Inaccessibility and Subinaccessibility. In two parts. Part II

TL;DR

This paper provides a detailed proof within ZF set theory demonstrating the nonexistence of inaccessible cardinals, utilizing advanced tools like subinaccessible cardinals, spectra, and matrices, with improved clarity and applications.

Contribution

It offers an enriched, simplified proof of inaccessible cardinals nonexistence in ZF, introducing refined tools and applications for set theory.

Findings

Proof of nonexistence of inaccessible cardinals in ZF

Development of simplified theories of spectra and matrices

Presentation of consequences and related results

Abstract

The work presents the second part of the second edition of its previous one published in 2000 under the same title, containing the proof (in ZF) of the inaccessible cardinals nonexistence, which is enriched and improved now. This part contains applications of the subinaccessible cardinals apparatus and its basic tools -- theories of reduced formula spectra and matrices, disseminators and others, which are used here in this proof and are 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. The proof of the theorem about inaccessible cardinals nonexistence is presented in its detailed exposition. Several easy consequences of this theorem and some well-known results are presented.