Applied Proof Theory - Proof Interpretations and their Use in Mathematics

@inproceedings{Kohlenbach2008AppliedPT,
title={Applied Proof Theory - Proof Interpretations and their Use in Mathematics},
author={Ulrich Kohlenbach},
booktitle={Springer Monographs in Mathematics},
year={2008}
}

Corrected version Nov.20: a confused slide on the functional interpretation of weak compactness as well as a slide stating a bound on Browder's theorem have been deleted as the latter has been superseded meanwhile: weak compactess can be bypassed resulting in a primitive recursive bound.Â

Conjecture: no uniformity w.r.t. C Baillon/Bruck (1996): Uniformity w.r.t. x0, f , C for Î» k Full uniformity for general Î» k

Goebel, Kirk

Conjecture: no uniformity w.r.t. C Baillon/Bruckâ€¦

1990

Uniformity w.r.t. x0 âˆˆ C (Î» k := Î») Uniformity for unif

O Edelstein, Brien

Uniformity w.r.t. x0 âˆˆ C (Î» k := Î») Uniformityâ€¦

1978

X unif. convex, general Î» k , X , no uniformity Ishikawa

Groetsch

1972

X unif.convex,Î» k = Î», no uniformity

Browder, Petryshyn

X unif.convex,Î» k = Î», no uniformity

1967

For a (Â·) : equivalent to adding a bound b on the metric d as input

For a (Â·) : equivalent to adding a bound b on theâ€¦

Further applications of proof theory to mathematics Numerous further applications in metric fixed point theory Convex Analysis (1), Abstr

Briseid, Gerhardy, Lambov, K Leustean

Fixed Point Theory (1), Proc. Fixed Point Theory

New results on Hilbert's 17th problem (Delzell, Inventiones Math. etc.) Proof theory and Ramsey's theorem for pairs: see the talk be A. Kreuzer at this meeting

New results on Hilbert's 17th problem (Delzellâ€¦

] : can treat statements involving an Îµ-net. If only the proof uses such a net: still much easier to formalize

T [ X Benefits, C Tot

] : can treat statements involving an Îµ-net. Ifâ€¦