Theorem: For any set P and 2-place function s if there is a number N
such that for all x the value of P(x) equals P(s(N,x)) then for any
r.e. set Q there is an M such that for all x, Q(x) equals P(s(M,x)).

Define a 2-place relation R as being a Base of Computing if for every
r.e. set P there is an N such that P(x) = R(N,x) for all x.  For
example, let R(x,y) be “Turing Machine x halts on input y.”  Then R is
a Base of Computing.  For each r.e. set P there is a Turing Machine N
that halts on just the elements of that set.

When is a given R a Base of Computing, without reference to Turing
Machines?  The first paragraph above is the answer.  R is of the form P
(s(a,b)) for some set P and function s.  R CONTAINS ITSELF by virtue
of a number N that defines P within P: P(s(N,x)) = P(x) for all x.

Godel proved that the relation “Wff x with input y is provable.”
contains itself.  Turing proved “Turing Machine x halts on input y.”
contains itself.

The theorem above generalizes these two observations.

This allows us to construct minimal Bases of Computing by devising  P
and s that meet the premise of the theorem.