Goodstein function ($\mathcal{G}(m)=\min \{s:m_s=0\}$)

The Goodstein function records the least step at which the Goodstein sequence from a given initial number reaches zero. With the initial step numbered zero, its values on $0,1,2,3$ are $0,1,3,5$. Goodstein's theorem makes this effective termination-time function total.