Abstract

Montalban and Marcone [1] verified that the principle ``if $WO(X)$
then $WO(\varepsilon_X)$'' is equivalent to $ACA_0^+$ ($ACA_0$ plus
the existence of the $\omega$-jump) over $RCA_0$. Their proof uses
degree-theoretic arguments and somehow has an ad hoc flavor. We give
a purely proof-theoretic and conceptually more satisfying proof for
this result.

[1] Alberto Marcone, Antonio Montalban, The epsilon function for
computability theorists, unpublished notes, 2007.