Abstract
It is known that the assumption that ``GCH first fails at \aleph_ω'' 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_ω) onto λ, where λ is any previously chosen cardinal in L greater than \aleph_ω, for instance, \aleph_{ω+17}. In other words, in such an extension GCH holds in proper sense for all cardinals \aleph_n but fails at \aleph_ω in Hartogs' sense. The goal of this note is to analyse the system of automorphisms involved in the Gitik -- Koepke proof.
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