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Mathematically Strong Subsystems of Analysis With Low Rate of Growth of Provably Recursive Functionals
, 1995
"... This paper is the first one in a sequel of papers resulting from the authors Habilitationsschrift [22] which are devoted to determine the growth in proofs of standard parts of analysis. A hierarchy (GnA # )n#I N of systems of arithmetic in all finite types is introduced whose definable objects of ..."
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Cited by 34 (21 self)
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This paper is the first one in a sequel of papers resulting from the authors Habilitationsschrift [22] which are devoted to determine the growth in proofs of standard parts of analysis. A hierarchy (GnA # )n#I N of systems of arithmetic in all finite types is introduced whose definable objects of type 1 = 0(0) correspond to the Grzegorczyk hierarchy of primitive recursive functions. We establish the following extraction rule for an extension of GnA # by quantifierfree choice ACqf and analytical axioms # having the form #x # #y ## sx#z # F0 (including also a `non standard' axiom F  which does not hold in the full settheoretic model but in the strongly majorizable functionals): From a proof GnA # +ACqf + # # #u 1 , k 0 #v ## tuk#w 0 A0(u, k, v, w) one can extract a uniform bound # such that #u 1 , k 0 #v ## tuk#w # #ukA0 (u, k, v, w) holds in the full settheoretic type structure. In case n = 2 (resp. n = 3) #uk is a polynomial (resp. an elementary recursive function) in k, u M := #x. max(u0, . . . , ux). In the present paper we show that for n # 2, GnA # +ACqf+F  proves a generalization of the binary Knig's lemma yielding new conservation results since the conclusion of the above rule can be verified in G max(3,n) A # in this case. In a subsequent paper we will show that many important ine#ective analytical principles and theorems can be proved already in G2A # +ACqf+# for suitable #. 1
Elimination of Skolem functions for monotone formulas in analysis
 ARCHIVE FOR MATHEMATICAL LOGIC
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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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Cited by 11 (4 self)
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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 Uniform Weak König's Lemma
, 1999
"... The socalled weak König's lemma WKL asserts the existence of an in nite path b in any in nite binary tree (given by a representing function f ). Based on this principle one can formulate subsystems of higherorder arithmetic which allow to carry out very substantial parts of classical mathematics b ..."
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Cited by 10 (5 self)
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The socalled weak König's lemma WKL asserts the existence of an in nite path b in any in nite binary tree (given by a representing function f ). Based on this principle one can formulate subsystems of higherorder arithmetic which allow to carry out very substantial parts of classical mathematics but are 2  conservative over primitive recursive arithmetic PRA (and even weaker fragments of arithmetic). In [10] we established such conservation results relative to nite type extensions PRA of PRA (together with a quanti erfree axiom of choice schema). In this setting one can consider also a uniform version UWKL of WKL which asserts the existence of a functional which selects uniformly in a given in nite binary tree f an in nite path f of that tree. This uniform version of WKL is of interest in the context of explicit mathematics as developed by S. Feferman. The elimination process in [10] actually can be used to eliminate even this uniform weak König's lemma provided that PRA only has a quanti erfree rule of extensionality QFER instead of the full axioms (E) of extensionality for all nite types. In this paper we show that in the presence of (E), UWKL is much stronger than WKL: whereas WKL remains to be 2 conservative over PRA, PRA + (E)+UWKL contains (and is conservative over) full Peano arithmetic PA.
Saturated models of universal theories
 Annals of Pure and Applied Logic
"... A notion called Herbrand saturation is shown to provide the modeltheoretic analogue of a prooftheoretic method, Herbrand analysis, yielding uniform modeltheoretic proofs of a number of important conservation theorems. A constructive, algebraic variation of the method is described, providing yet a ..."
