Generic absoluteness under projective forcing
Generic large cardinals: New axioms for mathematics?
Hanf numbers and well-ordering numbers.
Higher Tall axioms
HOD-supercompactness, Indestructibility, and Level by Level Equivalence
In an attempt to extend the property of being supercompact but not HOD-supercompact to a proper class of indestructibly supercompact cardinals, a theorem is discovered about a proper class of indestructibly supercompact cardinals which reveals a surprising incompatibility. However, it is still possible to force to get a model in which the property of being supercompact but not HOD-supercompact holds for the least supercompact cardinal κ₀, κ₀ is indestructibly supercompact, the strongly compact and...
Homological Dimension and Stationary Sets.
How many normal measures can carry?
We show that assuming the consistency of a supercompact cardinal with a measurable cardinal above it, it is possible for to be measurable and to carry exactly τ normal measures, where is any regular cardinal. This contrasts with the fact that assuming AD + DC, is measurable and carries exactly three normal measures. Our proof uses the methods of [6], along with a folklore technique and a new method due to James Cummings.
Hybrid Prikry forcing
We present a new forcing notion combining diagonal supercompact Prikry forcing with interleaved extender based forcing. We start with a supercompact cardinal κ. In the final model the cofinality of κ is ω, the singular cardinal hypothesis fails at κ, and GCH holds below κ. Moreover we define a scale at κ which has a stationary set of bad points in the ground model.
I teoremi di assolutezza in teoria degli insiemi: seconda parte
Questa è la seconda parte dell’articolo espositivo [A]. Qui vedremo come siapossibile utilizzare il forcinge gli assiomi forti dell’infinito per dimostrare nuovi teoremi sui numeri reali.
Ideals on the real line and Ulam's problem
Inaccessible cardinals without the axiom of choice
We consider four notions of strong inaccessibility that are equivalent in ZFC and show that they are not equivalent in ZF.
Indestructibility of generically strong cardinals
Foreman (2013) proved a Duality Theorem which gives an algebraic characterization of certain ideal quotients in generic extensions. As an application he proved that generic supercompactness of ω₁ is preserved by any proper forcing. We generalize portions of Foreman's Duality Theorem to the context of generic extender embeddings and ideal extenders (as introduced by Claverie (2010)). As an application we prove that if ω₁ is generically strong, then it remains so after adding any number of Cohen subsets...
Indestructibility, strong compactness, and level by level equivalence
We show the relative consistency of the existence of two strongly compact cardinals κ₁ and κ₂ which exhibit indestructibility properties for their strong compactness, together with level by level equivalence between strong compactness and supercompactness holding at all measurable cardinals except for κ₁. In the model constructed, κ₁'s strong compactness is indestructible under arbitrary κ₁-directed closed forcing, κ₁ is a limit of measurable cardinals, κ₂'s strong compactness is indestructible...
Indestructibility, strongness, and level by level equivalence
We construct a model in which there is a strong cardinal κ whose strongness is indestructible under κ-strategically closed forcing and in which level by level equivalence between strong compactness and supercompactness holds non-trivially.
Indestructible Strong Compactness and Level by Level Equivalence with No Large Cardinal Restrictions
We construct a model for the level by level equivalence between strong compactness and supercompactness with an arbitrary large cardinal structure in which the least supercompact cardinal κ has its strong compactness indestructible under κ-directed closed forcing. This is in analogy to and generalizes the author's result in Arch. Math. Logic 46 (2007), but without the restriction that no cardinal is supercompact up to an inaccessible cardinal.
Intensionality, reflection, and large cardinals.
Internally standard set theories
Invariant measures and ideals on discrete groups [Book]
Iterated ultrapower and Prikry's forcing
Iterating along a Prikry sequence
We introduce a new method which combines Prikry forcing with an iteration between the Prikry points. Using our method we prove from large cardinals that it is consistent that the tree property holds at ℵₙ for n ≥ 2, is strong limit and .