Inaccessibility and Subinaccessibility. In two parts. Part I

The work presents the first part of second edition of the previous edition of 2000 under the same title containing the proof (in ZF) of the nonexistence of inaccessible cardinals, now enriched and improved. This part contains the apparatus of subinaccessible cardinals and its basic tools -- theories of reduced formula spectra and matrices, disseminators and others -- which are used in this proof and is set forth now in their more transparent and simplified form. Much attention is devoted to the explicit and substantial development and cultivation of basic ideas, serving as grounds for all main constructions and reasonings.