  In this treatise on the theory of the continuum of the surreal numbers of
J.H. Conway, is proved ,that the three different techniques and hierarchies of
the continuums of the transfinite real numbers of Glayzal A. (1937) defined
through transfinite power series , of the surreal numbers of J.H. Conway (1976)
defined by Dedekind cuts ,and of the ordinal real numbers of K. E. Kyritsis
(1992) defined by fundamental Cauchy transfinite sequences, give by inductive
limit or union the same class and continuum of infinite numbers. This is quite
remarkable and is the analogue in the transfinite numbers, of that the real
numbers can be constructed either as decimal power series, or by Dedekind cuts,
or by Cauchy fundamental sequences.
