expression that yields a formula if it is applied to ..."
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Abstract. If � is conceived as an operator, i.e., an expression that gives applied to a formula another formula, the expressive power of the language is severely restricted when compared to a language where � is conceived as a predicate, i.e., an expression that yields a formula if it is applied to a term. This consideration favours the predicate approach. The predicate view, however, is threatened mainly by two problems: Some obvious predicate systems are inconsistent, and possibleworlds semantics for predicates of sentences has not been developed very far. By introducing possibleworlds semantics for the language of arithmetic plus the unary predicate �, we tackle both problems. Given a frame 〈W, R 〉 consisting of a set W of worlds and a binary relation R on W, we investigate whether we can interpret � at every world in such a way that ��A � holds at a world w ∈ W if and only if A holds at every world v ∈ W such that wRv. The arithmetical vocabulary is interpreted by the standard model at every world. Several ‘paradoxes ’ (like Montague’s Theorem, Gödel’s Second Incompleteness Theorem, McGee’s Theorem on the ωinconsistency of certain truth theories etc.) show that many frames, e.g., reflexive frames, do not
Finitary reductions for local predicativity, I: recursively regular ordinals
"... We define notation system for infinitary derivations arising from cutelimination for a theory T 1 \Sigma of recursively regular ordinals by the method of local predicativity. Using these notations, we derive finitary cutelimination steps together with corresponding ordinal assignments. Introductio ..."
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We define notation system for infinitary derivations arising from cutelimination for a theory T 1 \Sigma of recursively regular ordinals by the method of local predicativity. Using these notations, we derive finitary cutelimination steps together with corresponding ordinal assignments. Introduction There is an extensive literature connecting infinitary "Schuttestyle" and finitary "GentzenTakeutistyle" sides of proof theory. For example, in papers [Mi75, Mi75a, Mi79, Bu91, Bu97a] this was done for systems not exceeding in strength Peano Arithmetic. But most recently, there has been an interest to what one can get on the side of finitary proof theory from the methods which are used for prooftheoretical analysis of impredicative theories (see [Wei96, Bu97]). Especially we want to mention paper [Bu97], where it was shown that Takeuti's reduction steps for \Pi 1 1 \Gamma CA+ BI [Tak87, x27] can be derived from Buchholz' method of\Omega +1 rule ([BFPS, Ch. IVV], [BS88]). Here we ...