Related papers: Extended real number arithmetics via Dedekind cuts
We construct a topos in which the Dedekind reals are countable. The topos arises from a new kind of realizability, which we call parameterized realizability, based on partial combinatory algebras whose application depends on a parameter.…
This paper focuses on greedy expansions, one possible representation of numbers, and on arithmetical operations with them. Performing addition or multiplication some additional digits can appear. We study bounds on the number of such digits…
We introduce and study expansions of real numbers with respect to two integer bases.
A simultaneous extension of real numbers set and the class of real functions is discussed.
We study how well a real number can be approximated by sums of two or more rational numbers with denominators up to a certain size.
Recently uncovered second derivative discontinuous solutions of the simplest linear ordinary differential equation define not only an nonstandard extension of the framework of the ordinary calculus, but also provide a dynamical…
A new number system, the set of the non-Dedekindian numbers, is introduced and characterized axiomatically. It is then proved that any hypercontinous hyperreal number system is strictly included in the set of the Non-Dedekindian Numbers.…
We study first-order expansions of the reals which do not define the set of natural numbers. We also show that several stronger notions of tameness are equivalent to each others.
Using appropriate notation systems for proofs, cut-reduction can often be rendered feasible on these notations, and explicit bounds can be given. Developing a suitable notation system for Bounded Arithmetic, and applying these bounds, all…
In this paper we show how to apply various techniques and theorems (including Pincherle's theorem, an extension of Euler's formula equating infinite series and continued fractions, an extension of the corresponding transformation that…
In this paper we define "a continued fraction expansion of the exponential integral $E_{1}(x)$ at infinity", which is analogous to the regular continued fraction expansion of real numbers, and prove that this expansion gives the same…
Given a positive rational number $n/d$ with $d$ odd, its odd greedy expansion starts with the largest odd denominator unit fraction at most $n/d$, adds the largest odd denominator unit fraction so the sum is at most $n/d$, and continues as…
In this paper we model discontinuous extended real functions in pointfree topology following a lattice-theoretic approach, in such a way that, if $L$ is a subfit frame, arbitrary extended real functions on $L$ are the elements of the…
We introduce and study non-uniform expansions of real numbers, given by two non-integer bases.
Starting from a small number of well-motivated axioms, we derive a unique definition of sums with a noninteger number of addends. These "fractional sums" have properties that generalize well-known classical sum identities in a natural way.…
The paper studies a cluster of systems for fully disquotational truth based on the restriction of initial sequents. Unlike well-known alternative approaches, such systems display both a simple and intuitive model theory and remarkable…
We consider existential problems over the reals. Extended quantifier elimination generalizes the concept of regular quantifier elimination by providing in addition answers, which are descriptions of possible assignments for the quantified…
We give a construction of a real number that is normal to all integer bases and continued fraction normal. The computation of the first n digits of its continued fraction expansion performs in the order of n^4 mathematical operations. The…
The main results of this paper are the construction, both rigourous and intuitive, of "the" intrinsic extension of the set of non negative integers N and the smallest over-field of R set which is continue (according to R.Dedekind). The aim…
Entire functions in one complex variable are extremely relevant in several areas ranging from the study of convolution equations to special functions. An analog of entire functions in the quaternionic setting can be defined in the slice…