Abstract
Axiom systems are ubiquitous in mathematical logic, one famous example being first order Peano Arithmetic. Foundational questions asked about axiom systems comprise analysing their provable consequences, describing their class of provable recursive functions (i.e. for which programs can termination be proven from the axioms), and characterising their consistency strength. One branch of proof theory, called Ordinal Analysis, has been quite successful in giving answers to such questions, often providing a unifying approach to them. The main aim of Ordinal Analysis is to reduce such questions to the computation of so called proof theoretic ordinals, which can be viewed as abstract measures of the complexity inherent in axiom systems. Gentzen’s famous consistency proof of arithmetic [Gen35, Gen38] using transfinite induction up to (a notation of) Cantor’s ordinal ε 0, can be viewed as the first computation of the proof theoretic ordinal of Peano Arithmetic.
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Beckmann, A. (2007). Proofs, Programs and Abstract Complexity. In: Duparc, J., Henzinger, T.A. (eds) Computer Science Logic. CSL 2007. Lecture Notes in Computer Science, vol 4646. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74915-8_3
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