We show that universal indestructibility for both strong compactness and supercompactness is consistent with the existence of two strongly compact cardinals. This is in contrast to the fact that if κ is supercompact and universal indestructibility for either strong compactness or supercompactness holds, then no cardinal λ > κ is measurable.
For suitable topological spaces X and Y, given a continuous function f:X → Y and a point x ∈ X, one can determine the value of f(x) from the values of f on a deleted neighborhood of x by taking the limit of f. If f is not required to be continuous, it is impossible to determine f(x) from this information (provided |Y| ≥ 2), but as the author and Alan Taylor showed in 2009, there is nevertheless a means of guessing f(x), called the μ-predictor, that will be correct except on a small set; specifically,...
A subset of a topological space is said to be universally measurable if it is measured by the completion of each countably additive σ-finite Borel measure on the space, and universally null if it has measure zero for each such atomless measure. In 1908, Hausdorff proved that there exist universally null sets of real numbers of cardinality ℵ₁, and thus that there exist at least such sets. Laver showed in the 1970’s that consistently there are just continuum many universally null sets of reals....
A modification of Dempster's and Pawlak's constructs forms a new foundation for the identification of upper and lower sets formulas. Also, in this modified Dempster-Pawlak construct we require that subsets of the power set be restricted to the well-known information granules of the power set. An aggregation of upper information granules amongst each other and lower information granules amongst each other determine upper and lower set formulas for both crisp and fuzzy sets. The results are equivalent...
In this paper the concept of a fuzzy contraction mapping on a fuzzy metric space is introduced and it is proved that every fuzzy contraction mapping on a complete fuzzy metric space has a unique fixed point.