Semicontinuous real numbers in a topos

Some properties of fuzzy real numbers

In the mathematical analysis, there are some theorems and definitions that established for both real and fuzzy numbers. In this study, we try to prove  Bernoulli's inequality in fuzzy real numbers with some of its applications. Also, we prove two other theorems in fuzzy real numbers which are proved before, for real numbers.

Generating Families in a Topos

A generating family in a category C is a collection of objects {Ai|i ∈ I} such that if for any subobject Y // m //X, every Ai f //X factors through m, then m is an isomorphism – i.e. the functors C(Ai, ) are collectively conservative. In this paper, we examine some circumstances under which subobjects of 1 form a generating family. Objects for which subobjects of 1 do form a generating family a...

Lower and Upper Regularizations of Frame Semicontinuous Real Functions

As discovered recently, Li and Wang’s 1997 treatment of semicontinuity for frames does not faithfully reflect the classical concept. In this paper we continue our study of semicontinuity in the pointfree setting. We define the pointfree concepts of lower and upper regularizations of frame semicontinuous real functions. We present characterizations of extremally disconnected frames in terms of t...

How Real are Real Numbers?

We discuss mathematical and physical arguments against continuity and in favor of discreteness, with particular emphasis on the ideas of Emile Borel (1871–1956).

On approximation of real numbers by real algebraic numbers