Violating the Singular Cardinals Hypothesis Without Large Cardinals

TL;DR

This paper constructs a model of ZF set theory where the Generalized Continuum Hypothesis (GCH) fails at a high cardinal without assuming large cardinal axioms, extending previous results in set theory.

Contribution

It demonstrates the possibility of violating the Singular Cardinals Hypothesis at high cardinals without large cardinal assumptions, expanding understanding of set-theoretic models.

Findings

GCH holds below Alef_omega in the constructed model

A surjection from the power set of Alef_omega onto an arbitrary high cardinal exists

The model extends ZFC + GCH to ZF with specific GCH and surjection properties

Abstract

We extend a transitive model V of ZFC + GCH cardinal preservingly to a model N of ZF + "GCH holds below Alef_omega" + "there is a surjection from the power set of Alef_omega onto lambda" where lambda is an arbitrarily high fixed cardinal in V.