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What's so special about Kruskal's Theorem AND THE ORDINAL Γ0? A SURVEY OF SOME RESULTS IN PROOF THEORY
 ANNALS OF PURE AND APPLIED LOGIC, 53 (1991), 199260
, 1991
"... This paper consists primarily of a survey of results of Harvey Friedman about some proof theoretic aspects of various forms of Kruskal’s tree theorem, and in particular the connection with the ordinal Γ0. We also include a fairly extensive treatment of normal functions on the countable ordinals, an ..."
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Cited by 43 (3 self)
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This paper consists primarily of a survey of results of Harvey Friedman about some proof theoretic aspects of various forms of Kruskal’s tree theorem, and in particular the connection with the ordinal Γ0. We also include a fairly extensive treatment of normal functions on the countable ordinals, and we give a glimpse of Veblen hierarchies, some subsystems of secondorder logic, slowgrowing and fastgrowing hierarchies including Girard’s result, and Goodstein sequences. The central theme of this paper is a powerful theorem due to Kruskal, the “tree theorem”, as well as a “finite miniaturization ” of Kruskal’s theorem due to Harvey Friedman. These versions of Kruskal’s theorem are remarkable from a prooftheoretic point of view because they are not provable in relatively strong logical systems. They are examples of socalled “natural independence phenomena”, which are considered by most logicians as more natural than the metamathematical incompleteness results first discovered by Gödel. Kruskal’s tree theorem also plays a fundamental role in computer science, because it is one of the main tools for showing that certain orderings on trees are well founded. These orderings play a crucial role in proving the termination of systems of rewrite rules and the correctness of KnuthBendix completion procedures. There is also a close connection between a certain infinite countable ordinal called Γ0 and Kruskal’s theorem. Previous definitions of the function involved in this connection are known to be incorrect, in that, the function is not monotonic. We offer a repaired definition of this function, and explore briefly the consequences of its existence.
Partial realizations of Hilbert’s program
 Journal of Symbolic Logic
, 1988
"... JSTOR is a notforprofit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JS ..."
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Cited by 38 (8 self)
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JSTOR is a notforprofit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. Association for Symbolic Logic is collaborating with JSTOR to digitize, preserve and extend access to The
Higher Order Logic
 In Handbook of Logic in Artificial Intelligence and Logic Programming
, 1994
"... Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Definin ..."
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Cited by 19 (0 self)
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Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Defining data types : : : : : : : : : : : : : : : : : : : : : 6 2.4 Describing processes : : : : : : : : : : : : : : : : : : : : : 8 2.5 Expressing convergence using second order validity : : : : : : : : : : : : : : : : : : : : : : : : : 9 2.6 Truth definitions: the analytical hierarchy : : : : : : : : 10 2.7 Inductive definitions : : : : : : : : : : : : : : : : : : : : : 13 3 Canonical semantics of higher order logic : : : : : : : : : : : : 15 3.1 Tarskian semantics of second order logic : : : : : : : : : 15 3.2 Function and re
Gödel's program for new axioms: Why, where, how and what?
 IN GODEL '96
, 1996
"... From 1931 until late in his life (at least 1970) Gödel called for the pursuit of new axioms for mathematics to settle both undecided numbertheoretical propositions (of the form obtained in his incompleteness results) and undecided settheoretical propositions (in particular CH). As to the nature of ..."
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Cited by 16 (6 self)
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From 1931 until late in his life (at least 1970) Gödel called for the pursuit of new axioms for mathematics to settle both undecided numbertheoretical propositions (of the form obtained in his incompleteness results) and undecided settheoretical propositions (in particular CH). As to the nature of these, Gödel made a variety of suggestions, but most frequently he emphasized the route of introducing ever higher axioms of in nity. In particular, he speculated (in his 1946 Princeton remarks) that there might be a uniform (though nondecidable) rationale for the choice of the latter. Despite the intense exploration of the "higher infinite" in the last 30odd years, no single rationale of that character has emerged. Moreover, CH still remains undecided by such axioms, though they have been demonstrated to have many other interesting settheoretical consequences. In this paper, I present a new very general notion of the "unfolding" closure of schematically axiomatized formal systems S which provides a uniform systematic means of expanding in an essential way both the language and axioms (and hence theorems) of such systems S. Reporting joint work with T. Strahm, a characterization is given in more familiar terms in the case that S is a basic
Predicative Foundations of Arithmetic
 Journal of Philosophical Logic
, 1995
"... Predicative mathematics in the sense originating with Poincaré andWeylbegins by taking the natural number system for granted, proceeding immediately to real analysis and related fields. On the other hand, from a logicist or settheoretic standpoint, this appears problematic, for, as the story is usu ..."
