2025/06/04 by Sakaé Fuchino, Fuchino, Sakaé
Computer Science · Mathematics · #03E45 #03E50 #03E55 #03E57 #03E65 #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Logic (math.LO)
paper · pdf · doi:10.48550/arxiv.2506.03572
openalex publication_date 2025/06/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce (super-C(∞)-)Laver-generic large cardinal axioms for extendibility ((super-C(∞)-)LgLCAs for extendible, for short), and show that most of the previously known consequences of the (super-C(∞)-)LgLCAs for ultrahuge, in particular, general forms of Resurrection Principles, Maximality Principles, and Absoluteness Theorems, already follow from (super-C(∞)-)LgLCAs for extendible. The consistency of LgLCAs for extendible (for transfinitely iterable Σ2-definable classes of posets) follows from an extendible cardinal while the consistency of super-C(∞)-LgLCAs for extendible follows from a model with a strongly super-C(∞)-extendible cardinal. If μ is an almost-huge cardinal, there are cofinally many κ<μ such that Vμ\models``κ is strongly super-C(∞) extendible''. Most of the known reflection properties follow already from some of the LgLCAs for supercompact. We give a survey on the related results. We also show the separation between some of the LgLCAs as well as between LgLCAs and their consequences. LgLCAs are generic large cardinal axioms in terms of generic elementary embeddings with the critical point κ_\mathfrakrefl=max\ℵ2,2ℵ0\. We show that Laver generic large cardinal axioms for all posets in terms of generic elementary embeddings with the critical point 2ℵ0 is also possible. We abbreviate this type of axiom for the notion of extendibility as the LgLCAA for extendible and examine its consequences.