Kohlenbach, U., The use of a logical principle of uniform boundedness in analysis. To appear in: Proc. `Logic in Florence 1995'. Home/Search Document Not in Database Summary Related Articles Check |
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Proof Theory and Computational Analysis - Kohlenbach (1997) (1 citation) Self-citation (Kohlenbach) (Correct)
.... weakening which is true in S and even provable in G 3 A . Combined with AC qf, F proves a strong principle of uniform boundedness which allows to give very short proves of various non constructive analytical principles including a strong version of WKL (for details on this see [15] [17]) Definition 3.1 A term t[x ] of type 0 is called a polynomial in x; k if it is built up from 0 ; S; Delta; x; k only by application. Notation 3.2 1) For f we define f : Phi max f . Delta : f9V ffifl t8u ; w G 0 (u; V u; w) 8u 9v ffi tu8w G 0 (u; v; w) 2 ....
....appendix B and still denote the resulting system by G n A Here Phi means that F must not be used in the proof of the premise of an application of the quantifier free rule of extensionality QF ER. GnA satisfies the deduction theorem w.r.t Phi but not w.r.t . Theorem 3. 5 ( 14] 15] [17]) For suitable axioms Delta of the form 8u 9v 1 tu8w G 0 (u; v; w) contains a substantial part of analysis including: 1) Basic properties of the operations ; Gamma; Delta; Delta) j Delta j; max; min and the relations = for rational numbers and real numbers (which ....
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Kohlenbach, U., The use of a logical principle of uniform boundedness in analysis. To appear in: Proc. `Logic in Florence 1995'.
The Computational Strength of Extensions of Weak König's Lemma - Kohlenbach (1998) Self-citation (Kohlenbach) (Correct)
....Variables of type 2 however are necessary for a direct representation of analytical objects and sometimes for a representation of such objects which is faithful at all. That is why we couldn t use WKL in our development of analysis in weak fragments of arithmetic in all nite types in [12] 13] [14], where e.g. continuous functions : IR IR are represented directly as type 2 functionals and not as in the second order context of reverse mathematics but relied on certain non standard principles F and F instead which are not true in the full set theoretic type structure but can be ....
.... computational strength of In order to determine the e ect (or rather non e ect as it will turn out) of 1 b AC and 1 WKL on the provably recursive functionals when added to T , we make use of a certain non standard axiom F which was introduced rst in [12] and has been applied e.g. in [14]) F : 8 9y 0 1(0) y8k 8z 1 yk( kz 0 k(y 0 k) We call this axiom non standard since it does not hold in the full set theoretic type structure S . Nevertheless its use can be eliminated from certain proofs thereby yielding classically true results. This has been discussed ....
Kohlenbach, U., The use of a logical principle of uniform boundedness in analysis. To appear in: Proc. `Logic in Florence 1995'.
On the Arithmetical Content of Restricted Forms of.. - Kohlenbach (1997) Self-citation (Kohlenbach) (Correct)
....y; z) including a generalized 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 ....
....8v tu B ar (u; v) These results will be used also to prove new conservation results for k CP over EA Sigma k IA which strengthen the Friedman Paris Kirby result. Finally we consider generalizations Pi of the principle of uniform Sigma which was studied in [12] In [14] we showed that proves already relative to G 2 A AC qf many important analytical theorems (like Dini s theorem, the attainment of the maximum for f 2 C( 0; 1] IR) the sequential Heine Borel property for [0; 1] the existence of an inverse function for every strictly monotone ....
[Article contains additional citation context not shown here]
Kohlenbach, U., The use of a logical principle of uniform boundedness in analysis. To appear in: Proc. `Logic in Florence 1995'.
Things that can and things that can't be done in PRA - Kohlenbach (1998) (1 citation) Self-citation (Kohlenbach) (Correct)
....program which realizes A from a given proof. Typically the resulting program will have a quite restricted complexity or rate of growth (compared to merely being primitive recursive) In fact in a series of papers we have shown that in many cases even a polynomial bound is guaranteed (see [9] [11], 14] among others) 2) One can argue that PRA formalizes what has been called nitistic reasoning (see e.g. 26] If the conservation of T over PRA has been established nitistically (which is possible for mathematically strong systems T (see [22] 8] then all the mathematics which can be ....
Kohlenbach, U., The use of a logical principle of uniform boundedness in analysis. To appear in: Proc. `Logic in Florence 1995'.
Arithmetizing Proofs in Analysis - Kohlenbach (1 citation) Self-citation (Kohlenbach) (Correct)
.... In G 2 A Sigma 0 1 UB Gamma and hence in G 2 A F Gamma AC 1;0 qf one can give very short and perspicuous proofs of the analytical theorems listed above and since F Gamma has the form of an axiom Delta we can extract a polynomial bound from such a proof (see [17] for details) The verification of this so far still depends on the non standard axiom F Gamma which does not hold classically, i.e. it does not hold in the full set theoretic type structure S (but only in the type structure of all so called strongly majorizable functionals M ) ....
Kohlenbach, U., The use of a logical principle of uniform boundedness in analysis. To appear in: Proc. `Logic in Florence 1995'.
On the Arithmetical Content of Restricted Forms of.. - Kohlenbach (1998) Self-citation (Kohlenbach) (Correct)
....including a generalized 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 ....
....conservation results for EA Pi 0 k CP over EA Sigma 0 k IA which strengthen the Friedman Paris Kirby result. 5 Finally we consider generalizations Pi 0 k UB Gamma j n of the principle of uniform Sigma 0 1 boundedness Sigma 0 1 UB Gamma which was studied in [12] 6 In [14] we showed that Sigma 0 1 UB Gamma proves already relative to G 2 A ACqf many important analytical theorems (like Dini s theorem, the attainment of the maximum for f 2 C( 0; 1] d ; IR) the sequential Heine Borel property for [0; 1] d , the existence of an inverse function for ....
[Article contains additional citation context not shown here]
Kohlenbach, U., The use of a logical principle of uniform boundedness in analysis. To appear in: Proc. `Logic in Florence 1995'.
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