27 Nov 22:56 2013

## If all functions (N->N)->N are continuous, then 0=1.

If all functions (N->N)->N are continuous, then 0=1. Think of functions a : N -> N as sequences of natural numbers: a 0, a 1, a 2, a 3, ..., a n, ... The "continuity axiom" for functions f : (N -> N) -> N, that map sequences a : N -> N to numbers f(a), going back to Brouwer in his intuitionistic mathematics in the early 20th century, says that the value f(a) can depend only on a finite prefix of the infinite argument a. This makes sense computationally: there are no crystal balls in computational processes, able to see and grasp the infinite input at once. The input of f is computed bit-by-bit, in fact, often only when f queries it. After finitely many queries to its argument a : N -> N, the function f is supposed to deliver its answer, if it's ever going to produce an answer. When this finiteness condition holds, we say that f is continuous (never mind the reason for the terminology "continuous" in this message). The link(Continue reading)