000			03671nam a22005295i 4500
001			978-3-540-77533-1
003			DE-He213
005			20161121231215.0
007			cr nn 008mamaa
008			100301s2008 gw | s |||| 0|eng d
020			_a9783540775331 _9978-3-540-77533-1
024	7		_a10.1007/978-3-540-77533-1 _2doi
050		4	_aQA8.9-10.3
072		7	_aPBC _2bicssc
072		7	_aPBCD _2bicssc
072		7	_aMAT018000 _2bisacsh
082	0	4	_a511.3 _223
100	1		_aKohlenbach, Ulrich. _eauthor.
245	1	0	_aApplied Proof Theory: Proof Interpretations and Their Use in Mathematics _h[electronic resource] / _cby Ulrich Kohlenbach.
264		1	_aBerlin, Heidelberg : _bSpringer Berlin Heidelberg, _c2008.
300			_aXX, 536 p. _bonline resource.
336			_atext _btxt _2rdacontent
337			_acomputer _bc _2rdamedia
338			_aonline resource _bcr _2rdacarrier
347			_atext file _bPDF _2rda
490	1		_aSpringer Monographs in Mathematics, _x1439-7382
505	0		_aUnwinding proofs (‘Proof Mining’) -- Intuitionistic and classical arithmetic in all finite types -- Representation of Polish metric spaces -- Modified realizability -- Majorizability and the fan rule -- Semi-intuitionistic systems and monotone modified realizability -- Gödel’s functional (‘Dialectica’) interpretation -- Semi-intuitionistic systems and monotone functional interpretation -- Systems based on classical logic and functional interpretation -- Functional interpretation of full classical analysis -- A non-standard principle of uniform boundedness -- Elimination of monotone Skolem functions -- The Friedman A-translation -- Applications to analysis: general metatheorems I -- Case study I: Uniqueness proofs in approximation theory -- Applications to analysis: general metatheorems II -- Case study II: Applications to the fixed point theory of nonexpansive mappings -- Final comments.
520			_aUlrich Kohlenbach presents an applied form of proof theory that has led in recent years to new results in number theory, approximation theory, nonlinear analysis, geodesic geometry and ergodic theory (among others). This applied approach is based on logical transformations (so-called proof interpretations) and concerns the extraction of effective data (such as bounds) from prima facie ineffective proofs as well as new qualitative results such as independence of solutions from certain parameters, generalizations of proofs by elimination of premises. The book first develops the necessary logical machinery emphasizing novel forms of Gödel's famous functional ('Dialectica') interpretation. It then establishes general logical metatheorems that connect these techniques with concrete mathematics. Finally, two extended case studies (one in approximation theory and one in fixed point theory) show in detail how this machinery can be applied to concrete proofs in different areas of mathematics. .
650		0	_aMathematics.
650		0	_aApproximation theory.
650		0	_aFunctional analysis.
650		0	_aOperator theory.
650		0	_aMathematical logic.
650	1	4	_aMathematics.
650	2	4	_aMathematical Logic and Foundations.
650	2	4	_aMathematics, general.
650	2	4	_aApproximations and Expansions.
650	2	4	_aOperator Theory.
650	2	4	_aFunctional Analysis.
710	2		_aSpringerLink (Online service)
773	0		_tSpringer eBooks
776	0	8	_iPrinted edition: _z9783540775324
830		0	_aSpringer Monographs in Mathematics, _x1439-7382
856	4	0	_uhttp://dx.doi.org/10.1007/978-3-540-77533-1
912			_aZDB-2-SMA
950			_aMathematics and Statistics (Springer-11649)
999			_c510130 _d510130