From a suitable large cardinal hypothesis, we provide a model with a supercompact cardinal in which universal indestructibility holds: every supercompact and partially supercompact cardinal κ is fully indestructible by <κ-directed closed forcing. Such a state of affairs is impossible with two supercompact cardinals or even with a cardinal which is supercompact beyond a measurable cardinal. Lave...

Part 1 Cardinality of Sets 2 1. Our Purpose 2 2. Introduction 2 3. Notation 3 4. Basic Operations 3 5. Notation and definitions for classes of cardinal numbers 3 6. Basic properties of Cardinal Arithmetic 5 7. General Arithmetic Properties Of Transfinite Cardinals 5 7.1. Useful Special Cases 5 7.2. General Cases 5 8. Specific Arithmetic Relations 6 8.1. Cases involving א0 6 8.2. Cases involving...

Our perceptual systems are shaped by our natural environment. Statistical studies on natural scenes have provided important insights to the development of simple cell receptive fields ? and the connectivity among orientation selective neurons in the primary visual cortex 1,25, 28. Here, we extended such study to the neural encoding of statistical regularities of 3D surfaces in natural scenes. A...

The equiconsistency of a measurable cardinal with Mitchell order o(κ) = κ++ with a measurable cardinal such that 2κ = κ++ follows from the results by W. Mitchell [13] and M. Gitik [7]. These results were later generalized to measurable cardinals with 2κ larger than κ++ (see [8]). In [5], we formulated and proved Easton’s theorem [4] in a large cardinal setting, using slightly stronger hypothese...