Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Defining data types : : : : : : : : : : : : : : : : : : : : : 6 2.4 Describing processes : : : : : : : : : : : : : : : : : : : : : 8 2.5 Expressing convergence using second order validity : : : : : : : : : : : : : : : : : : : : : : : : : 9 2.6 Truth definitions: the analytical hierarchy : : : : : : : : 10 2.7 Inductive definitions : : : : : : : : : : : : : : : : : : : : : 13 3 Canonical semantics of higher order logic : : : : : : : : : : : : 15 3.1 Tarskian semantics of second order logic : : : : : : : : : 15 3.2 Function and re

A central theme running through all the main areas of Mathematical Logic is the classification of sets, functions or theories, by means of transfinite hierarchies whose ordinal levels measure their `rank' or `complexity' in some sense appropriate to the underlying context. In Proof Theory this is manifest in the assignment of `proof theoretic ordinals' to theories, gauging their `consistency strength' and `computational power'. Ordinal-theoretic proof theory came into existence in 1936, springing forth from Gentzen's head in the course of his consistency proof of arithmetic. To put it roughly, ordinal analyses attach ordinals in a given representation system to formal theories. Though this area of mathematical logic has is roots in Hilbert's "Beweistheorie " - the aim of which was to lay to rest all worries about the foundations of mathematics once and for all by securing mathematics via an absolute proof of consistency - technical results in pro...

The work reported in this thesis arises from the old idea, going back to the origins of constructive logic, that a proof is fundamentally a kind of program. If proofs can be

Abstract not found

We define an equivalent variant LK sp of the Gentzen sequent calculus LK. In LK sp weakenings or contractions can be performed in parallel. This modification allows us to interpret a symmetrical system of mix elimination rules by a finite rewriting system; the termination of this rewriting system can be machine checked. We give also a self-contained strong normalization proof by structural induction. We give another strong normalization proof by a strictly monotone subrecursive interpretation; this interpretation gives subrecursive bounds for the length of derivations. We give a strong normalization proof by applying orthogonal term rewriting results for a confluent restriction of the mix elimination system .

Abstract. The last section of “Lecture at Zilsel’s ” [9, §4] contains an interesting but quite condensed discussion of Gentzen’s first version of his consistency proof for P A [8], reformulating it as what has come to be called the no-counterexample interpretation. I will describe Gentzen’s result (in game-theoretic terms), fill in the details (with some corrections) of Gödel’s reformulation, and discuss the relation between the two proofs. 1. Let me begin with a description of Gentzen’s consistency proof. As had already been noted in [5], we may express it in terms of a game. 1 To simplify things, we can assume that the logical constants of the classical system of number theory, P A, are ∧, ∨, ∀ and ∃ and that negations are applied only to atomic formulas. ¬φ in general is represented by the complement φ of φ, obtained by interchanging ∧ with ∨, ∀ with ∃, and atomic sentences with their negations. The components of the sentences φ ∨ ψ and φ ∧ ψ are φ and ψ. The components of the sentences ∃xφ(x) and ∀xφ(x) are the sentences φ(¯n) for each numeral ¯n. A ∧- or ∀-sentence, called a �-sentence, is thus expressed by the conjunction of its components and a ∨- or ∃-sentence, called a �-sentence, is expressed by the disjunction of them; and so it follows that every sentence can be represented as an infinitary propositional formula built up from prime sentences— atomic or negated atomic sentences. Disjunctive and conjunctive sentences with the components φn (where the range of n is 1, 2 or ω) will be denoted respectively by

Abstract not found

The objective of this paper is to introduce an ordinal representation system which has been employed in the determination of the proof--theoretic strength of \Pi 1 2 comprehension and related systems. 1 Introduction The purpose of this paper is to provide an ordinal representation system for the system of \Pi 1 2 analysis, which is the subsystem of formal second order arithmetic, Z 2 , with comprehension confined to \Pi 1 2 -formulae. The ordinal representation can also be used to provide ordinal analyses for theories that are reducible to iterated \Pi 1 2 comprehension, e.g. \Delta 1 3 comprehension. The details will be laid out in the second part of this paper. Ordinal-theoretic proof theory came into existence in 1936, springing forth from Gentzen's head in the course of his consistency proof of arithmetic. Gentzen fostered hopes that with sufficiently large constructive ordinals one could establish the consistency of analysis, i.e., Z 2 . Considerable progress has been made...

Abstract. In this paper we introduce a notation system for the infinitary derivations occurring in the ordinal analysis of KP + Π3-Reflection due to Michael Rathjen. This allows a finitary ordinal analysis of KP+Π3-Reflection. The method used is an extension of techniques developed by Wilfried Buchholz, namely operator controlled notation systems for RS ∞-derivations. Similarly to Buchholz we obtain a characterisation of the provably recursive functions of KP+Π3-Reflection as <-recursive functions where < is the ordering on Rathjen’s ordinal notation system T (K). Further we show a conservation result for Π 0 2-sentences. §1. Introduction. Ordinal analysis uses cut-elimination techniques for proof theoretic investigations. The termination of the cut-elimination process is guaranteed by assigning decreasing ordinals to the proofs emerging in the process. Gerhard Gentzen was the first to form a relationship between an ordinal ε0 and a foundational mathematical theory (nowadays denoted Peano Arithmetic PA)

The objective of this paper is to present an ordinal analysis for the fragment of second order arithmetic with \Delta 1 2 -comprehension, bar induction and \Pi 1 2 -comprehension for formulae without set parameters.