BrainDen.com - Brain Teasers # Line vs Line segment paradox

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Suppose the point on a line segment is in a one to one correspondance relationship with a line. Both possess infinite points and so should fullly map onto each other however the line segment is of finite length while the line is of infinite length.

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The answer is that both the line and the line segments are made out of discrete relationships. Discrete relationships are container keeping things in and keeping things out. They also are objects. Being that a line is created from them distance is measured by the degree of separation the discrete relationships provide. Also being containers that contain each other they eventually contain themselves in fractal form so the line segment can be both discrete and of finite length AND infinite as it is fractally manifest you simply need a perspective on the combined object that contains both perspectives. This provides a easy and truthful resolution to this paradox.

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Consider the line segment [0, 1] and the entire real axis.

Both have the cardinality C of the reals.

f(x) = 1 / [ x (x-1) ] is a bijection.

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i like listening to a math teacher who argues that the real numbers are paradoxical.

the problem with them, say a number like square root 2, is that they require an infinite amount of information to "store". we can never know exactly what the square root of two is, we can only have an aproximate answer based on formulas.

consider the following irrational number. you take a list of intinite yes or no questions, and sort them alphabetically. then you express the answer to each question as a number in binary, containing a zero or no or one for yes. then you can answer an infinite number of questions just by knowing the position of the question in the sqeuence.

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