We give an affirmative answer to Brendle’s and Hrušák’s question of whether the club principle together with h > א1 is consistent. We work with a class of axiom A forcings with countable conditions such that q ≥n p is determined by finitely many elements in the conditions p and q and that all strengthenings of a condition are subsets, and replace many names by actual sets. There are two types o...

Proof. Without loss of generality, we only need to prove for the case s= 1, as ρ satisfies Axioms A1-A5 if and only if 1 s ρ satisfies Axioms A1-A5 (with s= 1 in Axiom A3). The “only if” part. First, we show that (2) holds for any X ∈L∞(Ω,F , P ). Define the set function ν(E) := ρ(1E),E ∈F . Then, it follows from Axiom A2 and A3 that ν is monotonic, ν(∅) = 0, and ν(Ω) = 1. ForM ≥ 1, define LM :...

We present instances of the following phenomenon: if a product of topological spaces satisfies some given compactness property then the factors satisfy a stronger compactness property, except possibly for a small number of factors. The first known result of this kind, a consequence of a theorem by A. H. Stone, asserts that if a product is regular and Lindelöf then all but at most countably many...

We prove that the Covering Property Axiom CPA prism, which holds in the iterated perfect set model, implies that there exists an additive discontinuous almost continuous function f : R → R whose graph is of measure zero. We also show that, under CPA prism, there exists a Hamel basis H for which, E+(H), the set of all linear combinations of elements from H with positive rational coefficients, is...

Inthispaper, (L,M)-fuzzy domain finiteness and (L,M)-fuzzy restricted hull spaces are introduced, and several characterizations of the category (L,M)-CS of (L,M)-fuzzy convex spaces are obtained. Then, (L,M)-fuzzy stratified (resp. weakly induced, induced) convex spaces are introduced. It is proved that both categories, the category (L,M)-SCS of (L,M)-fuzzy stratified convex spaces and the cate...

Strongly Unfoldable Cardinals Made Indestructible by Thomas A. Johnstone Advisor: Joel David Hamkins I provide indestructibility results for weakly compact, indescribable and strongly unfoldable cardinals. In order to make these large cardinals indestructible, I assume the existence of a strongly unfoldable cardinal κ, which is a hypothesis consistent with V = L. The main result shows that any ...

This paper contributes to the literature of pro-poor growth measurement by introducing a growth-rate consistency axiom. The axiom states that if one growth pattern is judged to be more pro-poor than another growth pattern at a given growth rate, then the pro-poor ranking between the two growth patterns should remain the same at a higher growth rate. We show that summary pro-poor measures such a...