Inaccessibility and Subinaccessibility. In two parts. Part I

TL;DR

This paper provides a detailed proof within ZF set theory demonstrating the nonexistence of inaccessible cardinals, introducing and refining concepts like subinaccessible cardinals and related tools.

Contribution

It offers the first part of a second edition with an improved, clearer proof of nonexistence of inaccessible cardinals, expanding on subinaccessible cardinals and their foundational tools.

Findings

Proof of nonexistence of inaccessible cardinals in ZF

Development of subinaccessible cardinals and associated tools

Enhanced clarity and simplification of key concepts

Abstract

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.