Ordinal analysis

• Q, Robinson arithmetic (although the definition of the proof-theoretic ordinal for such weak theories has to be tweaked).
• PA^–, the first-order theory of the nonnegative part of a discretely ordered ring.

• RFA, rudimentary function arithmetic.[1]
• IΔ₀, arithmetic with induction on Δ₀-predicates without any axiom asserting that exponentiation is total.

• EFA, elementary function arithmetic.
• IΔ₀ + exp, arithmetic with induction on Δ₀-predicates augmented by an axiom asserting that exponentiation is total.
• RCA^*
  ₀, a second order form of EFA sometimes used in reverse mathematics.
• WKL^*
  ₀, a second order form of EFA sometimes used in reverse mathematics.

Friedman's grand conjecture suggests that much "ordinary" mathematics can be proved in weak systems having this as their proof-theoretic ordinal.

• IΔ₀ or EFA augmented by an axiom ensuring that each element of the n-th level ${\displaystyle {\mathcal {E}}^{n}}$ of the Grzegorczyk hierarchy is total.

• RCA₀, recursive comprehension.
• WKL₀, weak König's lemma.
• PRA, primitive recursive arithmetic.
• IΣ₁, arithmetic with induction on Σ₁-predicates.

• PA, Peano arithmetic (shown by Gentzen using cut elimination).
• ACA₀, arithmetical comprehension.

• ATR₀, arithmetical transfinite recursion.
• Martin-Löf type theory with arbitrarily many finite level universes.

This ordinal is sometimes considered to be the upper limit for "predicative" theories.

• ID₁, the theory of inductive definitions.
• KP, Kripke–Platek set theory with the axiom of infinity.
• CZF, Aczel's constructive Zermelo–Fraenkel set theory.
• EON, a weak variant of the Feferman's explicit mathematics system T₀.

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.

• ${\displaystyle \Pi _{1}^{1}{\mbox{-}}{\mathsf {CA}}_{0}}$, Π₁¹ comprehension has a rather large proof-theoretic ordinal, which was described by Takeuti in terms of "ordinal diagrams", and which is bounded by ψ₀(Ω_ω) in Buchholz's notation. It is also the ordinal of ${\displaystyle ID_{<\omega }}$, the theory of finitely iterated inductive definitions. And also the ordinal of MLW, Martin-Löf type theory with indexed W-Types Setzer (2004).
• T₀, Feferman's constructive system of explicit mathematics has a larger proof-theoretic ordinal, which is also the proof-theoretic ordinal of the KPi, Kripke–Platek set theory with iterated admissibles and ${\displaystyle \Sigma _{2}^{1}{\mbox{-}}{\mathsf {AC}}+{\mathsf {BI}}}$.
• KPM, an extension of Kripke–Platek set theory based on a Mahlo cardinal, has a very large proof-theoretic ordinal ϑ, which was described by Rathjen (1990).
• MLM, an extension of Martin-Löf type theory by one Mahlo-universe, has an even larger proof-theoretic ordinal ψ_Ω1(Ω_{M + ω}).

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 second-order arithmetic and set theories with powersets including ZF and ZFC (as of 2019). The strength of intuitionistic ZF (IZF) equals that of ZF.