论文标题
Reinhardt红衣主教和V的迭代
Reinhardt cardinals and iterates of V
论文作者
论文摘要
假设ZF（$ j $），并且有一个莱因哈特红衣主教，如小学嵌入$ j：v \ to v $所见。我们调查了$（v，j）$的线性迭代$（n_α，j_α）$及其与$（v，j）$的关系，强迫和确定性，包括每个无限级序$α$，每个集合都超过$n_α$，但是$n_α$不是n_α$。 假设二阶ZF。我们证明，超级莱因哈特红衣主教的存在和总莱因哈特红衣主教不受小强迫的影响。而且，如果$ v [g] $具有一组不在$ v $中的列中，则$ v [g] $没有基本嵌入$ j：v [g] \ to m \ subseteq v $（甚至允许$ m $损失）。
Assume ZF($j$) and there is a Reinhardt cardinal, as witnessed by the elementary embedding $j:V\to V$. We investigate the linear iterates $(N_α,j_α)$ of $(V,j)$, and their relationship to $(V,j)$, forcing and definability, including that for each infinite ordinal $α$, every set is set-generic over $N_α$, but $N_α$ is not a set-ground. Assume second order ZF. We prove that the existence of super Reinhardt cardinals and total Reinhardt cardinals is not affected by small forcing. And if $V[G]$ has a set of ordinals which is not in $V$, then $V[G]$ has no elementary embedding $j:V[G]\to M\subseteq V$ (even allowing $M$ to be illfounded).
