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Good Math by Mark C. Chu-Carroll

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Where in Infinity?

Now we’ve got sets, and we can talk about the size of those sets using cardinal numbers. But even if we have a set where we know its size and we know that its elements are ordered by some kind of relation, we can’t talk about where in the set a particular value lives. A cardinal number describes the number of elements in a set. It’s a way of specifying a quantity.

In English, it’s easy to show why you can’t use ordinals in the place of cardinals, but it’s much harder to do the other way around. If you speak English and you try to say “I have seventh apples,” it’s obviously wrong. Going the other way, you can say “I want apple 3,” and it seems like you’re using a cardinal to specify an ordinal position. In English, you can ...

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