RNDr. Kateřina Trlifajová, Ph.D.

Article

10.1007/s13194-021-00429-7

Department of Applied Mathematics

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, Vopˇenka 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 Vopˇenka’s theory reverses the process: he models the finite in the infinite.

Book chapter

Department of Applied Mathematics

We examine a possibility to describe phenomenons of a process and of an event in mathematics. That can be done in Vopěnka's Alternative Set Theory involving such notions as semiset, pi-class or sigma-class.

Article

10.1007/s10699-018-9549-z

Department of Applied Mathematics

In his Foundations of a General Theory of Manifolds, Georg Cantor praised Bernard Bolzano as a clear defender of actual infinity who had the courage to work with infinite numbers. At the same time, he sharply criticized the way Bolzano dealt with them. Cantor’s concept was based on the existence of a one-to-one correspondence, while Bolzano insisted on Euclid’s Axiom of the whole being greater than a part. Cantor’s set theory has eventually prevailed, and became a formal basis of contemporary mathematics, while Bolzano’s approach is generally considered a step in the wrong direction. In the present paper, we demonstrate that a fragment of Bolzano’s theory of infinite quantities retaining the part-whole principle can be extended to a consistent mathematical structure. It can be interpreted in several possible ways. We obtain either a linearly ordered ring of finite and infinitely great quantities or a partially ordered ring containing infinitely small, finite and infinitely great quantities. These structures can be used as a basis of the infinitesimal calculus similarly as in non-standard analysis, whether in its full version employing ultrafilters due to Abraham Robinson or in the recent “cheap version” avoiding ultrafilters due to Terence Tao.

Proceedings paper

Department of Applied Mathematics

Since the 1960s, when Robinson non-standard analysis was established, several other non-standard models of natural and real numbers have been created. The not widely known theory of the Czech mathematician Petr Vopěnka, Alternative Set Theory, AST, was also developed. It is an alternative to Cantor Set Theory, which Vopěnka criticized for numerous reasons. Cantor’s justification for accepting the actual infinity was theological; in modern axiomatic systems it is expressed by the axiom of infinity. Infinite hierarchy of infinite cardinal and ordinal numbers finds minimal interpretation in the real world. The existence of independent theorems leads to dividing set theory into several branches, from which none can be considered the sole truth. Vopěnka’s AST relies on phenomenology and endeavours to interpret basic terms of infinite mathematics in the real world. It uses the infinite for the mathematization of indistinctness. Apart from classic sets and classes, here so-called semisets are introduced. AST can be partially formalized as the non-standard model. Similarly, as with other non-standard theories, it does not bring breakthrough mathematical results that have been impossible to describe in a standard manner. What is substantial is its philosophical interpretation, which attempts to retain correspondence with the real world. It offers the solution of certain old philosophical problems: Zeno's paradoxes, sorites, Leibniz’s conception of continuum, Pascal’s double infinity.

Book chapter

Department of Applied Mathematics

During the early 1830's Bernard Bolzano, working in Prague, wrote a manuscript giving a foundational account of numbers and their properties. In the final section of his work he described what he called 'infinite number expressions' and 'measurable numbers'. This work was evidently an attempt to provide an improved proof of the sufficiency of the criterion usually known as the 'Cauchy criterion' for the convergence of an infinite sequence. Bolzano had in fact published this criterion four years earlier than Cauchy who, in his work of 1821, made no attempt at a proof. Any such proof required the construction or definition of real numbers and this, in essence, was what Bolzano achieved in his work on measurable numbers. It therefore pre-dates the well-known constructions of Dedekind, Cantor and many others by several decades. Bolzano's manuscript was partially published in 1962 and more fully published in 1976. We give an account of measurable numbers, the properties Bolzano proved about them, and the controversial reception they have prompted since their publication.

Book chapter

Department of Applied Mathematics

When mathematicians speak about truth in mathematics today they considers mainly provability from axioms. Formerly scholars consider a special correspondence between mathematical objects and objects of the real world. How did it happen and in which sense can we speak about correspondence now?

Article

Department of Applied Mathematics

One of the key themes of Petr Vopenka was his understanding of mathematical in-finity. He put forward many objections to Cantor's established set theory. He worked out a new, alternative, theory in which he surprisingly interpreted infinity as a means of mathematising indeterminacy. He interpreted the continuum in a similar way. He drew on phenomenology, on Husserl's motto "Return to things themselves", and he employed a range of phenomenological concepts. At the same time, however, he did not give up the claim to mathematical precision in his theory. This claim brings with it certain pitfalls which centre on mathematical idealisation and its relation to the natural real world.

Article

Department of Applied Mathematics

Petr Vopenka was an important Czech mathematician. He was interested also in history and philosophy of mathematics, especially in questions concerning mathematical infinity and its relation to the natural world.

Article

Department of Applied Mathematics

Several words about the life and work of recently died professor Petr Vopenka.

Book chapter

10.1007/978-0-387-74759-0_12

Department of Applied Mathematics

Article

Department of Applied Mathematics

Only at the end of the 19th century a set theory was created that works with the actual infinite. The creator behind the theory, the German mathematician Georg Cantor, felt all the more the need to challenge the long tradition that only recognised potential infinite. In this he received strong support from the interest among German neothomist philosophers, who, under the influence of the Encyclical of Pope Leo XIII, Aeterni Patris, began to take an interest in Cantor's work.