Set-Theoretic Geology

A ground of the universe V is a transitive proper class W subset V, such that W is a model of ZFC and V is obtained by set forcing over W, so that V = W[G] for some W-generic filter G subset P in W . The model V satisfies the ground axiom GA if there are no such W properly contained in V . The model W is a bedrock of V if W is a ground of V and satisfies the ground axiom. The mantle of V is the intersection of all grounds of V . The generic mantle of V is the intersection of all grounds of all set-forcing extensions of V . The generic HOD, written gHOD, is the intersection of all HODs of all set-forcing extensions. The generic HOD is always a model of ZFC, and the generic mantle is always a model of ZF. Every model of ZFC is the mantle and generic mantle of another model of ZFC. We prove this theorem while also controlling the HOD of the final model, as well as the generic HOD. Iteratively taking the mantle penetrates down through the inner mantles to what we call the outer core, what remains when all outer layers of forcing have been stripped away. Many fundamental questions remain open.Comment: 44 pages; commentary concerning this article can be made at http://jdh.hamkins.org/set-theoreticgeology

Inner-Model Reflection Principles

We introduce and consider the inner-model reflection principle, which asserts that whenever a statement φ(a) in the first-order language of set theory is true in the set-theoretic universe V, then it is also true in a proper inner model W⊊V. A stronger principle, the ground-model reflection principle, asserts that any such φ(a) true in V is also true in some non-trivial ground model of the universe with respect to set forcing. These principles each express a form of width reflection in contrast to the usual height reflection of the Lévy–Montague reflection theorem. They are each equiconsistent with ZFC and indeed Π2-conservative over ZFC, being forceable by class forcing while preserving any desired rank-initial segment of the universe. Furthermore, the inner-model reflection principle is a consequence of the existence of sufficient large cardinals, and lightface formulations of the reflection principles follow from the maximality principle MP and from the inner-model hypothesis IMH. We also consider some questions concerning the expressibility of the principles.publishe

RV POSEIDON 523 Cruise Report / Fahrtbericht Malaga (Spain) 02.05.2018 Ponta Delgada (Azores) 23.05.2018 POS523 – ENERGY TRANSFER II

The RV Poseidon cruise 523 (POS523) is the second cruise to the work area as part of the observational program of the TRR 181 'Energy Transfers in Atmosphere and Ocean', and focussed on the energy transfer by low-mode internal waves. The goals of the cruise were to recover and redeploy a mooring to record the temporal variability of the internal wave field and associated energy fluxes, and to use time series CTD/LADCP/microstructure stations to assess locally the temporal variability of mixing, dissipation, and internal wave fluxes. The region south of the Azores in the east Atlantic is ideally suited for this kind of process study, because it is an area of a strong internal tide signal radiating away from the islands. The cruise track is located south of a chain of seamounts in a tidal beam formed by constructive interference of internal tides, and crosses the critical latitude for parametric subharmonic instability (PSI). During the cruise, we collected time series of CTD/LADCP and microstructure between 36 h and 52 h length on 5 stations in up to 4600 m water depth along the tidal beam between 29°20’N and 32°N latitude. In total 64 CTD/LADCP casts and 18 microstructure data sets were measured. The mooring equipped with current meter/temperature logger pairs and acoustic Doppler current profiler was successfully recovered and later redeployed along the track at 30°29’N, 30°12’W in a water depth of 4500 m (to be retrieved in 2019). All anticipated goals of the cruise were accomplished