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Provable wellorderings of formal theories for transfinitely iterated inductive definitions

Published online by Cambridge University Press:  12 March 2014

W. Buchholz  and
W. Pohlers
W. Buchholz
Affiliation:
Mathematisches Institut der Ludwig-Maximilians-Universität, München, Federal Republic of Germany
W. Pohlers
Affiliation:
Mathematisches Institut der Ludwig-Maximilians-Universität, München, Federal Republic of Germany
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Extract

By [12] we know that transfinite induction up to ΘεΩN+10 is not provable in IDN, the theory of N-times iterated inductive definitions. In this paper we will show that conversely transfinite induction up to any ordinal less than ΘεΩN+10 is provable in IDNi, the intuitionistic version of IDN, and extend this result to theories for transfinitely iterated inductive definitions.

In [14] Schütte proves the wellordering of his notational systems using predicates is wellordered) with Mκ ≔ {x ∈ and 0 ≤ κ ≤ N. Obviously the predicates are definable in IDNi with the defining axioms:

where Prog [Mκ, X] means that X is progressive with respect to Mκ, i.e.

The crucial point in Schütte's wellordering proof is Lemma 19 [14, p. 130] which can be modified to

where TI[Mκ + 1, a] is the scheme of transfinite induction over Mκ + 1 up to a.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 43 , Issue 1 , March 1978 , pp. 118 - 125
Copyright
Copyright © Association for Symbolic Logic 1978

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References

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