论文标题
累积层次结构的周期性
Periodicity in the cumulative hierarchy
论文作者
论文摘要
我们研究了等级至级基本嵌入的结构，在ZF集理论中起作用而无需选择公理。回想一下，累积层次结构的$v_α$是通过迭代应用程序的迭代应用定义的，从$ v_0 = \ emptyset $开始，并在极限阶段将工会占据。假设$ j：v_ {α+1} \ to v_ {α+1} $是（非平凡的）基本嵌入，我们表明$v_α$的结构与$ v_ {α+1} $的结构根本不同。我们表明，$ j $可以从$ v_ {α+1} $ iff $α+1 $的参数中定义。此外，如果$α+1 $是奇数的，则$ j $可以在$ v_ {α+1} $上定义，从参数$j``v_α= \ {此参数是最佳的，因为$ j $从$v_α$的元素的任何参数都无法定义。在$α=β+1 $的情况下，我们还通过某些超滤器在超能图中对这种$ j $进行了表征。 假设$λ$是限制级别，我们证明，如果$ j：v_λ\ tov_λ$是$σ_1$ - elementary，则$ j $在参数中不能超过$v_λ$，并且如果$β<λ$ and $ j：v_β\ w至v_λ$ in Blovely＆j $ in $ j $ in $ j $ iS $ - j $ iS $ cofinal，则;请注意，最后一个结果与一致性强度低于等级到等级的嵌入相关。 如果有一个莱因哈特红衣主教，那么对于所有足够大的序列$α$，确实有一个基本$ j：v_α\ tov_α$，因此累积层次结构最终是周期性的（带有周期2）。
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).
