Questo articolo offre una panoramica della ricerca sui fondamenti della teoria degli insiemi,discutendo due programmi che mirano a superare i risultati di indecidibilità, tra i quali l'indipendenza dell'ipotesi del continuo. I due programmi sono basati, rispettivamente, sugli Assiomi di Forcing e su una congettura di Woodin chiamata V = Ultimate-L. Nel presentare queste ricerche introdurremo brevemente le principali nozioni di teoria degli insiemi.
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Con il contributo del Ministero per i Beni e le Attività Culturali