The boundedness principle in ordinal recursion
The characterization of -compact elements in some lattices
The combinatorics of reasonable ultrafilters
We are interested in generalizing part of the theory of ultrafilters on ω to larger cardinals. Here we set the scene for further investigations introducing properties of ultrafilters in strong sense dual to being normal.
The consistency strength of the tree property at the double successor of a measurable cardina
The Main Theorem is the equiconsistency of the following two statements: (1) κ is a measurable cardinal and the tree property holds at κ⁺⁺; (2) κ is a weakly compact hypermeasurable cardinal. From the proof of the Main Theorem, two internal consistency results follow: If there is a weakly compact hypermeasurable cardinal and a measurable cardinal far enough above it, then there is an inner model in which there is a proper class of measurable cardinals, and in which the tree property holds at the...
The equiconsistency of two large cardinal axioms
The existence of universal invariant measures on large sets
The first order properties of Dedekind finite integers
The free one-generated left distributive algebra: basics and a simplified proof of the division algorithm
The left distributive law is the law a· (b· c) = (a·b) · (a· c). Left distributive algebras have been classically used in the study of knots and braids, and more recently free left distributive algebras have been studied in connection with large cardinal axioms in set theory. We provide a survey of results on the free left distributive algebra on one generator, A, and a new, simplified proof of the existence of a normal form for terms in A. Topics included are: the confluence of A, the linearity...
The gap between I₃ and the wholeness axiom
∃κI₃(κ) is the assertion that there is an elementary embedding with critical point below λ, and with λ a limit. The Wholeness Axiom, or WA, asserts that there is a nontrivial elementary embedding j: V → V; WA is formulated in the language ∈,j and has as axioms an Elementarity schema, which asserts that j is elementary; a Critical Point axiom, which asserts that there is a least ordinal moved by j; and includes every instance of the Separation schema for j-formulas. Because no instance of Replacement...
The homomorphic images of infinite symmetric groups.
The nonexistence of robust codes for subsets of ω₁
Several results are presented concerning the existence or nonexistence, for a subset S of ω₁, of a real r which works as a robust code for S with respect to a given sequence of pairwise disjoint stationary subsets of ω₁, where “robustness” of r as a code may either mean that whenever each is equal to modulo nonstationary changes, or may have the weaker meaning that for every club C ⊆ ω₁. Variants of the above theme are also considered which result when the requirement that S gets exactly...
The relative consistency of some consequences of the existence of measurable cardinal numbers [Book]
The structure of precipitous ideals
The tree property at both and
We force from large cardinals a model of ZFC in which and both have the tree property. We also prove that if we strengthen the large cardinal assumptions, then in the final model even satisfies the super tree property.
The tree property at the double successor of a measurable cardinal κ with large
Assuming the existence of a λ⁺-hypermeasurable cardinal κ, where λ is the first weakly compact cardinal above κ, we prove that, in some forcing extension, κ is still measurable, κ⁺⁺ has the tree property and . If the assumption is strengthened to the existence of a θ -hypermeasurable cardinal (for an arbitrary cardinal θ > λ of cofinality greater than κ) then the proof can be generalized to get .
The Wholeness Axioms and the Class of Supercompact Cardinals
We show that certain relatively consistent structural properties of the class of supercompact cardinals are also relatively consistent with the Wholeness Axioms.