2021/03/24 by Farmer Schlutzenberg, Schlutzenberg, Farmer
Computer Science · Mathematics · #03E25 #03E40 #03E45 #03E55 #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Limits and Structures in Graph Theory #Logic (math.LO)
paper · pdf · doi:10.48550/arxiv.2103.12925
openalex publication_date 2021/03/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Assume ZFC. Let κ be a cardinal. Recall that a <κ-ground is a transitive proper class W modelling ZFC such that V is a generic extension of W via a forcing ℙ∈ W of cardinality <κ, and the κ-mantle is the intersection of all <κ-grounds. Assume there is a Woodin cardinal and a proper class of measurables, and let x be a real of sufficiently high Turing degree. Let κ be a limit cardinal of L[x] of uncountable cofinality in L[x]. Using methods from Woodin's analysis of HODL[x,G], we analyze the κ-mantle of L[x], and show that it models ZFC + GCH + "There is a Woodin cardinal". Moreover, we show that it is a fully iterable strategy mouse (analogous to HODL[x,G]). We also analyze another form of "local mantle", partly assuming also a weak form of Turing determinacy. We also compute bounds on how much iteration strategy can be added to M1 before M1^# is added.