Mathematical paradoxes

upright=1.35|class=skin-invert-image|thumb|alt=Stylistic impression of the repeating decimal 0.9999..., representing the digit 9 repeating infinitely 0.999... is a repeating decimal that represents the number 1. The three dots represent an infinite list of "9" digits. Following the standard rules for representing real numbers in decimal notation, its value is the smallest number greater than or equal to every number in the sequence 0.9, 0.99, 0.999, and so on. It can be proved that this number is1; that is, 0.999\ldots = 1.

set of philosophical problems

infinite series

theorem that there exists a decomposition of a unit solid ball into a finite number of disjoint subsets, which can be put back together in a different way to yield two identical copies of the unit sphere

infinite surface of revolution with infinite surface area enclosing a finite volume, which contributed to 17th century debate on the nature of infinity

thought experiment proposed by David Hilbert

proposed explanation for how trying to improve traffic flow actually has the reverse effect

the infinite sum of alternating 1 and -1 terms

optical illusion used in mathematics

problem within the classical interpretation of probability theory: what is the probability that a randomly chosen chord on a circle is longer than a side of an equilateral triangle inscribed in the circle?

thought experiment about a game involving a reliable predictor of the player’s actions

divergent series

a paradox in which an arbitrary claim F is proved from the mere existence of a sentence C that says of itself “If C, then F”

logical contradiction in which every possible solution to the problem is exempt: if there were some uninteresting natural numbers, there would be a smallest uninteresting number, which would be therefore interesting

apparent contadiction in metamathematics

veridical paradox

self-referential paradox

apparent absurdity in rolling a coin along its edge

the paradox that there are countable models of theories that assert the existence of uncountable sets

paradox in game theory in which a combination of losing strategies becomes a winning strategy

topological operation of turning a sphere inside-out without creasing

paradox in mathematics

the statement that the number of points of intersection of two planar higher-order curves can be greater than the number of arbitrary points usually needed to define one such curve

polyhedral approximation to a cylinder obtained by stacking antiprisms, used as a pathological example to demonstrate that limits of surface areas of polyhedral approximations do not always equal the true surface area

discussion of paradoxes of set theory