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Dedicated to Prof. Dr. K. Schütte on his 75th birthday

The original version of this paper was presented as an invited address by Crossley at the Jubileum in honour of Professor Schütte's 75th birthday in Munich in June 1984. It was subsequently corrected and substantially revised in late 1985 by both authors in Oxford where Crossley held a Visiting Fellowship at All Souls College.

We are indebted to P.H.G. Aczel, W. Buchholz, S. Feferman, G. Jäger, H. Levitz, W. Pohlers, and S.S. Wainer for materials and comments which significantly helped in the preparation of this paper.

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Crossley, J.N., Kister, J.B. Natural well-orderings. Arch math Logik 26, 57–76 (1987). https://doi.org/10.1007/BF02017491

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