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In response to Is an empty set equal to another empty set?, I have another question.

In 'untyped' (regular) mathematics, any two empty sets are equal. The set of invisible hats and the set of straight bananas are the same set (eventhough this means that every invisible hat is a straight banana). In 'typed' mathematics, I would think that the sets are unequal because the first is a set of hats and the second is a set of bananas.

Is there such a thing as 'typed' mathematics?

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    $\begingroup$ +1 It's always bothered me that people were using my invisible hats for scale. $\endgroup$ – Axoren Feb 27 '15 at 6:45
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You're looking for something like Type Theory

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