Tuesday, May 30, 2006

CH in simple terms II

First, notice that a set has size N+1 if and only if it can be linearly ordered such that every element has at most N lesser elements. Similarly, a set is countable if and only if it can be linearly ordered such that every element has only finitely many lesser elements. The continuum hypothesis is equivalent to the existence of a (very strange) linear ordering of the reals with respect to which every real has only countably many lesser reals.

For comparison, consider the classical statement of CH: for every infinite set X of reals there is either a one-to-one correspondence between X and the integers, or a one-to-one correspondence between X and the set of all reals. Which version is easier for you to understand? I find linear orders easier to visualize than one-to-one correspondences, but others' imaginations might work very differently than mine.

(This post is the sequel to "CH in simple terms," which presents CH in terms of strange colorings of a cube.)

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