The next papers belong to the old mathematics and from the point of view of the 7-layers, in layer 6 and above. As I see it at the present time (2005) the value of using infinity (besides its historic role of an earlier phase in the evolution of civilization), is mainly that it permits a distance from the finite material ontology, and permits free thinking which was felt better . Otherwise  practicing is hardly separated from thinking anymore and hinders thinking with many emotional traps too Therefore  infinity should not be taken seriously in its  literal sense, as a new different ontology as this would lead to a collective paranoiac dead end without hope  for practical applications or with more and more difficult practical applications. Not to mention that it would become like a "computer worm"  a mental seduction that would consume more and more time and would lead more and more away from reality and life. In my way of appreciating it today infinite  can be considered  mainly as a  metaphor  or "encryption code" of facts about the finite. Probably facts about the finite that the civilization had  not been entirely ready at its time of introduction. Strange as it may seem, it holds that ,the creative world of finite has more choices and freedom for the mathematician, than the creative world of the infinite. Although the initial impression was that G. Cantor was leading mathematics to his paradise, it finally resulted to Cantor’s Hell. (Cantor himself, died mad in the sanatorium). If we try to discover the closest concept to infinite in the world of finite (as we shall see in the sequent) we immediately realize that infinite is the totalitarianism in mathematics, while the world of finite permits real conceptual democracy of creativity. As any totalitarianism seems attractive and might feel good at the beginning but sooner or later it results in to a totally wrong and destructive role by its users.

1) Alternative algebraic definitions of the Hessenberg      natural       operations   in the ordinal numbers (1990)

2) Free  algebrae and alternative definitions of the  Hessenberg         operations  in the ordinal numbers . (1990)

3) ORDINAL REAL NUMBERS 1. The ordinal characteristic.(1990)

4)  ORDINAL REAL  NUMBERS  2. The arithmetization of  order  types .(1990)

5) ORDINAL REAL  NUMBER 3. The techniques of transfinite real , surreal,  ordinal real, numbers ; unification . (1991)

6) Consistency problems and contradictions due to the additional axiom of ZFC-set theory introduced by   N. L. Alling  in his book "Foundations of Analysis over Surreal Number Fields"(1993)