2020/06/17 by Farmer Schlutzenberg, Schlutzenberg, Farmer · 1 citation
Mathematics · #03E25 #03E45 #03E55 #Advanced Topology and Set Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Logic (math.LO) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2006.10574
openalex publication_date 2020/06/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Assume ZF (without the Axiom of Choice). Let j:Vε→ Vδ be a non-trivial ∈-cofinal Σ1-elementary embedding, where ε,δ are limit ordinals. We prove some restrictions on the constructibility of j from Vδ, mostly focusing on the case ε=δ. In particular, if ε=δ and j∈ L(Vδ) then δ has cofinality ω. However, assuming ZFC+I3, with the appropriate ε=δ, one can force to get such j∈ L(VV[G]δ). Assuming Dependent Choice and that δ has cofinality ω (but not assuming V=L(Vδ)), and j:Vδ→ Vδ is Σ1-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→ M with M transitive. The main results rely on a development of extenders under ZF (which is most useful given such wLS cardinals).