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The µ Quantification Operator in Explicit Mathematics With Universes and Iterated Fixed Point Theories With Ordinals
, 1998
"... This paper is about two topics: 1. systems of explicit mathematics with universes and a nonconstructive quantification operator ¯; 2. iterated fixed point theories with ordinals. We give a prooftheoretic treatment of both families of theories; in particular, ordinal theories are used to get upper ..."
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This paper is about two topics: 1. systems of explicit mathematics with universes and a nonconstructive quantification operator ¯; 2. iterated fixed point theories with ordinals. We give a prooftheoretic treatment of both families of theories; in particular, ordinal theories are used to get upper bounds for explicit theories with finitely many universes. 1 Introduction The two major frameworks for explicit mathematics that were introduced in Feferman [4, 5] are the theories T 0 and T 1 . T 1 results from T 0 by strengthening the applicative axioms by the socalled nonconstructive ¯ operator. Although highly nonconstructive, ¯ is predicatively acceptable and makes quantification over the natural numbers explicit. While the proof theory of T 0 is wellknown since the early eighties (cf. Feferman [4, 5], Feferman and Sieg [10], Jager [14], Jager and Pohlers [17]), the corresponding investigations of subystems of T 1 have been completed only recently by Feferman and Jager [9, 8] and G...
The Unfolding of NonFinitist Arithmetic
, 2000
"... The unfolding of schematic formal systems is a novel concept which was initiated in Feferman [6]. This paper is mainly concerned with the prooftheoretic analysis of various unfolding systems for nonnitist arithmetic NFA. In particular, we examine two restricted unfoldings U 0 (NFA) and U 1 (NFA ..."
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The unfolding of schematic formal systems is a novel concept which was initiated in Feferman [6]. This paper is mainly concerned with the prooftheoretic analysis of various unfolding systems for nonnitist arithmetic NFA. In particular, we examine two restricted unfoldings U 0 (NFA) and U 1 (NFA), as well as a full unfolding, U(NFA). The principal results then state: (i) U 0 (NFA) is equivalent to PA; (ii) U 1 (NFA) is equivalent to RA<! ; (iii) U(NFA) is equivalent to RA< 0 . Thus U(NFA) is prooftheoretically equivalent to predicative analysis.
First Steps Into Metapredicativity in Explicit Mathematics
, 1999
"... The system EMU of explicit mathematics incorporates the uniform construction of universes. In this paper we give a prooftheoretic treatment of EMU and show that it corresponds to transfinite hierarchies of fixed points of positive arithmetic operators, where the length of these fixed point hierarc ..."
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The system EMU of explicit mathematics incorporates the uniform construction of universes. In this paper we give a prooftheoretic treatment of EMU and show that it corresponds to transfinite hierarchies of fixed points of positive arithmetic operators, where the length of these fixed point hierarchies is bounded by # 0 . 1 Introduction Metapredicativity is a new general term in proof theory which describes the analysis and study of formal systems whose prooftheoretic strength is beyond the FefermanSchutte ordinal # 0 but which are nevertheless amenable to purely predicative methods. Typical examples of formal systems which are apt for scaling the initial part of metapredicativity are the transfinitely iterated fixed point theories # ID # whose detailed prooftheoretic analysis is given by Jager, Kahle, Setzer and Strahm in [18]. In this paper we assume familiarity with [18]. For natural extensions of Friedman's ATR that can be measured against transfinitely iterated fixed point ...
Realization of analysis into Explicit Mathematics
 The Journal of Symbolic Logic
, 2000
"... We define a novel interpretation R of second order arithmetic into Explicit Mathematics. As a di#erence from standard Dinterpretation, which was used before and was shown to interpret only subsystems prooftheoretically weaker than T0 , our interpretation can reach the full strength of T0 . The ..."
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We define a novel interpretation R of second order arithmetic into Explicit Mathematics. As a di#erence from standard Dinterpretation, which was used before and was shown to interpret only subsystems prooftheoretically weaker than T0 , our interpretation can reach the full strength of T0 . The Rinterpretation is an adaptation of Kleene's recursive realizability, and is applicable only to intuitionistic theories. Introduction Systems of Explicit Mathematics were introduced by S. Feferman in the 70es as a logical framework for Bishopstyle constructive mathematics (see [Fef75], [Fef79]). In [Fef79] he gave an embedding of the basic theory T 0 into a subsystem # 1 2 CA+BI of 2nd order arithmetic and conjectured that the converse also holds. In [Ja83] G. Jager carried out a necessary wellordering proof in T 0 , which together with [JP82] completed its prooftheoretical analysis and established prooftheoretic equivalence of the system of Explicit Mathematics T 0 , system o...
