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In this treatise on the theory of the continuum of the surreal numbers ofJ.H. Conway, is proved ,that the three different techniques and hierarchies ofthe continuums of the transfinite real numbers of Glayzal A. (1937) definedthrough 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 inductivelimit or union the same class and continuum of infinite numbers. This is quiteremarkable and is the analogue in the transfinite numbers, of that the realnumbers can be constructed either as decimal power series, or by Dedekind cuts,or by Cauchy fundamental sequences.

The continuum of the surreal numbers revisited.The surreal numbers defined through transfinite Cauchy fundamental sequences