Theorem proving requires an axiom scheme yielding an enumerable set of axioms of a formal logic system whose formulas and theorems are symbol strings over some finite alphabet that may include traditional symbols of logic, such as , .

A proof is a sequence of theorems.
Each theorem is either an axiom or inferred from previous
theorems by applying one of several given inference rules such
as *modus ponens* combined with *unification*
[9].
It is obvious and well-known that theorems can be uniquely
encoded as bitstrings, and that one can write a program
that systematically enumerates all possible
proofs--compare
Gödel's
original paper [10] and
any textbook on logic or proof theory, e.g.,
[9].
In what follows, we do not have to specify all details
of a particular axiomatic encoding--it suffices to
recall that concepts such as
*elementary hardware operations*,
*computational costs*,
*utility functions*,
*probability*,
*provability*,
etc. can be formalized.

First consider a brute force proof searcher
systematically generating all proofs
in order of their sizes. To produce a certain proof,
this approach takes time exponential in proof size.
Instead our will produce many proofs
with low algorithmic complexity
[47,20,25]
much more quickly. It runs and evaluates
*proof techniques*
written in language
implemented within .
For example, may be a variant of PROLOG [7]
or the universal FORTH[26]-inspired
programming language used in recent experiments
with OOPS [38,40].
Certain long proofs can be produced by short programs.

Section 2.3 will present a general,
*bias-optimal* [38,40]
way of systematically testing proof techniques through
a convenient initialization of .
First, however, we need to specify
the precise goals of the proof search, details
of the proofs that can be derived by proof techniques,
and ways of verifying proofs and translating their results
into online changes of the Gödel machine software. This is most conveniently done
by describing the essential instructions for generating/checking
axioms/theorems and for transferring control to provably good
Gödel machine-changing programs.

- Instructions/Subroutines for Making & Verifying Axioms & Theorems and for Initiating Online Self-Improvements
- Globally Optimal Self-Changes: The Self-Improvement Strategy is not Greedy!

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