We discuss two general types of limitations on rationality:

i) (Computability) I can't use a number (price, quantity, etc.) if I can't compute it --- if I don't have an algorithm for computing its digits. And I can't use a preference that I can't compute.

ii) (Definability) I can't use a number if I can't define it --- if I don't have a name for it, or a way of describing it. And I can't use a preference that I can't define.

We ask whether the classical theorems of economics hold in computability-bounded worlds, or in definability-bounded worlds. We provide two general frameworks for answering these questions, and we show that many theorems survive in both contexts: utility representations, existence of consumer demand functions, the fundamental welfare theorems, and characterizations of excess demand functions. These positive results hold despite the fact that the commodity and price spaces of these worlds may not satisfy classical completeness properties.

Existence of competitive equilibrium survives in definability-bounded worlds, but not in computability-bounded worlds.

Beyond bounded rationality, the computability results can be interpreted as possibility and impossibility in computational economics.