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Complexities of Proof-Theoretical Reductions Complexities of Proof-Theoretical Reductions Michael Toppel Submitted in accordance with the requirements for the degree of Doctor of Philosophy The University of Leeds Department of Pure Mathematics September 2016 iii The candidate confirms that the work submitted is his own and that appropriate credit has been given where reference has been made to the work of others. This copy has been supplied on the understanding that it is copyright material and that no quotation from the thesis may be published without proper acknowledgement. c 2016 The University of Leeds and Michael Toppel iv v Abstract The present thesis is a contribution to a project that is carried out by Michael Rathjen 0 and Andreas Weiermann to give a general method to study the proof-complexity of Σ1- sentences. This general method uses the generalised ordinal-analysis that was given by Buchholz, Ruede¨ and Strahm in [5] and [44] as well as the generalised characterisation of provable-recursive functions of PA + TI(≺ α) that was given by Weiermann in [60]. The present thesis links these two methods by giving an explicit elementary bound for the i proof-complexity increment that occurs after the transition from the theory IDc ! + TI(≺ α), which was used by Ruede¨ and Strahm, to the theory PA + TI(≺ α), which was analysed by Weiermann. vi vii Contents Abstract . v Contents . vii Introduction 1 1 Justifying the use of Multi-Conclusion Sequents 5 1.1 Introduction . 5 1.2 A Gentzen-semantics . 8 1.3 Multi-Conclusion Sequents . 15 1.4 Comparison with other Positions and Responses to Objections . 19 1.5 Conservativity . 25 1.6 Conclusion . 27 2 Theory Reduction 29 2.1 Preliminaries . 30 2.2 Some Syntactic Notions of Theory Reduction . 40 2.2.1 Interpretability . 41 CONTENTS viii 2.2.2 Proof-Theoretical Reduction . 44 2.2.3 Translation . 52 2.2.4 Realisability . 54 2.3 An axiomatization of Theory Reduction . 67 2.4 Discussion . 71 3 A better Base Theory for Ordinal-Analysis 73 3.1 Ordinal-Analysis and Provable Conservativity . 74 3.2 Theories of Inductive Definitions and Fixed points . 82 i 3.3 Reducing IDc <!(strict) to HA . 86 3.3.1 A Proof by Buchholz . 88 3.3.2 A Proof by Ruede¨ and Strahm . 96 3.4 Conclusions . 101 i 4 A formulation of the Ω-Rule in IDc 2(strict) 105 4.1 A particular system including the Ω-rule . 106 i 4.2 Working out concepts in IDc 1 . 111 4.3 Formulating the Ω-rule . 112 4.4 Conclusion . 117 0 5 Proof-Complexity and the Deducibility of Σ1-Sentences 119 5.1 Proof-Complexity . 119 0 5.2 The Deducibility of Σ1-Sentences . 122 CONTENTS ix 5.2.1 A general characterisation of the provable recursive functions of PA + TI(≺ α) . 123 5.2.2 The classical approach . 127 0 5.2.3 A bound for the witnesses of Σ1-sentences . 129 i 6 IDc <!(strict) has at most an elementary speed-up over PA 133 6.1 Bounding Buchholz . 134 6.1.1 Formalised Recursion Theory . 134 6.1.2 Deduction-Complexity Increment for Basic Moves in Logic . 136 6.1.3 Realisability in HA . 143 6.2 Bounding Ruede¨ Strahm . 173 6.3 Sticking together and Conclusion . 186 Appendix 188 A Gentzen’s Method can be done Elementarily . 189 B The Proof-Theoretic Ordinal of EA . 206 C Deduction Systems . 214 Bibliography 218 CONTENTS x CONTENTS 1 Introduction The main purpose of the present thesis is to provide a technical lemma for a research project of Michael Rathjen and Andreas Weiermann that was outlined in Michael Rathjen’s talk at the Bertinoro International Center for Informatics in 2011. Rathjen’s and Weiermann’s work on Kruskal’s theorem in [41] foreshadows a general method to 0 bound the complexity of deductions of Σ1-sentences as well as a general method to prove 0 Π2-conservativity for arithmetical theories. Regarding the complexity of deductions of 0 Σ1-sentences, the idea is rather simple: 0 If the witnesses of a Σ1-sentence are very big, then the complexity of its deduction must lie above a certain limit. 