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The varieties of arboreal experience

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  1. First, let me simply update the references to my column on the Paris-Harrington Theorem: The papers of Erdös and Mills and of Ketonen and Solovay have since appeared, the former in the Journal of Combinatorial Theory, series A, vol. 30 (1981), pp. 53–70, and the latter in the Annals of Mathematics, vol. 113 (1981), pp. 267- 314. In addition, I note the bookRamsey Theory (J. Wiley, 1980) by R. Graham, B. Rothschild and J. Spencer has a nice exposition of a variant of the Ketonen-Solovay work.

  2. Friedman’s work reported above is not, at the time of writing, in manuscript form and I can only cite some background material. With respect to Kruskal’s Theorem and ordinals, I suggest: J. B. Kruskal, Well-quasi-ordering, the tree theorem, and Vázsonyi’s conjecture, Trans. AMS 95 (1960), pp. 210–225.

  3. C. St. J. A. Nash-Williams, On well-quasi-ordering finite trees, Proc. Cambridge Phil. Soc. 59 (1963), pp. 833–835.

  4. D. Schmidt, Well-partial-orderings and their maximal order types, Habilitationsschrift, Heidelberg, 1978.

  5. Finally, a good reference on Γ0, ordinals, and proof theory is K. Schütte,Proof Theory (Springer-Verlag, 1977).

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Correspondence to C. Smoryński.

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Smoryński, C. The varieties of arboreal experience. The Mathematical Intelligencer 4, 182–189 (1982). https://doi.org/10.1007/BF03023553

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  • Incompleteness Theorem
  • Finite Tree
  • Transfinite Induction
  • Countable Ordinal
  • Trivial Tree