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Cited by 7 (3 self)
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A notion called Herbrand saturation is shown to provide the modeltheoretic analogue of a prooftheoretic method, Herbrand analysis, yielding uniform modeltheoretic proofs of a number of important conservation theorems. A constructive, algebraic variation of the method is described, providing yet a third approach, which is finitary but retains the semantic flavor of the modeltheoretic version. 1
Harrington’s conservation theorem redone
 In: Archive for Mathematical Logic
, 2008
"... Leo Harrington showed that the secondorder theory of arithmetic WKL0 is Π 1 1conservative over the theory RCA0. Harrington’s proof is modeltheoretic, making use of a forcing argument. A purely prooftheoretic proof, avoiding forcing, has been eluding the efforts of researchers. In this short paper ..."
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Cited by 2 (1 self)
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Leo Harrington showed that the secondorder theory of arithmetic WKL0 is Π 1 1conservative over the theory RCA0. Harrington’s proof is modeltheoretic, making use of a forcing argument. A purely prooftheoretic proof, avoiding forcing, has been eluding the efforts of researchers. In this short paper, we present a proof of Harrington’s result using a cutelimination argument. 1
The Computational Strength of Extensions of Weak König's Lemma
, 1998
"... The weak Konig's lemma WKL is of crucial signi cance 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 quanti erfree. Whereas in g ..."
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Cited by 1 (0 self)
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The weak Konig's lemma WKL is of crucial signi cance 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 quanti erfree. 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 quanti er pre x 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 quanti ers `9g s' instead of `9y 1', where g s is de ned pointwise. The resulting schema is Basic Research in Computer Science, Centre of the Danish National Research Foundation. 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...
Primitive Recursive Selection Functions for Existential Assertions over Abstract Algebras
"... Abstract. We generalize to abstract manysorted algebras the classical prooftheoretic result due to Parsons, Mints and Takeuti that an assertion ∀x ∃y P(x,y) (where P is Σ0 1), provable in Peano arithmetic with Σ0 1 induction, has a primitive recursive selection function. This involves a correspond ..."
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Cited by 1 (1 self)
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Abstract. We generalize to abstract manysorted algebras the classical prooftheoretic result due to Parsons, Mints and Takeuti that an assertion ∀x ∃y P(x,y) (where P is Σ0 1), provable in Peano arithmetic with Σ0 1 induction, has a primitive recursive selection function. This involves a corresponding generalization to such algebras of the notion of primitive recursiveness. The main difficulty encountered in carrying out this generalization turns out to be the fact that equality over these algebras may not be computable, and hence atomic formulae in their signatures may not be decidable. The solution given here is to develop an appropriate concept of realizability of existential assertions over such algebras, generalized to realizability of sequents of existential assertions. In this way, the results can be seen to hold for classical proof systems. This investigation may give some insight into the relationship between specifiability and computability for data types such as the reals, where the atomic formulae, i.e., equations between terms of type real, are not computable. Key words and phrases: generalized computability, realizability, selection function2 1
Classical Proofs and Programs
"... Contents 1 Introduction 1 2 General Background 2 2.1 Godel's System T . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 Intuitionistic Arithmetic for Functionals . . . . . . . . . . . . . . 6 2.3 Program Extraction from Constructive Proofs . . . . . . . . . . . 7 2.4 Example: Fibonacci N ..."
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Contents 1 Introduction 1 2 General Background 2 2.1 Godel's System T . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 Intuitionistic Arithmetic for Functionals . . . . . . . . . . . . . . 6 2.3 Program Extraction from Constructive Proofs . . . . . . . . . . . 7 2.4 Example: Fibonacci Numbers . . . . . . . . . . . . . . . . . . . . 13 3 Computational Content of Classical Proofs 14 3.1 Definite and Goal Formulas . . . . . . . . . . . . . . . . . . . . . 14 3.2 Computational Content . . . . . . . . . . . . . . . . . . . . . . . 19 3.3 Example: Fibonacci Numbers Again . . . . . . . . . . . . . . . . 23 3.4 Example: Integer Square Roots . . . . . . . . . . . . . . . . . . . 26 3.5 Example: The Greatest Common Divisor . . . . . . . . . . . . . 28 3.6 Example: Dickson's Lemma . . . . . . . . . . . . . . . . . . . . . 35 3.7 Towards More Interesting Examples . . . . . . . . . . . . . . . . 38 1 Introduction It is