Gitik-Koepke模型不满足大基数基数假设的自同构
On automorphisms behind the Gitik -- Koepke model for violation of the Singular Cardinals Hypothesis w/o large cardinals
摘要：
众所周知，"GCH首先在\aleph_ω失败"的假设导致ZFC中的大基数。Gitik和Koepke证明了在ZF中并非如此：即存在L的一般基数保持扩张（或ZFC + GCH的任何论域，其中所有ZF公理成立，选择公理失败，GCH对所有基数\aleph_n成立，但存在从PowerSet（\aleph_ω）到λ的满射，其中λ是L中任何先前选择的大于\aleph_ω的基数，例如\aleph_{ω+17}。换句话说，在这样的扩展中，GCH对所有基数\aleph_n在正确意义上成立，但在Hartogs意义上的\aleph_ω处失败。本文的目的是分析Gitik-Koepke证明中涉及的自同构系统。
英文摘要：
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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