nLab
Russian constructivism

Introduction

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

Principles and results

• 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 $ℝ$ (the real line) to $ℝ$ is pointwise continuous (Ceitin's theorem?);
  • there exist continuous functions from $[0,1]$ (the unit interval) to $ℝ$ that are pointwise continuous but not uniformly continuous;
  • there exist bounded sets of real numbers with no supremum (given by Specker sequences).