Kohlenbach, U., Real growth in standard parts of analysis. Habilitationsschrift pp. xv+166, Frankfurt (1995).  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... added and for sentences involving higher types (instead 2 sentences) These results and a discussion of the reasons for the failure of the methods used in [6] and [9] which at least as they stand there can not be used to yield our positive results) are developed in chapters 11,12 of [5] and will be published in a paper under preparation. ....

Kohlenbach, U., Real growth in standard parts of analysis. Habilitationsschrift pp. xv+166, Frankfurt (1995).

.... of uniform bounds extractable from proofs which use them) in a subsequent paper using the method developed in this paper and discuss now only (PCM) in order to motivate the results of the present paper which is the second one in a sequence of papers resulting from the authors Habilitationsschrift [12]. All undefined notions are used in the sense of [14] on which this paper relies. A 0 , B 0 , C 0 , always denote quantifier free formulas. Using a convenient representation of real numbers, PCM) can be formalized as follows: PCM) 0 hk( a m ) PCM) immediately ....

.... and therefore causes an immense rate of growth (when added to e.g. G 2 A ) From the work in the context of reverse mathematics (see e.g. 3] 20] it is known that 1) 5) imply CA ar relatively to (a second order version of) AC qf (see [1] for the definition of ) In [12] it is shown that this holds even relatively to G 2 A . In contrast to these general facts we prove in this paper a meta theorem which in particular implies that if (PCM) is applied in a proof only to sequences (a n ) which are given explicitely in the parameters of the proposition (which is ....

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Kohlenbach, U., Real growth in standard parts of analysis. Habilitationsschrift, xv+166 pp., Frankfurt (1995).

....for PBA and w.r.t. to the great variety of theorems which can be expressed in the form (13) which is due to the fact that e.g. 0 f(x)dx and sup can be defined explicitly in PBA by certain functionals of type level 2 (see appendix A4 below) 3 The range of hereditarily polynomial analysis In [14], 15] we proposed a system G 2 A AC qf Delta for PBA. Here G 2 A is the second system in a hierarchy of subsystems (G n A ) n2IN of arithmetic in all finite types. The definable type 1 objects of G n A correspond to the well known Grzegorczyk hierarchy. Moreover G n A contains ....

....: Phi max f . Delta : f9V ffifl t8u ; w G 0 (u; V u; w) 8u 9v ffi tu8w G 0 (u; v; w) 2 Deltag: 3) G n A i denotes the intuitionistic variant of G n A 4) E G n A is the extension of G n A obtained by adding the extensionality implication for all types. Theorem 3. 3 ([14], 15] Let A 1 (x ) be a Sigma 1 formula which contains only x; k; y; z as free variables and let s be a closed term of G n A . Furthermore let Delta be a set of closed axioms of the form 8u 9u ffi 8w G 0 (u; v; w) with deg(ffi) 1. Then the following rule holds ....

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Kohlenbach, U., Real growth in standard parts of analysis. Habilitationsschrift, pp. xv+166, Frankfurt (1995).

....version of the binary Konig s lemma, allow to carry out a great deal of classical analysis even for n = 2; 3. The axioms Delta and AC qf do not contribute to the growth of extractable uniform bounds which in the particular case of G 2 A are polynomials (see [12] 14] and in particular [10] for more information) In contrast to this, fragments of arithmetical comprehension and choice as well as generalizations of our principle of uniform Sigma 1 boundedness (from [12] to more complex formulas do contribute significantly to the arithmetic strength of the base systems. In [13] we ....

.... 2 IA (which is known to prove the totality of the Ackermann function as well as the consistency of EA Sigma 1 IA) is a subsystem of EA Using a more involved argument one can show that already Phi it Pi proves the totality of the Ackermann function (see chapter 12 of [10] for details on this) So any proof of conservation of systems based on Pi over Sigma k IA has to take into account carefully the structure of the functionals of type level 2 which are definable in the given system. Things become of course even more complicated for the systems G n ....

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Kohlenbach, U., Real growth in standard parts of analysis. Habilitationsschrift, pp. xv+166, Frankfurt (1995).

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Kohlenbach, U., Real growth in standard parts of analysis. Habilitationsschrift, pp. xv+166, Frankfurt (1995).

....quantifier free formula containing only x; y; z free and s is a closed term. In particular for n = 2 (resp. n = 3) the extractability of a bound Phiuk which is a polynomial (resp. a finitely iterated exponential function) in u M x : max ix u(i) and k is guaranteed (see [15] for details) In [14] we have shown that for suitable Delta already G 2 A Delta AC qf covers a substantial part of standard analysis. In fact essentially only analytical axioms (4) having types ffi; ae 1; 0 are sufficient. The proof of the verification of the extracted bound Phi also relies on these ....

