Ordinal Diagrams II.
Ordinal Diagrams II.
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序数图 II.
DOI：
10.2969/jmsj/01240385
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发表时间：
1960
期刊：
影响因子：
--
通讯作者：
G. Takeuti
中科院分区：
文献类型：
--
作者：
G. Takeuti
在先前的一篇论文[1]中，作者发展了序数图理论，它表示了第二类数的某个“Abschnitt”中的序数，并对某些逻辑系统的一致性证明很有用。本文推广了序数图的概念及其序关系，证明了序数图的广义系统Od（I, $A,$ $S$）是良序的。如果我们将序数图的概念推广到以下方向，将会失去序数图系统的良序性质：a)在i $中使用序数图来代替$i\。b)用一个序数图代替$ s中的$ s，事实上，在a)的情况下，我们将得到一个严格降序序列：
In a former paper [1] the author developed the theory of ordinal diagrams, which represent the ordinal numbers in a certain “ Abschnitt ” of the second number class and are useful for the consistency proof of some logical systems. In this paper we shall generalize the notion of ordinal diagrams and their ordering relations, and we shall prove that the generalized system Od(I, $A,$ $S$ ) of ordinal diagrams is well-ordered. The well-ordering property of the system of ordinal diagrams will be lost if we generalize the notion of ordinal diagrams in the following directions: a) Making use of an ordinal diagram in place of $i\in I$. b) Making use of an ordinal diagram in place of $ s\in$ S. In fact, we shall have, in case of a), a strictly descending sequence: