## The proof-theoretic analysis of transfinitely iterated quasi least fixed points.(English)Zbl 1115.03084

The author gives a proof-theoretic analysis of the iterated fixed-point theories $$\mathsf{ID}_{\alpha}^*$$, $$\alpha$$ less than a certain ordinal $$\Phi_0$$. The theories $$\mathsf{ID}_{\alpha}^*$$ are defined as the well-known theories $$\mathsf{ID}_{\alpha},$$ but induction on the fixed points is restricted to formulas that contain fixed-point constants only positively. It is shown that the proof-theoretic ordinals $$| \mathsf{ID}_{\alpha}^*|$$ of $$\mathsf{ID}_{\alpha}^*$$ coincide with the ones of $$\widehat{\mathsf{ID}}_{\alpha}$$, the theories $$\mathsf{ID}_{\alpha}$$ but without any induction on fixed points.

### MSC:

 03F35 Second- and higher-order arithmetic and fragments

### Keywords:

Fixed-point theories; Iteration; Pseudo-hierarchies
Full Text:

### References:

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