| Description |
1 online resource (164 pages) |
| Contents |
Intro; 1 Preliminaries; 1.1 The System WE-HAw; 1.2 Fragments and Extensions of WE-HAw; 1.3 The ordered Field of rational Numbers within WE-HAw; 1.4 The Archimedean ordered Field of real Numbers within WE-HAw; 1.5 Kuroda's negative Translation; 1.6 Gödel's Functional Interpretation; 1.7 Majorizability; 1.8 Monotone Functional Interpretation; 1.9 A specific metatheorem; 1.10 Models of E-PAw; 1.11 Bar Recursion; 1.12 Law of excluded Middle; 1.13 Double Negation Shift; 1.14 Axiom of Choice; 1.15 Arithmetical Comprehension; 1.16 Weak König's Lemma; 1.17 Complexity classes and bar recursion |
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2 Effective learnability2.1 Fluctuations versus effective learnability; 2.2 Cauchy statements and unrestricted use of S01-LEM; 2.3 Which Cauchy statements are effectively learnable and which are not; 2.4 When learnability implies fluctuation bounds; 3 Interpreting a strong non-linear ergodic theorem; 3.1 Arithmetizing Wittmann's proof; 3.2 Obtaining a bound; 3.3 A general bound existence theorem; 3.4 Uniform bounds for Wittmann's ergodic theorems; 3.5 Proof of Proposition 3.9; 4 Interpreting Bolzano-Weierstraß; 4.1 Interpreting Bolzano-Weierstraß |
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4.2 Analysis of the Complexity of the Realizers5 Functional interpretation for IST; 5.1 Formalities; 5.2 Nonstandard principles; 5.3 A functional interpretation for E-HA; 5.4 The system E-PA and negative translation; 5.5 A functional interpretation for E-PA; 5.6 Countable saturation |
| Summary |
Annotation The Fields medalist, Terence Tao, recently emphasized the importance of ̀̀hard'' (or finitary) analysis and connected the finitisation to the methods we will employ in this thesis: ... The main advantage of working in a finitary setting ... is that the underlying dynamical system becomes extremely explicit. ... In proof theory, this finitisation is known as Gödel functional interpretation ...For convergence theorems Tao calls the finitary formulation metastability and the corresponding explicit content its rate(s). In the case of the mean ergodic theorem such a rate can be used to obtain even an effective bound on the number of fluctuations. We introduce effective learnability and three other natural kinds of such finitary information and analyze the corresponding proof-theoretic conditions.Effective learnability not only provides means to know when to expect a bound on the number of fluctuations but also explains a very common pattern in the realizers for strong ergodic theorems. Moreover, we will see how a most natural example for a non-learnable convergence theorem closely relates to a notable exception to this pattern, the strong nonlinear ergodic theorem due to Wittmann.Finally, we show how can computational content be extracted in the context of non-standard analysis.P. Safarik, born and studied at grammar school Grösslingová in Bratislava, Diplom (ger.) and PhD in mathematics at TU Darmstadt |
| Bibliography |
Includes bibliographical references |
| Notes |
Print version record |
| Subject |
Proof theory.
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Proof theory.
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| Form |
Electronic book
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| ISBN |
9783832591601 |
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3832591605 |
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