A barber shaves all men who don’t shave themselves. Does he shave himself? Mathematics offers explanations for this and other curious contradictions

Sometimes your gut feelings lead you astray—particularly in mathematics, in which one constantly comes across results that seem impossible. For example, infinity does not always equal infinity, and tortoises may outpace human athletes—at least from a certain mathematical point of view.

There are also many scenarios that appear contradictory at first glance (or second or third). These paradoxes can be explained, however. They are not errors but rather reminders that we should not rely too heavily on our intuition in mathematics. Here are three of the strangest paradoxes in the field.

Hilbert’s HotelHilbert’s Hotel

The Birthday ParadoxThe Birthday Paradox

Russell’s AntinomyRussell’s Antinomy

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