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Cited by 10 (3 self)
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Predicative mathematics in the sense originating with Poincaré andWeylbegins by taking the natural number system for granted, proceeding immediately to real analysis and related fields. On the other hand, from a logicist or settheoretic standpoint, this appears problematic, for, as the story is usually told, impredicative
The Realm of Ordinal Analysis
 SETS AND PROOFS. PROCEEDINGS OF THE LOGIC COLLOQUIUM '97
, 1997
"... A central theme running through all the main areas of Mathematical Logic is the classification of sets, functions or theories, by means of transfinite hierarchies whose ordinal levels measure their `rank' or `complexity' in some sense appropriate to the underlying context. In Proof Theory this is ma ..."
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Cited by 8 (3 self)
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A central theme running through all the main areas of Mathematical Logic is the classification of sets, functions or theories, by means of transfinite hierarchies whose ordinal levels measure their `rank' or `complexity' in some sense appropriate to the underlying context. In Proof Theory this is manifest in the assignment of `proof theoretic ordinals' to theories, gauging their `consistency strength' and `computational power'. Ordinaltheoretic proof theory came into existence in 1936, springing forth from Gentzen's head in the course of his consistency proof of arithmetic. To put it roughly, ordinal analyses attach ordinals in a given representation system to formal theories. Though this area of mathematical logic has is roots in Hilbert's "Beweistheorie "  the aim of which was to lay to rest all worries about the foundations of mathematics once and for all by securing mathematics via an absolute proof of consistency  technical results in pro...
Challenges to Predicative Foundations of Arithmetic
 in Between Logic and Intuition Essays in Honor of Charles Parsons
, 1996
"... This paper was written while the first author was a Fellow at the Center for Advanced Study in the Behavioral Sciences (Stanford, CA) whose facilities and support, under grants from the Andrew W. Mellon Foundation and the National Science Foundation, have been greatly appreciated. ..."
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Cited by 4 (0 self)
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This paper was written while the first author was a Fellow at the Center for Advanced Study in the Behavioral Sciences (Stanford, CA) whose facilities and support, under grants from the Andrew W. Mellon Foundation and the National Science Foundation, have been greatly appreciated.
The modeltheoretic ordinal analysis of theories of predicative strength
 Journal of Symbolic Logic
, 1999
"... We use modeltheoretic methods described in [3] to obtain ordinal analyses of a number of theories of first and secondorder arithmetic, whose prooftheoretic ordinals are less than or equal to Γ0. 1 ..."
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Cited by 4 (1 self)
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We use modeltheoretic methods described in [3] to obtain ordinal analyses of a number of theories of first and secondorder arithmetic, whose prooftheoretic ordinals are less than or equal to Γ0. 1
A Correspondence between MartinLöf Type Theory, the Ramified Theory of Types and Pure Type Systems
 Journal of Logic, Language and Information
, 2001
"... In Russell's Ramified Theory of Types rtt, two hierarchical concepts dominate: orders and types. The use of orders has as a consequence that the logic part of rtt is predicative. The concept of order however, is almost dead since Ramsey eliminated it from rtt. This is why we find Church's simple the ..."
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Cited by 3 (1 self)
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In Russell's Ramified Theory of Types rtt, two hierarchical concepts dominate: orders and types. The use of orders has as a consequence that the logic part of rtt is predicative. The concept of order however, is almost dead since Ramsey eliminated it from rtt. This is why we find Church's simple theory of types (which uses the type concept without the order one) at the bottom of the Barendregt Cube rather than rtt. Despite the disappearance of orders which have a strong correlation with predicativity, predicative logic still plays an influential role in Computer Science. An important example is the proof checker Nuprl, which is based on MartinLöf's Type Theory which uses type universes. Those type universes, and also degrees of expressions in Automath, are closely related to orders. In this paper, we show that orders have not disappeared from modern logic and computer science, rather, orders play a crucial role in understanding the hierarchy of modern systems. In order to achieve our goal, we concentrate on a subsystem of Nuprl. The novelty of our paper lies in: 1) a modest revival of Russell's orders, 1 2) the placing of the historical system rtt underlying the famous Principia Mathematica in a context with a modern system of computer mathematics (Nuprl) and modern type theories (MartinLöf's type theory and PTSs), and 3) the presentation of a complex type system (Nuprl) as a simple and compact PTS.