In order to avoid well-know paradoxes associated with self-referential definitions, higher-order dependent type theories stratify the theory using a countably infinite hierarchy of universes (also known as sorts), Type0 : Type1 : · · · . Such type systems are called cumulative if for any type A we have that A : Typei implies A : Typei+1. The predicative calculus of inductive constructions (pCIC...

After a summary of earlier work it is shown that elementary or Kalmar arithmetic can be interpreted within the system of Russell’s Principia Mathematica with the axiom of infinity but without the axiom of reducibility. 1 Historical introduction After discovering the inconsistency in Frege’s Grundgesetze der Arithmetik, Russell proposed two changes: first, dropping the assumption that to every h...

The article is devoted to the problem of applying the formal data mining tool – forecasting – for the developing of new software and for reengineering the present software. We propose the algorithm adjustments of the time series forecasting. This algorithm takes into account the dependence of the current state of time series from the previous one, the influence of basic fuzzy projected trends i...

Borrowings and Adaptations. First of all, the author would like to clarify the meanings of the terms "predicative" and "nonpredicative". As is well known, praedicatum in Late Latin means "what has been said (previously)". In Aristotelian and subsequent forms of traditional logic [8,9,11,12,13,14,15,16] a predicate was understood to be one (the one in which something is said about the subject of...

Minimalist grammars are a rich lexicalized syntactic formalism which inherits the depth and wide covering of the generative tradition. Nevertheless up to now there is no straightforward way to extract the predicative structure of a sentence from syntactic analyses and from the predicative structure of the lexical items. In this paper we try to apply to this richer syntactic model the traditiona...

Formal topology is today an established topic in the development of constructive, that is intuitionistic and predicative, mathematics. Many classical results of general topology have been already brought into the realm of constructive mathematics by using formal topology and also new light on basic topological notions was gained with this approach which allows distinction which are not sensible...

Predicative analysis of recursion schema is a method to characterize complexity classes like the class of polynomial time functions. This analysis comes from the works of Bellantoni and Cook, and Leivant. Here, we refine predicative analysis by using a ramified Ackermann’s con10 struction of a non-primitive recursive function. We obtain an hierarchy of functions which characterizes exactly func...

This is the third installment in a series of papers on algebraic set theory. In it, we develop a uniform approach to sheaf models of constructive set theories based on ideas from categorical logic. The key notion is that of a “predicative category with small maps” which axiomatises the idea of a category of classes and class morphisms, together with a selected class of maps whose fibres are set...

A novel analysis of the syntax of utterance particles in Chinese is proposed in this paper. In terms of their diachronic origin, there are two types of utterance particles in Chinese, namely predicative utterance particles and non-predicative utterance particles, both of which are derived by a generalized syntactic schema. It is proposed that utterance particles are the complement of a function...

We provide machine-independent characterizations of some complexity classes, over an arbitrary structure, in the model of computation proposed by L. Blum, M. Shub and S. Smale. We show that the levels of the polynomial hierarchy correspond to safe recursion with predicative minimization and the levels of the digital polynomial hierarchy to safe recursion with digital predicative minimization. A...