Sizes of Countable Sets

The paper introduces the notion of the size of countable sets that preserves the Part-Whole Principle and generalizes the notion of the cardinality of finite sets. The sizes of natural numbers, integers, rational numbers, and all their subsets, unions, and Cartesian products are algorithmically enumerable up to one element as sequences of natural numbers. The method is similar to that of Theory of Numerosities of Benci and Di Nasso 2019) but in comparison, it is motivated by Bolzano's concept of infinite series from his Paradoxes of the Infinite, it is constructive because it does not use ultrafilters, and set sizes are uniquely determined. The results mostly agree with those of Theory of Numerosities, but some differ, such as the size of rational numbers. However, set sizes are just partially and not linearly ordered. \emph{Quid pro quo.

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