Ordinal analysis

Ordinal analysis assigns transfinite ordinals to formal theories to measure their proof-theoretic strength, often by identifying the ordinals whose well-foundedness a theory can establish.

Proof-theoretic ordinal: An ordinal marking a formal theory’s strength, commonly the supremum of order types whose well-foundedness it can prove. This is the central measure ordinal analysis assigns to a theory.
Formal theory: A set of sentences specified by axioms and rules of inference in a formal language. Ordinal analysis compares the deductive strength of such axiom systems.
Ordinal analysis of Peano arithmetic: The analysis assigning the ordinal ε₀ to the proof-theoretic strength of Peano arithmetic. This is the classic case connecting induction in arithmetic with a transfinite ordinal.
Reverse mathematics: A program that classifies mathematical theorems by the axioms needed to prove them over weak base theories. It analyzes the strength needed for theorems rather than assigning a transfinite measure to a theory.
Ordinal analysis of set theory: The study of proof-theoretic measures for set theories, including systems with strong reflection and large-cardinal principles. Extending familiar ordinal methods to strong set theories remains technically demanding.
Well-founded relation: A relation is well-founded when every nonempty subset has a minimal element under that relation. Ordinal analyses often measure which well-foundedness claims a theory can establish.
Proof theory: The mathematical study of formal proofs, their structure, and the systems that generate them. Ordinal analysis is a proof-theoretic method for classifying theories.
Epsilon nought: The least ordinal ε satisfying ω^ε = ε. It is the proof-theoretic ordinal of Peano arithmetic.
Model theory: The study of mathematical structures through the formal languages and sentences they satisfy. It examines theories through their models, offering a different perspective on formal strength.
Ordinal notation system: A formal calculus for representing and comparing a specified range of ordinals. Reliable notation systems become harder to construct as target ordinals grow.

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