The Solution of the Invariant Subspace Problem in Complex Hilbert Space: External non-Archimedean eld *Rc by Cauchy Completion of the internal Non-Archimedean eld *R (Part I)

The incompleteness of set theory ZFC leads one to look for natural extensions of ZFC in which one can prove statements wich appear as independent of ZFC but which look to be \true". In this paper we deal with set theory NC based on hyper infinitary logic with Restricted Modus Ponens Rule. Set theory
NCcontains Aczel's anti-foundation axiom. We present a new approach to the invariant subspace problem for complex Hilbert spaces.This approach based on nonconservative extension of the model theoretical NSA. Our main result will be that: if T is a bounded linear operator on an infinite-dimensional separable complex Hilbert space H;it follow that T has a non-trivial closed invariant subspace.