RNDr. Kateřina Trlifajová, Ph.D.

Článek

10.1007/s13194-021-00429-7

Katedra aplikované matematiky

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.

Článek

10.1007/s10699-018-9549-z

Katedra aplikované matematiky

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.

Stať ve sborníku

Katedra aplikované matematiky

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.

Kapitola v knize

Katedra aplikované matematiky

Zabýváme se možností popsat fenomény procesů a událostí matematicky. To je možné ve Vopěnkově Alternativní teorii množina, která obsahuje takové pojmy polomnožina, pi-třída a sigma-třída, které mají svůj zvláštní význam.

Kapitola v knize

Katedra aplikované matematiky

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.

Článek

Katedra aplikované matematiky

Jedna z věcí, na nichž Petru Vopěnkovi obzvlášť záleželo, byl způsob uchopení nekonečna v matematice. Z řady důvodů odmítal klasickou Cantorovu teorii množin. Vytvořil teorii novou, alternativní, v níž nekonečno překvapivě využil k matematizaci neurčitosti. Vyložil zároveň novým způsobem kontinuum. Opřel se o fenomenologii, o Husserlovo heslo „Návrat k věcem samým“, a použil některé její pojmy. Přitom ani v nejmenším nevzdal nárok na matematickou přesnost své teorie. To přináší jistá úskalí, která se týkají zejména vztahu přirozeného reálného světa a jeho matematické idealizace.

Kapitola v knize

Katedra aplikované matematiky

Dnes se pod pravdivostí v matematice rozumí hlavně dokazatelnost z axiomů. Dříve se opírala o jistou korespondence mezi objekty přirozeného světa a matematickými objekty. Jak došlo k tomuto posunu a v jakém smyslu můžeme dnes mluvit o této korespondenci?

Článek

Katedra aplikované matematiky

Petr Vopěnka byl nejen matematik, ale zabýval se také filosofií a historií matematiky. Jedním z nejdůležitějších témat pro něj bylo nekonečno a jeho vztah k přirozenému světu.

Článek

Katedra aplikované matematiky

Několik slov o životě a díle nedávno zemřelého profesora Petra Vopěnky.

Kapitola v knize

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

Katedra aplikované matematiky

Článek

Katedra aplikované matematiky

Až na konci 19. století bylo přijato aktuální nekonečno do matematiky pomocí teorie množin. Tvůrce této teorie Georg Cantor pociťoval potřebu se vyrovnat s dlouhou tradicí, která uznávala pouze nekonečno potenciální. Velkou podporou pro něj byl zájem německých neotomistických myslitelů, kteří se pod vlivem encykliky papeže Lva XIII. Aeterni Patris začali zabývat Cantorovým dílem.