## On the intuitionistic strength of monotone inductive definitions.(English)Zbl 1070.03040

Summary: We prove here that the intuitionistic theory $${\mathbf T}_0\!\!\upharpoonright \!\!+\,\, \mathbf{ UMID}_N$$, or even $$\mathbf{EETJ}\!\!\upharpoonright\!\! +\,\, \mathbf{ UMID}_N$$, of Explicit Mathematics has the strength of $$\Pi^1_2-\mathbf{CA}_0$$. In Section 1 we give a double-negation translation for the classical second-order $$\mu$$-calculus, which was shown by M. Möllerfeld [Generalized inductive definitions. The $$\mu$$-calculus and $$\Pi^1_2$$-comprehension. Thesis. Münster: Univ. Münster (2002; Zbl 1050.03040)] to have the strength of $$\Pi^1_2-\mathbf{CA}_0$$. In Section 2 we interpret the intuitionistic $$\mu$$-calculus in the theory $$\mathbf{EETJ}\!\!\upharpoonright +\,\, \mathbf{ UMID}_N$$. The question about the strength of monotone inductive definitions in $$\mathbf T_0$$ was asked by S. Feferman in 1982, and – assuming classical logic – was addressed by M. Rathjen.

### MSC:

 03F50 Metamathematics of constructive systems 03D70 Inductive definability 03F35 Second- and higher-order arithmetic and fragments 03F55 Intuitionistic mathematics

Zbl 1050.03040
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### References:

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