累积层次中的周期性
Periodicity in the cumulative hierarchy
摘要：
我们研究的秩到秩的基本嵌入的结构，工作在ZF集理论没有公理的选择。回想一下，累积层次结构的级别$V_α$是通过幂集运算的迭代应用来定义的，从$V_0=\emptyset$开始，并在极限阶段取并集。假设$j:V_{α+1}\to V_{α+1}$是一个（非平凡的）基本嵌入，我们证明了$V_α$的结构与$V_{α+1}$的结构有根本不同。证明了$j$是$V_{α+1}$上的参数可定义的当且仅当$α+1$是奇序数.此外，如果$α+1$是奇数，则$j$可以从参数$j`` V_α=\{j(x)\bigm|x\in V_α\}$在$V_{α+1}$上定义，并且一致地如此。该参数是最佳的，因为$j$不能从作为$V_α$的元素的任何参数定义。在这种情况下，$α=β+1$，我们也给出了这样的$j$的超幂映射通过某些超滤子的特征。假设$λ$是一个极限序数，我们证明了，如果$j:V_λ\to V_λ$是$Σ_1$ -elementary，那么$j$是不可定义的$V_λ$从参数，如果$β<λ$和$j:V_β\to V_λ$是完全小学和$\in$ -共尾，那么$j$同样是不可定义的;请注意，这最后一个结果是相关的嵌入的一致性强度低得多的秩到秩。如果存在莱因哈特基数，那么对于所有足够大的序数$α$，确实存在初等$j:V_α\to V_α$，因此累积层次最终是周期的（周期为2）。
英文摘要：
We investigate the structure of rank-to-rank elementary embeddings, working in ZF set theory without the Axiom of Choice. Recall that the levels $V_α$ of the cumulative hierarchy are defined via iterated application of the power set operation, starting from $V_0=\emptyset$, and taking unions at limit stages. Assuming that $j:V_{α+1}\to V_{α+1}$ is a (non-trivial) elementary embedding, we show that the structure of $V_α$ is fundamentally different to that of $V_{α+1}$. We show that $j$ is definable from parameters over $V_{α+1}$ iff $α+1$ is an odd ordinal. Moreover, if $α+1$ is odd then $j$ is definable over $V_{α+1}$ from the parameter $j`` V_α=\{j(x)\bigm|x\in V_α\}$, and uniformly so. This parameter is optimal in that $j$ is not definable from any parameter which is an element of $V_α$. In the case that $α=β+1$, we also give a characterization of such $j$ in terms of ultrapower maps via certain ultrafilters. Assuming $λ$ is a limit ordinal, we prove that if $j:V_λ\to V_λ$ is $Σ_1$-elementary, then $j$ is not definable over $V_λ$ from parameters, and if $β<λ$ and $j:V_β\to V_λ$ is fully elementary and $\in$-cofinal, then $j$ is likewise not definable; note that this last result is relevant to embeddings of much lower consistency strength than rank-to-rank. If there is a Reinhardt cardinal, then for all sufficiently large ordinals $α$, there is indeed an elementary $j:V_α\to V_α$, and therefore the cumulative hierarchy is eventually periodic (with period 2).
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