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Some transfinite natural sums. (English) Zbl 1521.03144

Summary: We study a transfinite iteration of the ordinal Hessenberg natural sum obtained by taking suprema at limit stages. We show that such an iterated natural sum differs from the more usual transfinite ordinal sum only for a finite number of iteration steps. The iterated natural sum of a sequence of ordinals can be obtained as a mixed sum (in an order-theoretical sense) of the ordinals in the sequence; in fact, it is the largest mixed sum which satisfies a finiteness condition. We introduce other infinite natural sums which are invariant under permutations and show that all the sums under consideration coincide in the countable case.

MSC:

03E10 Ordinal and cardinal numbers
06A05 Total orders
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