Book contents
- Frontmatter
- Contents
- Preface
- An Introduction To Finitary Analyses Of Proof Figures
- What Mathematical Truth Could Not Be – II
- Proof Search in Constructive Logics
- David's Trick
- A Semantical Calculus for Intuitionistic Propositional Logic
- An Iteration Model Violating the Singular Cardinals Hypothesis
- An Introduction to Core Model Theory
- Games of Countable Length
- On the Complexity of the Propositional Calculus
- The Realm of Orinal Analysis
- Covering Properties of Core Models
- Ordinal Systems
- Polish Group Topologies
- Forcing Closed Unbounded Subsets of Nw+1
- First Steps into Metapredicativity in Explicit Mathematics
- What Makes A (Pointwise) Subrecursive Hierarchy Slow Growing?
- Minimality Arguments for Infinite Time Turing Degrees
An Introduction To Finitary Analyses Of Proof Figures
Published online by Cambridge University Press: 05 September 2013
- Frontmatter
- Contents
- Preface
- An Introduction To Finitary Analyses Of Proof Figures
- What Mathematical Truth Could Not Be – II
- Proof Search in Constructive Logics
- David's Trick
- A Semantical Calculus for Intuitionistic Propositional Logic
- An Iteration Model Violating the Singular Cardinals Hypothesis
- An Introduction to Core Model Theory
- Games of Countable Length
- On the Complexity of the Propositional Calculus
- The Realm of Orinal Analysis
- Covering Properties of Core Models
- Ordinal Systems
- Polish Group Topologies
- Forcing Closed Unbounded Subsets of Nw+1
- First Steps into Metapredicativity in Explicit Mathematics
- What Makes A (Pointwise) Subrecursive Hierarchy Slow Growing?
- Minimality Arguments for Infinite Time Turing Degrees
Summary
Abstract
In this paper we expound the approach to proof theory due to Gentzen-Takeuti: the finitary analysis of proof figures using ordinal diagrams. As an example we take up theories of strength measured by the Howard ordinal. First we recall Takeuti's consistency proof for a subsystem BI of second order arithmetic in a retrospective analysis of the genesis of ordinal diagrams. Second a theory of Π2-reflecting ordinals is analysed in the spirit of Gentzen-Takeuti. This paper is intended to be an introduction to an ongoing project of the author's, in which a proof theory for theories of recursively large ordinals is developed.
G. Gentzen published his new version of a consistency proof for first order number theory in 1938 [26]. He had already given two consistency proofs [24] and [25]. The first used a constructive but rather abstract notion of Junctionals. In the second he had first introduced transfinite ordinals in proof theory. Although he formulated the result as a consistency proof, his interest seems to involve a divergence from Hilbert's program. Concerning this development G. Kreisel [28] p. 262 wrote:
…, by introducing a quantitative ordinal measure he (=Gentzen) forces us to pay attention to combinatorial complexity and thereby makes it at least more difficult for us to slip into an abstract reading.
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- Information
- Sets and Proofs , pp. 1 - 26Publisher: Cambridge University PressPrint publication year: 1999
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