On automorphisms behind the Gitik -- Koepke model for violation of the Singular Cardinals Hypothesis w/o large cardinals

TL;DR

This paper analyzes the automorphism structure in the Gitik-Koepke model, which demonstrates the violation of the Singular Cardinals Hypothesis without relying on large cardinals, within a ZF framework.

Contribution

It provides a detailed analysis of the automorphisms underlying the Gitik-Koepke construction that shows GCH can fail at l_{\u00a0} without large cardinals.

Findings

Identifies key automorphisms in the Gitik-Koepke model.

Clarifies how automorphisms preserve certain cardinal properties.

Enhances understanding of symmetry in models violating SCH

Abstract

It is known that the assumption that ``GCH first fails at \aleph_{\omega}'' leads to large cardinals in ZFC. Gitik and Koepke have demonstrated that this is not so in ZF: namely there is a generic cardinal-preserving extension of L (or any universe of ZFC + GCH in which all ZF axioms hold, the axiom of choice fails, GCH holds for all cardinals \aleph_n, but there is a surjection from PowerSet(\aleph_{\omega}) onto {\lambda}, where {\lambda} is any previously chosen cardinal in L greater than \aleph_{\omega}, for instance, \aleph_{\omega +17}. In other words, in such an extension GCH holds in proper sense for all cardinals \aleph_n but fails at \aleph_{\omega} in Hartogs' sense. The goal of this note is to analyse the system of automorphisms involved in the Gitik -- Koepke proof.