Well-ordering Principles, ω-models and $$ \varPi_{1}^{1} $$ -comprehension

In this paper we show that over the weak theory RCA0 the existence of ω-models of $$ \varPi_{1}^{1} $$ -comprehension is equivalent to saying that a certain elementary endomorphism of the category of linear orders preserves well-orderings. This result continues a long line of research. Technically, among other methods, we use Schutte’s search trees and a relativized version of the ordinal analysis of $$ \varPi_{1}^{1} $$ -comprehension via the Ωn-rules akin to the presentation in Buchholz and Schutte’s 1988 book [3]. A crucial challenge one faces is to show that the latter approach can be made to work in a weak background theory.