Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 085.03802
Autor: Erdös, Pál; Rado, R.
Title: A theorem on partial well-ordering of sets of vectors. (In English)
Source: J. London Math. Soc. 34, 222-224 (1959).
Review: In a earlier paper (Zbl 057.04302) R. Rado considered, for any abstract set S and any ordinal n, the set Ws(n) of all vectors of "length" n over S. If one puts further Ws(< n) equal to the union of all Ws(m) for all m < n, then any quasi-order \leq on S induces a quasi-order on Ws(< n) defined by: X \leq Y if and only if the sequences of components of X and Y satisfy xi \leq yt(i) for each i and an increasing sequence of subscripts t(i). Graham Higman (Zbl 047.03402) showed that if S is partially well-ordered, then so is Ws(< \omega), and R. Rado showed (loc. cit.) that this is not generally true for Ws (\omega). He conjectured, however, that for the set Vs(n) of all vectors with only a finite number of distinct components, and the corresponding set Vs(< n), it is true that Vs (< n) is partially well-ordered if S is partially well-ordered, whatever the ordinal n. He obtained some partial results in this direction. In the present note the authors prove this conjecture for all n less than \omega\omega. They state that a – longer and unpublished – proof by J. Kruskal stimulated their present proof.
Reviewer: Hanna Neumann
Classif.: * 06A99 Ordered sets
Index Words: set theory
© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag