Large cardinals need not be large in HOD

Abstract

Abstract. We prove that large cardinals need not generally exhibit their large cardinal nature in HOD. For example, a supercompact cardinal κ need not be weakly compact in HOD, and there can be a proper class of supercompact cardinals in V , none of them weakly compact in HOD, with no supercompact cardinals in HOD. Similar results hold for many other types of large cardinals, such as measurable and strong cardinals.

The consistency of level by level equivalence with $V = {\rm HOD}$, the Ground Axiom, and instances of square and diamond

“…In particular, we assume that W ⊆ V is such that V = W [h], where h is W -generic for some set partial ordering Q ∈ W . By [7,Lemma 19], for κ > |Q| a singular cardinal, the models W and V will agree on the properties "♦ * κ ++ holds" and "♦ * κ ++ fails". As the proof of Lemma 2.3 shows, every set of ordinals x ∈ V is coded using the oracle "Either ♦ * κ ++ holds or fails for κ a singular cardinal".…”

The Hypothesis and a supercompact cardinal

“…In , very large cardinals such as supercompact cardinals in are forced not to exhibit their large cardinal properties in : they can be very small (not even weakly compact) in . A reasonable natural question would be how far this can be taken, that is whether there exists a supercompact cardinal in which is not only not even weakly compact in but also has no other cardinals in which exhibit large cardinal behavior.…”

“…Examining under which hypothesis and are close to each other and how and can be pushed apart via forcing is a very interesting area of research. From , via forcing, behaviors of large cardinals from can become disordered in . A natural question is whether the Hypothesis has some effect on the behavior of large cardinals from in .…”

Indestructibility properties of remarkable cardinals

“…One application of the main theorem is that any GCH pattern can be forced above a remarkable cardinal. Another application uses a recent forcing construction of [2], to produce a remarkable cardinal that is not weakly compact in HOD. Using techniques from the proof of the main theorem, we also show that remarkability is preserved by the canonical forcing of the GCH.…”

“…We show that it is consistent to realize any possible continuum pattern above a remarkable cardinal. Using techniques developed recently in [2], we show that it is consistent to have a remarkable cardinal that is not remarkable, and indeed not even weakly compact in HOD.…”