Introduction

The Medvedev lattice was introduced in [5] as an attempt to make precise the idea, due to Kolmogorov, of identifying true propositional formulas with identically “solvable” problems. A mass problem is any set of functions (throughout this paper “function” means total function from ω to ω; the small Latin letters f, g, h,… will be used as variables for functions). Mass problems correspond to informal problems in the following sense: given any “informal problem”, a mass problem corresponding to it is a set of functions which “solve” the problem, and at least one such function can be “obtained” by any “solution” to the problem (see [10]).

Example 1.1 If A, B ⊆ ω are sets, and φ is a partial function, then the following are mass problems:

1. {CA} (where CA is the characteristic function of A): this is called the problem of solvability of A; this mass problem will be denoted by the symbol SA;

2. {f : range(f) = A}: the problem of enumerability of A; this mass problem will be denoted by the symbol εA;

3. (Other examples) The problem of separability of A and B, i.e. {f : f−1(0) = A & f−1(1) = B}; of course, this mass problem is empty if A∩B ≠ Ø: it is absolutely impossible to “solve” the problem in this case. The problem of many-one reducibility of A to B: {f : f−l(B) = A}. The problem of extendibility of φ: {f : f ⊇ φ}.