Cantorův paradox

Cantorův paradox je poznatek publikovaný Georgem Cantorem roku 1899, který spolu s dalšími výsledky podobného typu (označovanými jako antinomie nebo paradoxy naivní teorie množin) vedl ke krizi klasické naivní teorie množin a jejímu následnému nahrazení axiomatickým systémem.

Podstata paradoxu

Uvažujme o množině všech množin. Podle Cantorovy věty má množina všech jejích podmnožin (její potenční množina) větší mohutnost než samotná . Proto nemůže existovat prosté zobrazení do , ovšem takovým zobrazením je například identické zobrazení - což je spor.

Řešení paradoxu

V době publikování nebyla Cantorovu paradoxu přikládána příliš velká váha s tím, že se odehrává na příliš velkých množinách (množina všech podmnožin množiny všech množin). Proto se také vžilo označení paradox, ačkoliv ve skutečnosti se jednalo o spor v klasické definici množiny jako „souboru objektů (prvků) vymezených pomocí operace náležení“.

Teprve později, společně s dalšími „paradoxy“, z nichž jako nejdůležitější se ukázal Russellův paradox, vedl tento výsledek ke kompletnímu přepracování základů teorie množin na axiomatickém základě - vizte Zermelova–Fraenkelova teorie množin.

V axiomatické teorii množin žádným způsobem nelze zkonstruovat výše uvedenou množinu - soubor všech množin není množina, ale vlastní třída, a o její potenční množině tedy nemá ve světě teorie množin vůbec smysl mluvit.

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