Giant and illusionary giant Goodstein principles

We analyze several natural Goodstein principles which themselves are defined with respect to the Ackermann function and the extended Ackermann function. These Ackermann functions are well established canonical fast growing functions labeled by ordinals not exceeding $\varepsilon_0$. Among the Goodsteinprinciples under consideration, the giant ones, will be proof-theoretically strong (being unprovable in $\mathrm{PA}$ in the Ackermannian case and being unprovable in $\mathrm{ID}_1$ in the extended Ackermannian case) whereas others, the illusionary giant ones, will turn out to be comparatively much much weaker although they look strong at first sight.

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