A tapered floating point encoding is proposed which uses the redundant signed radix 2 system and is based on the canonical recoding. By making use of ternary technology, the encoding has a dynamic range exceeding that of the IEEE 754-1985 Standard for Floating Point Arithmetic (IEEE-754-1985, and precision equal to or better than that of the IEEE-754-1985 system and the recently proposed Posit system when equal input sizes are compared. In addition, the encoding is capable of supporting several proposed extensions, including extensions to integers, boolean values, complex numbers, higher number systems, low-dimensional vectors, and system artifacts such as machine instructions. A detailed analytic comparison is provided between the proposed encoding, the IEEE-754-1985 system, and the recently proposed Posit number system.
This article is an introduction to the basic generalized category theory used in the recent work [14] studying an extension of the theory of categories and categorical logic, including parts of topos theory. We discuss functors, equivalences, natural transformations, adjoints, and limits in a generalized setting, giving a concise outline of these frequently arising constructions.
We present a variant of the calculus of deductive systems developed in [5,6], and give a generalization of the Curry-Howard-Lambek theorem giving an equivalence between the category of typed lambda-calculi and the category of cartesian closed categories and exponentialpreserving morphisms that leverages the theory of generalized categories [13]. We discuss potential applications and extensions.
Abstract. We introduce an extension of the propositional calculus to include abstracts of predicates and quantifiers, employing a single rule along with a novel comprehension schema and a principle of extensionality, which are substituted for the Bernays postulates for quantifiers and the comprehension schemata of ZF and other set theories. We prove that it is consistent in any finite Boolean subset lattice. We investigate the antinomies of Russell, Cantor, Burali-Forti, and others, and discuss the relationship of the system to other set theoretic systems ZF, NBG, and NF. We discuss two methods of axiomatizing higher order quantification and abstraction, and then very briefly discuss the application of one of these methods to areas of mathematics outside of logic.
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