论文标题
ZF下的扩展器以及等级到级嵌入的构造性
Extenders under ZF and constructibility of rank-to-rank embeddings
论文作者
论文摘要
假设ZF（没有选择的公理）。令$ j：v_ \ varepsilon \ tov_δ$是$ -Cofinal $σ_1$ -Elementary嵌入中的非平凡$ \，其中$ \ varepsilon，δ$是极限级别。我们证明了从$v_δ$的$ j $的构建性限制，主要集中在$ \ varepsilon =δ$上。特别是，如果$ \ varepsilon =δ$和$ j \ in L（v_δ）$，则$δ$具有cofinality $ω$。但是，假设ZFC+i $ _3 $，则使用适当的$ \ varepsilon =δ$，可以强迫在l（v^{v [g]}_δ）$中获得这样的$ j \。假设有依赖的选择，并且$δ$具有cofinality $ω$（但不假定$ v = l（v_δ）$），而$ j：v_Δ\ tov_Δ$ as $σ_1$ - elementary，我们表明有“完美的”此类$ J $，没有“孤立”。假设有一类适当的弱洛恩海姆 - 科莱姆红衣主教，我们还给出了嵌入式$ j：v \ to m $ tim $ m $ transitive的一阶表征。主要结果取决于ZF下的扩展器的开发（考虑到此类WLS红衣主教最有用）。
Assume ZF (without the Axiom of Choice). Let $j:V_\varepsilon\to V_δ$ be a non-trivial $\in$-cofinal $Σ_1$-elementary embedding, where $\varepsilon,δ$ are limit ordinals. We prove some restrictions on the constructibility of $j$ from $V_δ$, mostly focusing on the case $\varepsilon=δ$. In particular, if $\varepsilon=δ$ and $j\in L(V_δ)$ then $δ$ has cofinality $ω$. However, assuming ZFC+I$_3$, with the appropriate $\varepsilon=δ$, one can force to get such $j\in L(V^{V[G]}_δ)$. Assuming Dependent Choice and that $δ$ has cofinality $ω$ (but not assuming $V=L(V_δ)$), and $j:V_δ\to V_δ$ is $Σ_1$-elementary, we show that there are "perfectly many" such $j$, with none being "isolated". Assuming a proper class of weak Lowenheim-Skolem cardinals, we also give a first-order characterization of critical points of embeddings $j:V\to M$ with $M$ transitive. The main results rely on a development of extenders under ZF (which is most useful given such wLS cardinals).
