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.
