Theorem: General Intelligence Entails Creativity, Assuming - Computational Creativity Research: Towards Creative Machines - page 53

But we must be a bit more systematic about what arithmetic is, and how much of
it must be mastered if an agent is to earn the right to be classified as gi. The next step
is to quickly review some standard formal machinery from mathematical logic and
computability. 8

3.2.2 Basic Machinery

Let A be some axiomatic theory of arithmetic based on some corresponding formal
language be the natural and received model-theoretic interpretation of

ordinary arithmetic with which you and I are intimately familiar. (Warning: I use A
as a variable for a given axiom system, instances to be visited below; but I use

L A .Let I A

A to

refer to arithmetic, period.) A is simply a set of formulae from L A : viz., a certain

set of axioms of arithmetic. Now let ʱ A ∈ L A be some arbitrary formula about A

arithmetic. (When the context is clear, we shall sometimes drop the subscript A and
refer to a given arithmetical formula as simply ʱ .) To say that ʱ is true on some

interpretation I , we write the customary:

I |= ʱ

φ A (here too we sometimes omit the subscript A ) is a set of formulae based

where

L A , ʱ

is a consequence of φ iff

on

For every I , if all of φ aretrueon I , then I |= ʱ.

A |= denotes the set of all formulae that are consequences of A . We assume a standard
finitary proof-theory

based in first-order logic (e.g., resolution-, natural deduction-,
or equational-based), and write the usual

˄

φ ˄ ʱ

(Footnote 7 continued)
I find this scenario unlikely, but that will hardly impress skeptics; and the fact of the matter is that
such a scenario is mathematically and, it would seem, even technologically, possible. However,
there is clearly an analogue to my theorem-and-proof based in axioms, theories, and theorems cast
in systems that yield both natural-number and real-number arithmetics: viz., axiomatic set theory. I
recommend that any readers interested in pursuing this route, or the narrow route of a real-number
version of what the case I give herein, consult [ 17 ], which works backwards from all of mathematics,
to axioms systems.
8 My notation and focus is devised for the purposes at hand, but in general nice coverage is provided
in the venerable [ 8 ], which I have long used in classroom teaching of intermediate mathematical
logic. But there is an especially good background provided in [ 18 ], which has the added benefit of a
learned discussion of potential ways of distinguishing between an understanding of basic arithmetic,
versus understanding more. Unlike [ 18 ], I have high-ish standards: I interpret basic arithmetic to
include truths of arithmetic beyond ordinary, mechanical proof in first-order logic.