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Good Math by Mark C. Chu-Carroll

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Cantor’s Diagonalization

The original motivation behind the ideas that ended up growing into set theory was Cantor’s recognition of the fact that there’s a difference between the size of the set of natural numbers and the size of the set of real numbers. They’re both infinite, but they’re not the same!

Cantor’s original idea was to abstract away the details of numbers. Normally when we think of numbers, we think of them as being things that we can do arithmetic with, things that can be compared and manipulated in all sorts of ways. Cantor said that for understanding how many numbers there were, none of those properties or arithmetic operations were needed. The only thing that mattered was that a kind of number like the natural numbers was ...

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