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A ModelTheoretic Approach to Ordinal Analysis
 Bulletin of Symbolic Logic
, 1997
"... . We describe a modeltheoretic approach to ordinal analysis via the finite combinatorial notion of an #large set of natural numbers. In contrast to syntactic approaches that use cut elimination, this approach involves constructing finite sets of numbers with combinatorial properties that, in no ..."
Abstract

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. We describe a modeltheoretic approach to ordinal analysis via the finite combinatorial notion of an #large set of natural numbers. In contrast to syntactic approaches that use cut elimination, this approach involves constructing finite sets of numbers with combinatorial properties that, in nonstandard instances, give rise to models of the theory being analyzed. This method is applied to obtain ordinal analyses of a number of interesting subsystems of first and secondorder arithmetic. x1. Introduction. Two of proof theory's defining goals are the justification of classical theories on constructive grounds, and the extraction of constructive information from classical proofs. Since Gentzen, ordinal analysis has been a major component in these pursuits, and the assignment of recursive ordinals to theories has proven to be an illuminating way of measuring their constructive strength. The traditional approach to ordinal analysis, which uses cutelimination procedures to transfor...
Brief introduction to unprovability
"... Abstract The article starts with a brief survey of Unprovability Theory as of autumn 2006. Then, as an illustration of the subject's modeltheoretic methods, we reprove exact versions of unprovability results for the ParisHarrington Principle and the KanamoriMcAloon Principle using indiscernibles. ..."
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Abstract The article starts with a brief survey of Unprovability Theory as of autumn 2006. Then, as an illustration of the subject's modeltheoretic methods, we reprove exact versions of unprovability results for the ParisHarrington Principle and the KanamoriMcAloon Principle using indiscernibles. In addition, we obtain a short accessible proof of unprovability of the ParisHarrington Principle. The proof employs old ideas but uses only one colouring and directly extracts the set of indiscernibles from its homogeneous set. We also present modified, abridged statements whose unprovability proofs are especially simple. These proofs were tailored for teaching purposes. The article is intended to be accessible to the widest possible audience of mathematicians, philosophers and computer scientists as a brief survey of the subject, a guide through the literature in the field, an introduction to its modeltheoretic techniques and, finally, a modeltheoretic proof of a modern theorem in the subject. However, some understanding of logic is assumed on the part of the readers. The intended audience of this paper consists of logicians, logicaware mathematicians andthinkers of other backgrounds who are interested in unprovable mathematical statements.