Vol. 59, No. 2, 1975

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Reducing series of ordinals

J. L. Hickman

Vol. 59 (1975), No. 2, 461–473

If s is a sequence of ordinals, we denote by “S(s)” the set of sums (of the corresponding series) obtainable by permuting the terms of s in such a way that the length o(s) is unchanged. If o(s) = ω, the first tran sfinite ordinal, then a fairly well-known result of Sierpiński’s states that S(s) is finite, which immediately raises the question of whether there is a finite sequence r such that S(r) = S(s).

It turns out in fact that such a sequence r always exists: and we are concerned in this note with proving certain generalizations of this latter result. The general problem, that of determining criteria that must be satisfled by an infinite sequence s in order that a sequence r exist with o(r) < o(s) and S(r) = S(s), is to the best of our knowledge still open and would appear to be no easy one.

Mathematical Subject Classification
Primary: 04A10
Received: 9 September 1974
Published: 1 August 1975
J. L. Hickman