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What notions of universe does predicative type theory admit?

Palmgren (1997), On universes in type theory, discusses work of several theorists that provide what we might call a family of Large Universe Axioms (LUAs) for predicative type theory, culminating in R …
Charles Stewart's user avatar
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Can dependent sums be encoded as dependent products?

This is certainly not what the question was after, but since the answers seem to be something along the lines of yes, sometimes, but not unless... Multi-sorted first-order logic (i.e., classical logi …
Charles Stewart's user avatar