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 latter ..."
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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...
Weak theories of operations and types
"... This is a survey paper on various weak systems of Feferman’s explicit mathematics and their proof theory. The strength of the systems considered in measured in terms of their provably terminating operations typically belonging to some natural classes of computational time or space complexity. Keywor ..."
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This is a survey paper on various weak systems of Feferman’s explicit mathematics and their proof theory. The strength of the systems considered in measured in terms of their provably terminating operations typically belonging to some natural classes of computational time or space complexity. Keywords: Proof theory, Feferman’s explicit mathematics, applicative theories, higher types, types and names, partial truth, feasible operations 1
Metapredicative And Explicit Mahlo: A ProofTheoretic Perspective
"... After briefly discussing the concepts of predicativity, metapredicativity and impredicativity, we turn to the notion of Mahloness as it is treated in various contexts. Afterwards the ..."
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After briefly discussing the concepts of predicativity, metapredicativity and impredicativity, we turn to the notion of Mahloness as it is treated in various contexts. Afterwards the
Is the Continuum Hypothesis a definite mathematical problem?
"... [t]he analysis of the phrase “how many ” unambiguously leads to a definite meaning for the question [“How many different sets of integers do their exist?”]: the problem is to find out which one of the א’s is the number of points of a straight line … Cantor, after having proved that this number is gr ..."
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[t]he analysis of the phrase “how many ” unambiguously leads to a definite meaning for the question [“How many different sets of integers do their exist?”]: the problem is to find out which one of the א’s is the number of points of a straight line … Cantor, after having proved that this number is greater than א0, conjectured that it is א1. An equivalent proposition is this: any infinite subset of the continuum has the power either of the set of integers or of the whole continuum. This is Cantor’s continuum hypothesis. … But, although Cantor’s set theory has now had a development of more than sixty years and the [continuum] problem is evidently of great importance for it, nothing has been proved so far relative to the question of what the power of the continuum is or whether its subsets satisfy the condition just stated, except that … it is true for a certain infinitesimal fraction of these subsets, [namely] the analytic sets. Not even an upper bound, however high, can be assigned for the power of the continuum. It is undecided whether this number is regular or singular, accessible or inaccessible, and (except for König’s negative result) what its character of cofinality is. Gödel 1947, 516517 [in Gödel 1990, 178]
Relationships between constructive, predicative, and classical systems of analysis
 In Hendricks et al
"... Both the constructive and predicative approaches to mathematics arose during the period of what was felt to be a foundational crisis in the early part of this century. Each critiqued an essential logical aspect of classical mathematics, namely concerning the unrestricted use of the law of excluded ..."
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Both the constructive and predicative approaches to mathematics arose during the period of what was felt to be a foundational crisis in the early part of this century. Each critiqued an essential logical aspect of classical mathematics, namely concerning the unrestricted use of the law of excluded middle on the one hand, and of apparently circular \impredicative " de nitions on the other. But the positive redevelopment of mathematics along constructive, resp. predicative grounds did not emerge as really viable alternatives to classical, settheoretically based mathematics until the 1960s. Now wehave a massive amount of information, to which this lecture will constitute an introduction, about what can be done by what means, and about the theoretical interrelationships between various formal systems for constructive, predicative and classical analysis. In this nal lecture I will be sketching some redevelopments of classical analysis on both constructive and predicative grounds, with an emphasis on modern approaches. In the case of constructivity, Ihave very little to say about Brouwerian intuitionism, which has been discussed extensively in other lectures at this conference, and concentrate instead on the approach since 1967 of Errett Bishop and his school. In the case of predicativity, I concentrate on developmentsalso since the 1960swhich take up where Weyl's work left o, as described in my second lecture. In both cases, I rst look at these redevelopments from a more informal, mathematical, point This is the last of my three lectures for the conference, Proof Theory: History and