User blog comment:Hyp cos/Monotonic FS systems/@comment-35470197-20180630140801/@comment-11227630-20180630163428

We build a pair of "inverse functions" f and g such that f maps ordinals using T into ordinals using S. Since \(\omega^\lambda>\omega^\lambda[]_S\) and it's closed under addition, we have \(f(\omega^\lambda)=\omega^\lambda\), so \(\omega^\lambda[]_T=g(f(\omega^\lambda)[]_S)=g(\omega^\lambda[]_S)=0\).