Ordinal analysis
In proof theory, ordinal analysis assigns ordinals (often large countable ordinals) to mathematical theories as a measure of their strength. If theories have the same proof-theoretic ordinal they are often equiconsistent, and if one theory has a larger proof-theoretic ordinal than another it can often prove the consistency of the second theory. History. The field of ordinal analysis was formed when Gerhard Gentzen in 1934 used cut elimination to prove, in modern terms, that the proof-theoretic ordinal of Peano arithmetic is ε. See Gentzen's consistency proof. Definition. 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 formula_1 is the supremum of the order types of all ordinal notations (necessarily recursive, see next section) that the theory can prove are well founded—the supremum of all ordinals formula_2 for which there exists a notation formula_3 in Kleene's sense such that formula_1 proves that formula_3 is an ordinal notation. Equivalently, it is the supremum of all ordinals formula_2 such that there exists a recursive relation formula_7 on formula_8 (the set of natural numbers) that well -orders it with ordinal formula_2 and such that formula_1 proves transfinite induction of arithmetical statements for formula_7. Ordinal notations. Some theories, such as subsystems of second-order arithmetic, have no conceptualization of or way to make arguments about transfinite ordinals. For example, to formalize what it means for a subsystem of Z formula_1 to "prove formula_2 well-ordered", we instead construct an ordinal notation formula_14 with order type formula_2. formula_1 can now work with various transfinite induction principles along formula_14, which substitute for reasoning about set-theoretic ordinals. However, some pathological notation systems exist that are unexpectedly difficult to work  with. For example, Rathjen gives a primitive recursive notation system formula_18 that is well-founded iff PA is consistent, despite having order type formula_8 - including such a notation in the ordinal analysis of PA would result in the false equality formula_20. Upper bound. The existence of a recursive ordinal that the theory fails to prove is well-ordered follows from the formula_21 bounding theorem, as the set of natural numbers that an effective theory proves to be ordinal notations is a formula_22 set (see Hyperarithmetical theory). Thus the proof-theoretic ordinal of a theory will always be a (countable) recursive  ordinal, that is, less than the Church–Kleene ordinal formula_23. Examples. Theories with proof-theoretic ordinal ω. Friedman's grand conjecture suggests that much "ordinary" mathematics can be proved in weak systems having this as their proof-theoretic ordinal. Theories with proof-theoretic ordinal the Feferman–Schütte ordinal Γ. This ordinal is sometimes considered to be the upper limit for "predicative" theories. Theories with proof-theoretic ordinal the Bachmann–Howard ordinal. The Kripke-Platek or CZF set theories are weak set theories without axioms for the full powerset given as set of all subsets. Instead, they tend to either have  axioms of restricted separation and formation of new sets, or they grant existence of certain function spaces (exponentiation) instead of carving them out from bigger relations. Theories with larger proof-theoretic ordinals. Most theories capable of describing the power set of the natural numbers have proof-theoretic ordinals that are so large that no explicit combinatorial description has yet been given. This includes formula_45, full second-order arithmetic (formula_46) and set theories with powersets including ZF and ZFC. The strength of intuitionistic ZF (IZF) equals that of ZF. Table of ordinal analyses. Key. This is a list of symbols used  in this table: This is a list of the abbreviations used in this table: A superscript zero indicates that formula_106-induction is removed (making the theory significantly weaker).