Theorem: General Intelligence Entails Creativity, Assuming - Computational Creativity Research: Towards Creative Machines - page 56

We come now to the third axiom system: PA I . This is standard Peano Arithmetic,
which most readers will at least have heard something about. It includes the first six
of the seven axioms composing Q (see note 9), plus one additional axiom schema:
Induction Schema Every sentence that is the universal closure of an instance of this schema:

[ ˆ( 0 ) ∧∀ x (ˆ( x ) ₒ ˆ( s ( x )) ]ₒ∀ x ˆ( x )

PA I is what I shall take as the springboard from which to launch to a completion
of the definition of alien-fair gi. 10 We can jump from the circle containing all the
theorems provable from PA I (which is the same set as all the consequences of PA I )
to a larger class: namely, all the truths of standard first-order arithmetic. (Circles
beyond this one involve second-order logic (as indicated by the subscript II ), and
are left for future expansions of my case for the gi-implies-creativity theorem.) But
notice that here the arrow designed to reflect the “travel” of our agent is not labeled
with a check, but rather with a question-mark. This is so because making this jump
requires some impressive intelligence. What do you reach if you make this jump,
specifically? I give two examples in the diagram, each one marked with a . And what

are the examples? The first is , and is a label for the formula that Gödel pointed to

via his first incompleteness theorem. Each of these formulae is of course such that
neither it nor its negation can be proved from PA I —but each such formula is true on
I A

G

proposition I call out here is a particular number-theoretic fact:
Goodstein's Theorem [ 11 ], and is indicated by 'GT.' While GT and instances of

. The second

G
are all true on the standard interpretation of arithmetic, they are beyond the theorems
of PA I , a nice result first proved by Kirby and Paris [ 13 ]. 11

3.2.4 Key Definitions

I define a truly gi agent, whether human, alien, or machine, to be one that under-
stands 12 not only basic arithmetic (i.e., that understands PA ˄
I

and below, to include

Q ˄ and EA / BA ˄ ), but also at least one ∴ truth.

10 Alert readers will note that in jumping from this board I pass straight through ACA 0 without
comment, and they will have already have noticed the dotted circle I drew for this axiom system.
What gives? Ultimately, ACA 0 supports a class of first-order theorems that doesn't exceed those
provable from PA I ; hence the dotted circle rather than a solid one. As to what ACA 0 is, we shall
have to rest content with the highly informal piece of information that it's a restricted form of
second-order arithmetic. A philosophically rich presentation of ACA 0 is provided in [ 18 ].
11 GT is simply the fact that a particular sequence of natural numbers, the Goodstein sequence ,start-
ing with any natural number n , eventually terminates at zero. But many folks who first understand
the sequence are utterly convinced that it's both astonishingly fast-growing and never terminates,
and simply returns larger and larger numbers as the sequence progresses, forever. See [ 14 ]fora
nice version of the proof, which makes use of infinitary concepts and techniques, and turns these
intuitions upside down to yield a result that a truly general-intelligent agent can appreciate.
12 As alert readers will have noted, multiple times above I've made use of the concept of under-
standing , and here I sustain the practice. This concept does carry a lot of baggage, yes; but on