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Some uses of dilators in combinatorial problems. II

Published online by Cambridge University Press:  12 March 2014

V. Michele Abrusci ,
Jean-Yves Girard  and
Jacques van de Wiele
V. Michele Abrusci
Affiliation:
Dipartimento di Scienze Filosofiche, Università di Bari, 70121 Bari, Italy Équipe de Logique Mathematique, Université Paris-VII, 75251 Paris, France
Jean-Yves Girard
Affiliation:
Dipartimento di Scienze Filosofiche, Università di Bari, 70121 Bari, Italy Équipe de Logique Mathematique, Université Paris-VII, 75251 Paris, France
Jacques van de Wiele
Affiliation:
Dipartimento di Scienze Filosofiche, Università di Bari, 70121 Bari, Italy Équipe de Logique Mathematique, Université Paris-VII, 75251 Paris, France
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Abstract

We study increasing F-sequences, where F is a dilator: an increasing F-sequence is a sequence (indexed by ordinal numbers) of ordinal numbers, starting with 0 and terminating at the first step x where F(x) is reached (at every step x + 1 we use the same process as in decreasing F-sequences, cf. [2], but with “+ 1” instead of “−1”). By induction on dilators, we shall prove that every increasing F-sequence terminates and moreover we can determine for every dilator F the point where the increasing F-sequence terminates.

We apply these results to inverse Goodstein sequences, i.e. increasing (1 + Id)(ω)-sequences. We show that the theorem

every inverse Goodstein sequence terminates

(a combinatorial theorem about ordinal numbers) is not provable in ID1.

For a general presentation of the results stated in this paper, see [1].

We use notions and results concerning the category ON (ordinal numbers), dilators and bilators, summarized in [2, pp. 25–31].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 55 , Issue 1 , March 1990 , pp. 32 - 40
Copyright
Copyright © Association for Symbolic Logic 1990

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References

REFERENCES

[1]Abrusci, V. M., Dilators, generalized Goodstein sequences, independence results, Logic and combinatorics (Simpson, S. G., editor), Contemporary Mathematics, vol. 65, American Mathematical Society, Providence, Rhode Island, 1987, pp. 1–23.CrossRefGoogle Scholar
[2]Abrusci, V. M., Girard, J.-Y. and van de Wiele, J., Some uses of dilators in combinatorial problems. I, Logic and combinatorics (Simpson, S. G., editor), Contemporary Mathematics, vol. 65, American Mathematical Society, Providence, Rhode Island, 1987, pp. 25–53.CrossRefGoogle Scholar
[3]Aczel, P., Describing ordinals using functionals of transfinite type, this Journal, vol. 37 (1972), pp. 35–47.Google Scholar
[4]Girard, J.-Y., -togic. Part 1: Dilators, Annals of Mathematical Logic, vol. 21 (1981), pp. 75–219.CrossRefGoogle Scholar
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