At the suggestion of J. C. Shepherdson the author first attempted to
construct an analogue for order types of DEKKER and MYHILL’S monograph
(1960) where they studied a recursive analogue of the theory of
cardinal numbers. (We give a brief survey of their work, insofar as we
parallel it, in chapter 13.) Their work to some extent corresponds to
TARSKI (1949) and the reader will find that the present work is related in
a similar way to TARSKI (1956). This latter monograph and Dekker and
Myhill’s were the chief influences in the development of the present
theory; for constructive order types satisfy all the finitary postulates for
an ordinal algebra. (Incidentally, we note that if everywhere below we
replace “recursive” by “arithmetic” then the results are even closer to
those of TARSKI (1956).)