U. Kohlenbach. Foundational and mathematical uses of higher types. In W. Sieg et al., editors, Reflections on the Foundations of Mathematics: Essay in Honor of Solomon Feferman, volume 15 of Lecture Notes in Logic, pages 92--116. A. K. Peters, Ltd., 2002. Home/Search Document Details and Download
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Proof Interpretations and the Computational Content of Proofs - Kohlenbach (2002) (1 citation) Self-citation (Kohlenbach) (Correct)
....elimination procedure of # 0 1 UB was established and a combination of monotone functional interpretation with a model theoretic argument was used to show that # 0 1 UB does not create new provably recursive functionals of type 2 even relative to weak systems as G 2 A # . In [86] is was observed that in the presence of full extensionality, F already implies F . In [78] we showed that # 0 1 UB allows (even relative to systems as weak as G 2 A # ) to give short proofs of important analytic theorems as Dini s, theorem, the existence of an inverse for strongly monotone ....
....require any encoding of functions f # C[0, 1] as type 1 objects which would be 2 Note that our definition of fx implies that V i n (zi # 0 yki) # zn # 0 (yk)n 0 for n # 0 n 0 . CHAPTER 8: A NON STANDARD PRINCIPLE 71 necessary to obtain the same results by WKL (see [117] and [86]) In [79] a combintation of (function parameter free) # 0 1 comprehension and # 0 1 UB was used to derive fixed instances of the Bolzano Weierstra principle relative to G 2 A # which allowed a precise calibration of the contribution of the use of such instances to the provably recursive ....
Kohlenbach, U., Foundational and mathematical uses of higher types. In: Sieg, W., Sommer, R., Talcott, C. (eds.), Reflections on the foundations of mathematics. Essays in honor of Solomon Feferman, Lecture Notes in Logic 15, A.K. Peters, pp. 92-120 (2001).
Things that can and things that can't be done in PRA - Kohlenbach (2000) (1 citation) Self-citation (Kohlenbach) (Correct)
....systems of arithmetic in all nite types. Such systems which are on the one hand mathematically very strong but on the other hand are still conservative over PRA (and even much weaker systems) have been developed by the author in a series of papers (see e.g. 8] 9] and for a general survey [15]) These facts are of interest for mainly two reasons 1) If a 0 2 sentence A is provable in T and the conservation of T over PRA has been established proof theoretically, then one can extract a primitive recursive program which realizes A from a given proof. Typically the resulting program ....
.... has been called nitistic reasoning (see e.g. 27] If the conservation of T over PRA has been established nitistically (which is possible for mathematically strong systems T (see [23] 8] then all the mathematics which can be carried out in T has a nitistic justi cation (see [25] 26] and [15] for discussions of this point) In this paper we exhibit a sharp boundary between nistically reducible parts of analysis and extensions which provably go beyond the strength of PRA. More precisely we study the (proof theoretical and numerical) strength of function parameter free schematic forms ....
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Kohlenbach, U., Foundational and mathematical uses of higher types. Preprint 21 pp., June 1999.
Higher Order Reverse Mathematics - Kohlenbach (2000) (1 citation) Self-citation (Kohlenbach) (Correct)
....by H. Friedman, S. Simpson and others (see [16] for a comprehensive treatment) focuses on the language of second order arithmetic because that language is the weakest one that is rich enough to express and develop the bulk of core mathematics ( 16] p. viii) However, as we have argued in [14], already the treatment of continuous functions f : X # Y between Polish spaces X,Y not only requires a quite complicated encoding. Even more importantly, the restricted language makes it necessary (already for X = IN IN , Y = IN) to use a constructively slightly enriched definition of ....
....the presence of arithmetical comprehension the di#erence between the encoding of continuous functionals and their direct treatment disappears. For functions f : 2 IN # IN, already the binary Konig s lemma WKL su#ces for this but it is open whether this holds e.g. in E PA # QF AC 1,0 (see [14] for all this) Thus already for those parts of analysis which only deal with continuous functions, there are reasons to extend the context of reverse mathematics to the language of arithmetic in all finite types. This need becomes even more urgent if one considers principles involving ....
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Kohlenbach, U., Foundational and mathematical uses of higher types. To appear in forthcoming Festschrift in honor of Professor Solomon Feferman.
Proof Mining in Subsystems of Analysis - Oliva (2003) (Correct)
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U. Kohlenbach. Foundational and mathematical uses of higher types. In W. Sieg et al., editors, Reflections on the Foundations of Mathematics: Essay in Honor of Solomon Feferman, volume 15 of Lecture Notes in Logic, pages 92--116. A. K. Peters, Ltd., 2002.
"Clarifying the Nature of the Infinite": the development of.. - Avigad, Reck (2001) (Correct)
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Ulrich Kohlenbach. Foundational and mathematical uses of higher types. Preprint.
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