0 The basic methods to make this precise are well known. In the case of a Σ1-sentence σ that is deducible in a theory T with a deduction of complexity n, in symbols T `n σ; one tries to give an ordinal-analysis of T . Under the assumption that this ordinal-analysis 0 gives the ordinal α one can usually prove that T is Π2-conservative over PA + TI(≺ α): T ≡ 0 PA + TI(≺ α): Π2 Since the theories PA + TI(≺ α) are well-studied, one can use classical subrecursion theory to bound the witnesses of σ by a Hardy function Hβn (0), where βn is smaller than α and depends on the complexity of the deduction of σ in T . Hence, if the witnesses of σ are bigger than Hβn (0), then the complexity of any deduction of σ must be bigger than n. Generalising this argument faces two issues even in the case when a natural ordinal notation system is used in the ordinal-analysis. First, one has to find a recipe to prove that T ≡ 0 PA + TI(≺ α) from an ordinal-analysis that is general enough to draw Π2 general conclusions from it. Here we face the problems that one has to unify the methods CONTENTS 2 of formalising an ordinal-analysis and that there are ordinal-analyses in the literature that include methods that hinder proving conservativity in the standard ways, e.g. the Ω-rule. Second, one has to find a general approach in subrecursion theory to bound the witnesses 0 of a deducible Σ1-sentence. Here we face the problem that the assignment of fundamental sequences to an ordinal notation system must be generalised in order to define a version of the Hardy hierarchy that is independent enough from the particular ordinal notation system, which was used in the ordinal-analysis, to give meaningful bounds. Both of these issues have been already resolved. Buchholz discovered in [5] that an 0 intuitionistic version of the theory of inductive definitions is Π2-conservative over PA and can therefore be used to replace PA in PA + TI(≺ α). Since all infinite deduction systems that are used in ordinal-analysis are inductively defined, this solves the first problem for a large number of ordinal-analyses that are present in the literature. In [44] 0 Ruede¨ and Strahm strengthened Buchholz’s result by showing that Π2-conservativity over PA is preserved, when certain inductive definitions are iterated; hence also the missing ordinal-analyses, those which include the Ω-rule, are covered as well. The second issue was solved by Weiermann in [60] by drawing on Buchholz’s, Cichon’s and Weiermann’s work, which is presented in [7]. Using Cichon’s notion of a norm that can be defined on all ordinal notation systems which are present in the literature, in [60] Weiermann was able to generalise the Hardy functions in a way that supports the previously given aims. However it remained open how these two solutions work together; in order to use 0 Weiermann’s approach of subrecursion theory in the case of the Π2-conservativity that was proved for Ruede’s¨ and Strahm’s theories of inductive definitions to study the 0 complexity of deductions of Σ1-sentences, one has to ensure that the function Hβn , that is found by Weiermann’s approach when working directly with PA + TI(≺ α), is not too far away from Hβm , that is found by Weiermann’s approach when working indirectly via theories of inductive definitions. In other words: one has to ensure that the proof- 0 theoretical reduction that Ruede¨ and Strahm used to prove their Π2-conservativity result gives a rather weak speed-up for the theories of inductive definitions over PA+TI(≺ α). CONTENTS 3 This is done in the present thesis. The plan of the thesis is as follows. Chapter 1 has no connection to the rest of the thesis. It is a philosophical paper that I tried to publish during my PhD. I developed a method for philosopy inspired by proof theory by following a Carnapian line, in contrast to American pragmatism. I hoped that it might be a philosophically fruitful alternative to the contemporary idealists and pragmatists in modern inferentialism. I tried this by solving a central problem in proof-theoretical semantic by working out the differences between the requirement for harmony and that for compositionality, and focusing on the latter without ignoring the former.1 I hoped that a purely syntax-based semantic, that is also an alternative to the commonly used holistic syntax-based semantic, would show the community of philosophy of logic that there is a fruitful way of doing philosophy syntactically without relying on common human behaviour, because common human behaviour seems to be a bad starting point, when a highly academic field like logic is the subject of philosophical theorising. For a certain amount of “educational brain-washing” has to take place before people consider studying logic in a certain manner, which usually differs from ordinary human behaviour.2 Chapter 2 starts with most of the bookkeeping that has to be done; hence most of the standard notions that are used in metamathematics are defined. Also the most common notions of theory reduction are presented and compared. Primarily the notion of proof- theoretical reduction is introduced and it is emphasised that most of the techniques that are used in proof-theory, including some other notions of theory reduction, fall under this notion.
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