Tanaka's theorem revisited

Tanaka (Ann Pure Appl Log 84:41-49, 1997) proved a powerful generalization of Friedman's self-embedding theorem that states that given a countable nonstandard model (M, A) of the subsystem WKL0 of second order arithmetic, and any element m of M, there is a self-embedding j of (M, A) onto a proper initial segment of itself such that j fixes every predecessor of m. Here we extend Tanaka's work by establishing the following results for a countable nonstandard model (M, A) of WKL0 and a proper cut I of M: Theorem A. The following conditions are equivalent: (a) I is closed under exponentiation. (b) There is a self-embedding j of (M, A) onto a proper initial segment of itself such that I is the longest initial segment of fixed points of j. Theorem B. The following conditions are equivalent: (a) I is a strong cut of Mand I. 1 M. (b) There is a self-embedding j of (M, A) onto a proper initial segment of itself such that I is the set of all fixed points of j.