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1

Aksjomat multiplikatywny Russela

100%

Lewandowska K.

Zagadnienia Filozoficzne w Nauce

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2012

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issue 50

152-166

PL

We present the history of two parallel (and equivalent) discoveries: the axiom of choice and the multiplicative axiom. Firstly, we consider the origins of the formulation of the multiplicative axiom. Next, we concentrate on Russell’s attitude towards the role of this axiom, which is closely related to his philosophy of mathematics. We also highlight some diﬀerences between Russell’s and Zermelo’s propositions.

2

Rola aksjomatu w matematyce współczesnej oraz w perspektywie dociekań nad aksjomatem wyboru

100%

Lewandowska K.

Zagadnienia Filozoficzne w Nauce

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2011

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issue 48

148-165

PL

We show how philosophy effected the shape of mathematics when the proof of Well-Ordering Principle was formulated by Ernst Zermelo. We also consider the significance of philosophy of mathematics today. We concentrate on Solomon Feferman and Penelope Maddy attitude in the recent debate on the need of new axioms in mathematics.

3

Aksjomat wyboru w pracach Wacława Sierpińskiego

88%

Lewandowska K.

Zagadnienia Filozoficzne w Nauce

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2013

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issue 52

9-33

PL

This paper presents Wacław Sierpiński – the first advocate of the axiom of choice. We focus on the philosophical and mathematical topics related to the axiom of choice which were considered by Sierpiński. We analyze some of his papers to show how his results effected the debate over Zermelo’s axiom. Sierpiński’s impact on this discussion is of particular importance since he was the first who tried to explore consequences of the axiom of choice thoroughly and asserted its undoubted significance to mathematics as a whole.

4

Axiom výběru a hypotéza kontinua - souvislosti a rozdíly

75%

Slabá T.

Teorie vědy (Theory of Science)

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2023

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vol. 45

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issue 1

67-93

EN

We compare two well-known set-theoretical statements, namely the axiom of choice and the continuum hypothesis, with regard to their historical development and formulation, as well as their consequences in mathematics. It is known that both statements are independent from the other axioms of set theory (if they are consistent). The axiom of choice – despite initial controversies – is today almost universally accepted as an axiom. However, the status of the continuum hypothesis is more complex and no agreement has been found so far: both the continuum hypothesis and its negation (often as consequences of stronger statements) decide several mathematical problems differently, but in contrast with the axiom of choice it is not clear which of the two solutions should be the “correct” one (in the sense of an agreement within the community).

CS

V práci porovnáváme dvě známá množinově-teoretická tvrzení, totiž axiom výběru a hypotézu kontinua, z hlediska jejich historického vývoje a formulace a rovněž z hlediska jejich důsledků v matematice. Obě tvrzení jsou nezávislá na ostatních axiomech teorie množin (pokud jsou tyto axiomy konzistentní). Axiom výběru – přes počáteční váhání a někdy i odpor – je dnes téměř univerzálně přijímán. Naproti tomu status hypotézy kontinua je mnohem složitější a nepanuje shoda ohledně její platnosti: hypotéza kontinua i její negace (často jako důsledky silnějších tvrzení) rozhodují odlišně mnohá matematicky zajímavá tvrzení, ale na rozdíl od axiomu výběru není zřejmé, které řešení je to „správné“ (ve smyslu shody v matematické komunitě).

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Aksjomat multiplikatywny Russela

Rola aksjomatu w matematyce współczesnej oraz w perspektywie dociekań nad aksjomatem wyboru

Aksjomat wyboru w pracach Wacława Sierpińskiego

Axiom výběru a hypotéza kontinua - souvislosti a rozdíly