Russian constructivism

The Russian school of constructive mathematics, associated principally with Andrey Markov Jr, was (is?) a variety of constructive mathematics focussing on recursion theory?.

- Classically true principles:
- Markov's principle for natural numbers: if an infinite sequence of binary digits is not all $0$, then it has a least one $1$;
- dependent choice.

- Classically false principles:
- every partial function from $\mathbb{N}$ to $\mathbb{N}$ is computable;
- every set is a subquotient of $\mathbb{N}$.

- Classically false results:
- every total function from $\mathbb{R}$ (the real line) to $\mathbb{R}$ is pointwise continuous (Ceitin's theorem?);
- there exist continuous functions from $[0,1]$ (the unit interval) to $\mathbb{R}$ that are pointwise continuous but not uniformly continuous;
- there exist bounded sets of real numbers with no supremum (given by Specker sequences).

Revised on July 29, 2011 16:50:13
by Toby Bartels
(64.89.48.241)