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Analyzing Proofs in Analysis
 LOGIC: FROM FOUNDATIONS TO APPLICATIONS. EUROPEAN LOGIC COLLOQUIUM (KEELE
, 1993
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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)
On the Arithmetical Content of Restricted Forms of Comprehension, Choice and General Uniform Boundedness
 PURE AND APPLIED LOGIC
, 1997
"... In this paper the numerical strength of fragments of arithmetical comprehension, choice and general uniform boundedness is studied systematically. These principles are investigated relative to base systems T n in all finite types which are suited to formalize substantial parts of analysis but ..."
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In this paper the numerical strength of fragments of arithmetical comprehension, choice and general uniform boundedness is studied systematically. These principles are investigated relative to base systems T n in all finite types which are suited to formalize substantial parts of analysis but nevertheless have provably recursive function(al)s of low growth. We reduce the use of instances of these principles in T n proofs of a large class of formulas to the use of instances of certain arithmetical principles thereby determining faithfully the arithmetical content of the former. This is achieved using the method of elimination of Skolem functions for monotone formulas which was introduced by the author in a previous paper. As
Things that can and things that can't be done in PRA
, 1998
"... It is wellknown by now that large parts of (nonconstructive) mathematical reasoning can be carried out in systems T which are conservative over primitive recursive arithmetic PRA (and even much weaker systems). On the other hand there are principles S of elementary analysis (like the BolzanoW ..."
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It is wellknown by now that large parts of (nonconstructive) mathematical reasoning can be carried out in systems T which are conservative over primitive recursive arithmetic PRA (and even much weaker systems). On the other hand there are principles S of elementary analysis (like the BolzanoWeierstra principle, the existence of a limit superior for bounded sequences etc.) which are known to be equivalent to arithmetical comprehension (relative to T ) and therefore go far beyond the strength of PRA (when added to T ). In this paper
The Computational Strength of Extensions of Weak König's Lemma
, 1998
"... The weak König's lemma WKL is of crucial significance in the study of on the other hand have a low prooftheoretic and computational strength. In addition to the restriction to binary trees (or equivalently bounded trees), WKL is also `weak' in that the tree predicate is quantifierfree. W ..."
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The weak König's lemma WKL is of crucial significance in the study of on the other hand have a low prooftheoretic and computational strength. In addition to the restriction to binary trees (or equivalently bounded trees), WKL is also `weak' in that the tree predicate is quantifierfree. Whereas in general the computational and prooftheoretic strength increases when logically more complex trees are allowed, we show that this is not the case for trees which are given by formulas in a class 1 where we allow an arbitrary function quantifier prefix over bounded functions in front of a 1 formula. This results in a schema 1WKL. Another way of looking at WKL is via its equivalence to the principle 8x9y 18z A 0 (x; y; z) ! 9f x:18x; z A 0 (x; fx; z); where A 0 is a quanti erfree formula (x; y; z are natural number variables). We generalize this to 1 formulas as well and allow function quantifiers `9g s' instead of `9y 1', where g s is defined pointwise. The resulting schema is called 1 bAC In the absence of functional parameters (so in particular in a second order context), the corresponding versions of 1WKL and 1 bAC turn out to be equivalent to WKL. This changes completely in the presence of functional variables of type 2 where we get proper hierarchies of principles n WKL and . Variables of type 2 however are necessary for a direct representation of analytical objects and { sometimes { for a faithful representation of such objects at all as we will show in a subsequent paper. By a reduction of 1WKL and 1 bAC to a nonstandard axiom F (introduced in a previous paper) and a new elimination result for F relative to various fragment of arithmetic in...
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"... this le with prentcsmacro.sty for your meeting, or with entcsmacro.sty for your meeting. Both can be found at the ENTCS Macro Home Page. A logical uniform boundedness principle for abstract metric and hyperbolic spaces ..."
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this le with prentcsmacro.sty for your meeting, or with entcsmacro.sty for your meeting. Both can be found at the ENTCS Macro Home Page. A logical uniform boundedness principle for abstract metric and hyperbolic spaces
This document in subdirectoryRS/98/18/ Things
, 1998
"... Reproduction of all or part of this work is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a list of recent BRICS Report Series publications. Copies may be obtained by contacting: BRICS ..."
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Reproduction of all or part of this work is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a list of recent BRICS Report Series publications. Copies may be obtained by contacting: BRICS
Bounded Functional Interpretation
"... We present a new functional interpretation, based on a novel assignment of formulas. In contrast with Gödel’s functional “Dialectica ” interpretation, the new interpretation does not care for precise witnesses of existential statements, but only for bounds for them. New principles are supported by o ..."
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We present a new functional interpretation, based on a novel assignment of formulas. In contrast with Gödel’s functional “Dialectica ” interpretation, the new interpretation does not care for precise witnesses of existential statements, but only for bounds for them. New principles are supported by our interpretation, including (a version of) the FAN theorem, weak König’s lemma and the lesser limited principle of omniscience. Conspicuous among these principles are also refutations of some laws of classical logic. Notwithstanding, we end up discussing some applications of the new interpretation to theories of classical arithmetic and analysis.