Transcript Mathematical Paradoxes

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!