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Mathematical Paradoxes Jasmine Neal Infinite Regress • Which came first the chicken or the egg? Hilbert Hotel • Hilbert Hotel has an infinite number of rooms and all the rooms are occupied. A new guest arrives can he be accommodated? • The Infinite Circle Nicholas of Cusa (1401-1464) made the following interesting point regarding the shape of an infinite circle. The curvature of a circle's circumference decreases as the size of the circle increases. For example, the curvature of the earth's surface is so negligible that it appears flat. The limit of decrease in curvature is a straight line. • An infinite circle is therefore... a straight line! Russell’s Paradox • The Barber shaves all those men in town, and only those men, who do not shave themselves • Who shaves the Barber? Russell’s Paradox • The set M is the set of all sets that do not contain themselves as numbers. Does M contain itself? • M = {A | A is not an element of A} Solution: Type Theory and axiomatic set theory It's official: 1=2 • And here is the proof... • (1) X = Y Given (2) X2 = XY Multiply both sides by X (3) X2 - Y2 = XY - Y2 Subtract Y2 from both sides (4) (X+Y)(X-Y) = Y(X-Y) Factor both sides (5) (X+Y) = Y Cancel out common factors (6) Y+Y = Y Substitute in from line (1) (7) 2Y = Y Collect the Y's (8) 2 = 1 Divide both sides by Y • Can you spot the error that has led to this curious conclusion? Paradoxes in Geometry • Curry’s puzzle • How can this be true!