Goodstein principles of intermediate strength

The original Goodstein principle is a number-theoretic statement known to be independent from $\mathsf{PA}$. It proceeds by writing numbers in so-called hereditary exponential form, then applying various operations to produce a finite (but very long) sequence. In recent work with T. Arai and S. Wainer, we have shown that a similar process based on the Ackermann function leads to independence results for $\mathsf{ATR}_0$. In this talk we discuss how modifications in the representation of such numbers can lead to independence results for theories of intermediate strength.