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Mark Pilgrim offers an entertaining refresher course on aleph-naught, cardinality of infinite sets, and Cantor’s
diagonal argument (Wikipedia
entry). Those damnable &#211D; — they always fall into
the cracks between natural numbers!

But there are ℵ₁ real numbers. ℵ₁ is, pardon the pun, infinitely larger than ℵ₀. Asking exactly how much larger is a good way to get mathematicians to swear at each other, and combined with a few rounds at the bar, can lead to predictably disastrous results.