Set theory without choice: not everything on cofinality is possible

极限滤子与可数支持对称迭代中的依赖选择

Limit Filters and Dependent Choice in Countable-Support Symmetric Iterations

本文研究了可数支持对称迭代中极限滤子的构造及其在ZF和DC中的应用，证明了特定模型满足ZF和DC，并展示了ω₁-完备滤子对控制AC失败程度的重要性。

Extenders under ZF and constructibility of rank-to-rank embeddings

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 $Σ_1$-e…