  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.
