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Russells paradoks

Russells paradoks ble formulert av Bertrand Russell i 1901, og viser at den naive mengdelæren til Cantor og Frege inneholder motsigelser. La M være mengden av alle mengder som ikke inneholder seg selv, :

M = \.

Denne definisjonen fører til en motsigelse, når man prøver å avgjøre om M selv er et element i M. Antar en at M ikke er et element i M, følger det av definisjonen at M er et element i M, og antar en at M er et element i M, medfører det at M ikke er et element i M. :

M \in M \Rightarrow M \notin M

M \notin M \Rightarrow M \in M

Det finnes mange formuleringer av Russells paradoks. En kjent versjon er barbereren som barberer alle som ikke barberer seg selv. Barberer barbereren seg selv? At det var mulig å formulere et paradoks i den naive mengdelæren, gjorde at man så det nødvendig å bygge opp mengdelæren på en annen måte. I aksiomatisk mengdelære kan ikke «mengden av alle mengder som ikke inneholder seg selv» være en mengde, og paradokset blir dermed unngått. Kategori:Matematikk

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