Alternative set theory with elementary classes

summary:In this paper we sketch the development and give a model of the formal version of a generalization of the Alternative Set Theory

Biequivalence vector spaces in the alternative set theory

summary:As a counterpart to classical topological vector spaces in the alternative set theory, biequivalence vector spaces (over the field $Q$ of all rational numbers) are introduced and their basic properties are listed. A methodological consequence opening a new view towards the relationship between the algebraic and topological dual is quoted. The existence of various types of valuations on a biequivalence vector space inducing its biequivalence is proved. Normability is characterized in terms of total convexity of the monad and/or of the galaxy of $0$. Finally, the existence of a rather strong type of basis for a fairly extensive area of biequivalence vector spaces, containing all the most important particular cases, is established

Infinity and Continuum in the Alternative Set Theory

Alternative set theory was created by the Czech mathematician Petr Vop\v enka in 1979 as an alternative to Cantor's set theory. Vop\v enka criticised Cantor's approach for its loss of correspondence with the real world. Alternative set theory can be partially axiomatised and regarded as a nonstandard theory of natural numbers. However, its intention is much wider. It attempts to retain a correspondence between mathematical notions and phenomena of the natural world. Through infinity, Vop\v enka grasps the phenomena of vagueness. Infinite sets are defined as sets containing proper semisets, i.e. vague parts of sets limited by the horizon. The new interpretation extends the field of applicability of mathematics and simultaneously indicates its limits. This incidentally provides a natural solution to some classic philosophical problems such as the composition of a continuum, Zeno's paradoxes and sorites. Compared to strict finitism and other attempts at a reduction of the infinite to the finite Vop\v enka's theory reverses the process: he models the finite in the infinite.Comment: 25 page

Infinity and Continuum in the Alternative Set Theory

Alternative set theory was created by the Czech mathematician Petr Vopěnka in 1979 as an alternative to Cantor's set theory. Vopěnka criticised Cantor's approach for its loss of correspondence with the real world. Alternative set theory can be partially axiomatised and regarded as a nonstandard theory of natural numbers. However, its intention is much wider. It attempts to retain a correspondence between mathematical notions and phenomena of the natural world. Through infinity, Vopěnka grasps the phenomena of vagueness. Infinite sets are defined as sets containing proper semisets, i.e. vague parts of sets limited by the horizon. The new interpretation extends the field of applicability of mathematics and simultaneously indicates its limits. Compared to strict finitism and other attempts at a reduction of the infinite to the finite Vopenka's theory reverses the process: he models the finite in the infinite

Revealed automorphisms

summary:We study automorphisms in the alternative set theory. We prove that fully revealed automorphisms are not closed under composition. We also construct some special automorphisms. We generalize the notion of revealment and Sd-class