The Functor Bext under the Negation of CH.
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 measure algebra does not always embed
The Open Colouring Axiom implies that the measure algebra cannot be embedded into P(ℕ)/fin. We also discuss errors in previous results on the embeddability of the measure algebra.
The nonabsolute boundedness model of the theory of semisets
The non-existence of well-orderings of the Cantor set
The null ideal restricted to some non-null set may be ℵ₁-saturated
Our main result is that possibly some non-null set of reals cannot be divided into uncountably many non-null sets. We also deal with a non-null set of real, the graph of any function from which is null, and deal with our iterations somewhat more generally.
The power set of ω Elementary submodels and weakenings of CH
We define a new principle, SEP, which is true in all Cohen extensions of models of CH, and explore the relationship between SEP and other such principles. SEP is implied by each of CH*, the weak Freeze-Nation property of (ω), and the (ℵ₁,ℵ₀)-ideal property. SEP implies the principle , but does not follow from , or even .
The quantifier complexity of NF.
The regular open algebra of βRR is not equal to the completion of P(ω)/fin
Two compact spaces are co-absoluteif their respective regular open algebras are isomorphic (i.e. homeomorphic Gleason covers). We prove that it is consistent that βω and βℝ are not co-absolute.
The regularity properties on the real line
The relative consistency of some consequences of the existence of measurable cardinal numbers [Book]
The size of maximal almost disjoint families [Book]
The splitting number can be smaller than the matrix chaos number
Let χ be the minimum cardinality of a subset of that cannot be made convergent by multiplication with a single Toeplitz matrix. By an application of a creature forcing we show that < χ is consistent. We thus answer a question by Vojtáš. We give two kinds of models for the strict inequality. The first is the combination of an ℵ₂-iteration of some proper forcing with adding ℵ₁ random reals. The second kind of models is obtained by adding δ random reals to a model of for some δ ∈ [ℵ₁,κ). It...
The strength of the projective Martin conjecture
We show that Martin’s conjecture on Π¹₁ functions uniformly -order preserving on a cone implies Π¹₁ Turing Determinacy over ZF + DC. In addition, it is also proved that for n ≥ 0, this conjecture for uniformly degree invariant functions is equivalent over ZFC to -Axiom of Determinacy. As a corollary, the consistency of the conjecture for uniformly degree invariant Π¹₁ functions implies the consistency of the existence of a Woodin cardinal.
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 Tree Property at ω₂ and Bounded Forcing Axioms
We prove that the Tree Property at ω₂ together with BPFA is equiconsistent with the existence of a weakly compact reflecting cardinal, and if BPFA is replaced by BPFA(ω₁) then it is equiconsistent with the existence of just a weakly compact cardinal. Similarly, we show that the Special Tree Property for ω₂ together with BPFA is equiconsistent with the existence of a reflecting Mahlo cardinal, and if BPFA is replaced by BPFA(ω₁) then it is equiconsistent with the existence of just a Mahlo cardinal....
The undecidability of the existence of a non-separable normal Moore space satisfying the countable chain condition
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.
Théorèmes de consistance en théorie de la mesure de R. Solovay