.... an immense rate of growth (when added to e.g. G 2 A ) From the work in the context of reverse mathematics (see e.g. 6] 22] it is known that 1) 5) imply CA ar relatively to (a second order version of) c PA j n AC 0;0 qf (see [5] for the definition of c PA j n ) In [14] it is shown that this holds even relatively to G 2 A . In contrast to these general facts on huge growth we prove in this paper a theorem which in particular implies that if (PCM) is applied in a proof only to sequences (a n ) which are given explicitely in the parameters of the proposition ....

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Kohlenbach, U., Real growth in standard parts of analysis. Habilitationsschrift, xv+166 p., Frankfurt (1995).

....version of the binary Konig s lemma WKL, allow to carry out a great deal of classical analysis even for n = 2; 3. The axioms Delta and AC qf do not contribute to the growth of extractable uniform bounds which in the particular case of G 2 A are polynomials (see [12] 14] and in particular [10] for more information) In contrast to this, fragments of arithmetical comprehension and choice as well as generalizations of our principle of uniform Sigma 0 1 boundedness (from [12] to more complex formulas do contribute significantly to the arithmetical strength of the base systems. In [13] ....

....as well as the consistency of EA Sigma 0 1 IA) is a subsystem of EA 2 Phi it AC 0;0 qf Pi 0 1 AC Gamma . Using a more involved argument one can show that already EA 2 Phi it Pi 0 1 AC Gamma proves the totality of the Ackermann function (see chapter 12 of [10] for details on this) So any proof of conservation of systems based on Pi 0 k AC Gamma over Sigma 0 k IA has to take into account carefully the structure of the functionals of type 2 which are definable in the given system. Things become of course even more complicated for the systems ....

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Kohlenbach, U., Real growth in standard parts of analysis. Habilitationsschrift, pp. xv+166, Frankfurt (1995).

....In a previous paper [10] we introduced a hierarchy (GnA ) n2IN of subsystems of classical arithmetic in all finite types where the growth of definable functions of GnA corresponds to the well known Grzegorczyk hierarchy. Let AC qf denote the schema of quantifier free choice. [8], 10] and subsequent papers (under preparation) study various analytical principles Gamma in the context of the theories GnA AC qf (mainly for n = 2) and use proof theoretic tools like e.g. monotone functional interpretation (which was introduced in [9] to determine their impact on the ....

.... AC qf the extractability of bounds Phiuk which are polynomials in u M n : max(u0; un) k is guaranteed (or if the proof relies on certain functions of exponential growth which are not iterated in the proof, then the bound will be of polynomial growth relative to these functions, see [8], 10] 12] This paper essentially contains material from chapter 8 of the author s Habilitationschrift. Some of the results were presented at the Logic Colloquium 94 at Clermont Ferrand (see [7] 1 As is well known (cf. the discussion at the end of x3 of [10] the use of classical logic ....

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Kohlenbach, U., Real growth in standard parts of analysis. Habilitationsschrift, xv+166 p., Frankfurt (1995). 20

....subsystems of analysis with low rate of growth of provably recursive functionals Ulrich Kohlenbach Fachbereich Mathematik J.W. Goethe Universitt D 60054 Frankfurt, Germany September 1995 Abstract 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 ....

....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 # AC qf # for suitable #. 1 Introduction 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. Let U be a complete separable metric space, K a compact metric space and A # # 0 1 . As we have elaborated in [21] many numerically interesting theorems in analysis can be transformed into sentences having the ....

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Kohlenbach, U., Real growth in standard parts of analysis. Habilitationsschrift, pp. xv+166, Frankfurt (1995).

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Kohlenbach, U., Real growth in standard parts of analysis. Preprint xv+166 pp., Frankfurt (1995).

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Kohlenbach, U., Real growth in standard parts of analysis. Habilitationsschrift, pp. xv+166, Frankfurt (1995).

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Kohlenbach, U., Real growth in standard parts of analysis. Habilitationsschrift, pp. xv+166, Frankfurt (1995).

.... use of a logical principle of uniform boundedness in analysis Ulrich Kohlenbach Department of Mathematics University of Michigan Ann Arbor MI 48109, USA December 1996 1 Introduction This paper is part of a sequence of papers ( 9] 10] 11] 12] resulting from our Habilitation thesis [8] addressing the following question: What is the impact on the growth of extractable uniform bounds the use of various analytical principles Gamma in a given proof of an 89 sentence might have In particular we are interested in analyzing proofs of sentences having the form (1) 8u 1 ; k 0 8v ....

....for n = 2 (resp. n = 3) one can extract a bound Phiuk which is a polynomial (resp. a finitely iterated exponential function) in u M x : max ix u(i) and k (see [9] for details) For suitable Delta already G 2 A Delta AC qf covers a substantial part of standard analysis (see [8], 9] In [9] also we introduced new axioms F and F Gamma which both (essentially) have the form (4) and are true in the type structure of all strongly majorizable functionals (see [1] but are false in the full set theoretic model (weaker versions of these axioms were studied already in [7] ....

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Kohlenbach, U., Real growth in standard parts of analysis. Habilitationsschrift, xv+166 p., Frankfurt (1995).

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