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The metamathematics of Fraïssé's order type conjecture

Part of the Lecture Notes in Mathematics book series (LNM,volume 1432)

Abstract

A well ordering has the property that any non-empty subset has a minimum element. In [Girard

  1. (i)

    ATR0 proves that the collection S n of countable scattered linear orderings at level n of the Hausdorff hierarchy is better quasi ordered (bqo), and

  2. (ii)

    ATR0 proves that "if α is an ordinal and Q is bqo then Q α is bqo".

We conjecture that the techniques introduced will eventually allow a proof in ATR0 of Fraïssé's order type conjecture (proved by R. Laver) which states that the collection L of all countable linear orderings is wqo under embeddability.

Key words and phrases

  • Fraïssé's order type conjecture
  • scattered linear ordering
  • arithmetic transfinite recursion

Research partially supported by NSF grant # DCR-8606165.

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© 1990 Springer-Verlag

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Clote, P. (1990). The metamathematics of Fraïssé's order type conjecture. In: Ambos-Spies, K., Müller, G.H., Sacks, G.E. (eds) Recursion Theory Week. Lecture Notes in Mathematics, vol 1432. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0086113

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  • DOI: https://doi.org/10.1007/BFb0086113

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