某些超限自然和
Some transfinite natural sums
摘要：
我们研究了超限迭代的顺序Hessenberg自然和通过采取在极限阶段的上确界，并表明，这样的迭代自然和不同于更常见的超限序数和只有有限数量的迭代步骤。序数序列的迭代自然和可以作为序列中序数的“混合和”（在序理论意义上）得到，事实上，它是满足有限性条件的最大混合和，相对于序列的序。我们引入了其他的无穷自然和，它们在置换下是不变的，并证明了它们在可数情形下都是一致的。最后，在最后一节中，我们使用上面的无穷自然和来定义良基树的大小，以及可数情况下的序理论特征。这种序理论特征的证明大多独立于本文的其余部分。
英文摘要：
We study a transfinite iteration of the ordinal Hessenberg natural sum obtained by taking suprema at limit stages and 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, relative to the ordering of the sequence. We introduce other infinite natural sums which are invariant under permutations and show that they all coincide in the countable case. Finally, in the last section we use the above infinitary natural sums in order to provide a definition of size for a well-founded tree, together with an order-theoretical characterization in the countable case. The proof of this order-theoretical characterization is mostly independent from the rest of this paper.
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