This is a tutorial for mathematically inclined functional programmers, based on previously published, peered reviewed theoretical work. We discuss a higher-type functional, written here in the functional programming language Haskell, which (1) optimally plays sequential games, (2) implements a computational version of the Tychonoff Theorem from topology, and (3) realizes the Double-Negation Shift from logic and proof theory. The functional makes sense for finite and infinite (lazy) lists, and in the binary case it amounts to an operation that is available in any (strong) monad. In fact, once we define this monad in Haskell, it turns out that this amazingly versatile functional is already available in Haskell, in the standard prelude, called sequence, which iterates this binary operation. Therefore Haskell proves that this functional is even more versatile than anticipated, as the function sequence was introduced for other purposes by the language designers, in particular the iteration of a list of monadic effects (but effects are not what we discuss here). D.1.1 [Programming tech-

Assuming the propositional axiom of extensionality, we show that a Martin-Löf universe à la Russell is indiscrete in its intrinsic topology. This doesn’t invoke Brouwerian continuity principles. As a corollary, we derive Rice’s Theorem for the universe: the existence of a non-trivial, decidable, extensional property of the universe implies the weak limited principle of omniscience. This is a theorem in type theory. Without assuming extensionality, we deduce the following metatheorem: in intensional Martin-Löf type theory with a universe, there is no closed term defining a non-trivial, decidable, extensional property of the universe. 1