The starting point of this article is an old question asked by Feferman in his paper on Hancock's conjecture [6] about the strength of ${\rm ID}_{1}^{\ast}$. This theory is obtained from the well-known theory ID₁ by restricting fixed point induction to formulas that contain fixed point constants only positively. The techniques used to perform the proof-theoretic analysis of ${\rm ID}_{1}^{\ast}$ also permit to analyze its transfinitely iterated variants ${\rm ID}_{\alpha}^{\ast}$. Thus, we eventually know that $|\widehat{{\rm ID}}_{\alpha}|=|{\rm ID}_{\alpha}^{\ast}|$.