Ordinal analysis

In proof theory, ordinal analysis assigns ordinals (often large countable ordinals) to mathematical theories as a measure of their strength. The field was formed when Gerhard Gentzen in 1934 used cut elimination to prove, in modern terms, that the proof-theoretic ordinal of Peano arithmetic is ε₀.

Ordinal analysis concerns true, effective (recursive) theories that can interpret a sufficient portion of arithmetic to make statements about ordinal notations. The proof-theoretic ordinal of such a theory ${\displaystyle T}$ is the smallest recursive ordinal that the theory cannot prove is well founded—the supremum of all ordinals ${\displaystyle \alpha }$ for which there exists a notation ${\displaystyle o}$ in Kleene's sense such that ${\displaystyle T}$ proves that ${\displaystyle o}$ is an ordinal notation. Equivalently, it is the supremum of all ordinals ${\displaystyle \alpha }$ such that there exists a recursive relation ${\displaystyle R}$ on ${\displaystyle \omega }$ (the set of natural numbers) that well-orders it with ordinal ${\displaystyle \alpha }$ and such that ${\displaystyle T}$ proves transfinite induction of arithmetical statements for ${\displaystyle R}$.

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