On ordinal diagrams
Akiko Kino · Journal of the Mathematical Society of Japan · 1961
G. Takeuti developed the theory of ordinal diagrams of order $n$ (where $2l$ is a positive integer) in [2], and generalized it to the theory of ordinal dia- grams constructed from well-ordered sets $I,$ $A$ , and $S$ in [3].It was necessary to consider $S$ in order to prove the accessibility for Od(I, $A,$ $S$ ) (the system of ordinal diagrams constructed from $I,$ $A$ and S) given in [3].But $S$ did not serve to extend the system of ordinal diagrams.In fact, if we denote $Od(LA, S)$ and $O(I, A, S)$ with empty $S$ by Od(I, $A$ ) and $O(I, A)$ respectively, we$where*is$ distinct form any element of $I,$ $A$ and $S$ ; the notation $A$ $US$ means the well-ordered set obtained from $A$ and $S$ by keeping the orders in them- selves and setting the elements of $A$ before the elements of S. The embed- ding is defined as follows:1 Now we can simplify the proof of the accessibility of Od(I,In this paper, we shall construct a system Od(I), namely " the system of ordinal diagrams constructed from a well-ordered set $l$ ' (in \S 1), and prove that the system is well-ordered for the given orderings in a similar way as in [2] (in \S 2).Then we shall show that the present system is a generalization of pre- vious systems.In fact, $Od(LA)$ is embedded into Od(I $UA$) in \S 3.By the way, we shall show that a formal theory of Od(I, $A$ ) can be formalized in the system developed in [5] and is consistent.The author wishes to express her heart-felt thanks to Prof. G. Takeuti for his valuable advice and kind encouragement in the preparation of this paper.\S 1. Ordinal diagrams constructed from $I$ .Let $I$ be a well-ordered set with the order $<*ando$ be the first element of $I$ .In this section, we shall construct a kind of system of ordinal diagrams, called ordinal diagrams constructed from $I$ and denoted by Od(I).Though