Well-Ordering Principles in Proof Theory and Reverse Mathematics

Several theorems about the equivalence of familiar theories of reverse mathematics with certain well-ordering principles have been proved by recursion-theoretic and combinatorial methods (Friedman, Marcone, Montalbán et al.) and with far-reaching results by proof-theoretic technology (Afshari, Freund, Girard, Rathjen, Thomson, Valencia Vizcaíno, Weiermann et al.), employing deduction search trees and cut elimination theorems in infinitary logics with ordinal bounds in the latter case. At type level 1, the well-ordering principles are of the form $$ \text { (*) } ``\text {if}\,X \text { is well-ordered then } \, f(X) \, \text { is well-ordered''} $$ (*) ` ` if X is well-ordered then f ( X ) is well-ordered'' where f is a standard proof-theoretic function from ordinals to ordinals (such f’s are always dilators). One aim of the paper is to present a general methodology underlying these results that enables one to construct omega-models of particular theories from $$(*)$$ ( ∗ ) and even $$\beta $$ β -models from the type 2 version of $$(*)$$ ( ∗ ) . As $$(*)$$ ( ∗ ) is of complexity $$\Pi ^1_2$$ Π 2 1 such a principle cannot characterize stronger comprehensions at the level of $$\Pi ^1_1$$ Π 1 1 -comprehension. This requires a higher order version of $$(*)$$ ( ∗ ) that employs ideas from ordinal representation systems with collapsing functions used in impredicative proof theory. The simplest one is the Bachmann construction. Relativizing the latter construction to any dilator f and asserting that this always yields a well-ordering turns out to be equivalent to $$\Pi ^1_1$$ Π 1 1 -comprehension. This result has been conjectured and a proof strategy adumbrated roughly 10 years ago, but a detailed proof has only been worked out in recent years.