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copied!<p>Forgive the poorly written metaphors below.</p> <p>I personally think of the countability/uncountability dichotomy as being very closely related to Zeno's paradox of the arrow.</p> <p>The set of all natural numbers is countable, there is a specific method of generating the "next" integer, and it will get you a step forward. Countable sets are forward-moving in that sense. It's almost as if it has a <i>velocity</i>, it keeps moving forward.</p> <hr> <p>The set of all real numbers is uncountable, like <a href="http://en.wikipedia.org/wiki/Zeno's_paradox#The_arrow_paradox" rel="nofollow noreferrer">zeno's arrow</a>.</p> <p>If you have to move between the origin (0) and the destination (1 == 2<sup>-0</sup>), you must first go through the midpoint (1/2 == 2<sup>-1</sup>).</p> <p>Now your destination is 1/2; If you must then go between the origin (0) and the (1/2), you must go through the midpoint (1/4 == 2<sup>-2</sup>)</p> <p>So on and so forth, so to get between 0 and 1, you must first get between something inbetween, which you must first get between something inbetween. There is no finite method of calculating the "next" step, so the <i>velocity</i> (in contrast to the velocity of natural numbers) doesn't really exist, your next step is not going to take you anywhere.</p> <p><b>Edit:</b></p> <p>I realize now that this probably has to do with the total ordering and mapping of the set of natural numbers to any countable sets. <b>If you can't totally order the items in a set, or you can't create a method to determine what the next item is in a set, chances are it's uncountable.</b></p>

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