We present old and new characterizations of core spaces, alias worldwide web spaces, originally defined by the existence of supercompact neighborhood bases. The patch spaces of core spaces, obtained by joining the original topology with a second topology having the dual specialization order, are the so-called sector spaces, which have good convexity and separation properties and determine the o...

In this paper we explicate a very weak version of the principle discovered by Jensen who proved it holds in the constructible universe L. This principle is strong enough to include many of the known applications of , but weak enough that it is consistent with the existence of very large cardinals. In this section we show that this principle is equivalent to a common combinatorial device, which ...

We show that there is a class-sized partial order P with the property that forcing with P preserves ZFC, supercompact cardinals, inaccessible cardinals and the value of 2κ for every inaccessible cardinal κ and, if κ is an inaccessible cardinal and A is an arbitrary subset of κκ, then there is a P-generic extension of the ground model V in which A is definable in 〈H(κ+)V[G],∈〉 by a Σ1-formula wi...

This paper deals with the question whether the assumption that for every inaccessible cardinal κ there is a well-order of H(κ+) definable over the structure 〈H(κ+),∈〉 by a formula without parameters is consistent with the existence of (large) large cardinals and failures of the GCH. We work under the assumption that the SCH holds at every singular fixed point of the i-function and construct a c...

Versions of Laver sequences are known to exist for supercompact and strong cardinals. Assuming very strong axioms of infinity, Laver sequences can be constructed for virtually any globally defined large cardinal not weaker than a strong cardinal; indeed, under strong hypotheses, Laver sequences can be constructed for virtually any regular class of embeddings. We show here that if there is a reg...

The authors show, by means of a finitary version fin λ,D of the combinatorial principle b ∗ λ of [6], the consistency of the failure, relative to the consistency of supercompact cardinals, of the following: for all regular filters D on a cardinal λ, if Mi and Ni are elementarily equivalent models of a language of size ≤ λ, then the second player has a winning strategy in the Ehrenfeucht-Fräıssé...

A Σ21 truth for λ is a pair 〈Q,ψ〉 so that Q ⊆ Hλ, ψ is a first order formula with one free variable, and there exists B ⊆ H λ+ such that (H λ+ ;∈, B) |= ψ[Q]. A cardinal λ is Σ21 indescribable just in case that for every Σ 2 1 truth 〈Q,ψ〉 for λ, there exists λ̄ < λ so that λ̄ is a cardinal and 〈Q ∩ Hλ̄, ψ〉 is a Σ 2 1 truth for λ̄. More generally, an interval of cardinals [κ, λ] with κ ≤ λ is Σ21 in...

We show that assuming the consistency of certain large cardinals (namely a supercompact cardinal with a measurable cardinal above it), it is possible to force and construct choiceless universes of ZF in which the first two uncountable cardinals א1 and א2 are both measurable and carry certain fixed numbers of normal measures. Specifically, in the models constructed, א1 will carry exactly one nor...

After small forcing, almost every strongness embedding is the lift of a strongness embedding in the ground model. Consequently, small forcing creates neither strong nor Woodin cardinals. The widely known Levy-Solovay Theorem [LevSol67] asserts that small forcing does not affect the measurability of any cardinal. If a forcing notion P has size less than κ, then κ is measurable in V P if and only...

In this paper we continue work from a previous paper on the fragility and indestructibility of the tree property. We present the following: (1) A preservation lemma implicit in Mitchell’s PhD thesis, which generalizes all previous versions of Hamkins’ Key lemma. (2) A new proof of theorems the ‘superdestructibility’ theorems of Hamkins and Shelah. (3) An answer to a question from our previous p...