Russian constructivism in nLab

Russian constructivism

Russian constructivism

Summary

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 (not always accepted by other constructivists):
  • Markov's principle for natural numbers: if an infinite sequence of binary digits is not all 00, then it has a least one 11;
  • dependent choice for natural numbers.
• Classically false principles:
  • every partial function from ℕ\mathbb{N} to ℕ\mathbb{N} is computable;
  • every set is a subquotient of ℕ\mathbb{N}.
• Classically false results (false w.r.t. classical functions and sets):
  • every total function from ℝ\mathbb{R} (the real line) to ℝ\mathbb{R} is pointwise continuous (Ceitin's theorem?);
  • there exist continuous functions from [0,1][0,1] (the unit interval) to ℝ\mathbb{R} that are pointwise continuous but not uniformly continuous;
  • there exist bounded sets of real numbers with no supremum (given by Specker sequences).