computable functions according to our original assumption, there **is a natural**
number n such that g(x) = fn(x) for all natural numbers x**. This immediately** yields the contradiction g(n) = fn(n) and g(n) = fn(n**) + 1 because of the** definition of the function g**. This means that our original assumption is false,**
that is, there is no Turing machine producing all computable functions f1, f2**, ... on the set of natural numbers.**
The construction of the computable function g in the proof can be represented by a Turing machine T. Of course, T **can be incorporated into any**
Turing machine. The proof implies that no Turing machine can produce all computable functions f1, f2**, ... on the set of natural numbers no matter** how T is incorporated. In particular, T **can be incorporated into the Turing**
machine whose existence is assumed in the proof so that the extended Turing machine produces the sequence f1, f2, ... and the function g**. But the**
extended machine is different from the original machine and thus its application would contradict our assumption that there is a (single) Turing machine producing all computable functions on the set of natural numbers.

Because any formal system can simply be defined as a theorem-proving Turing machine (see, for example, [4, p. 72]), the theorem also implies that any formal theory is incomplete in the sense that it cannot "capture" all computable functions on the set of natural numbers.

## 3 Creative Systems

Let C **be a system capable of performing the reasoning processes required for**
proving my simple incompleteness theorem. Thus, C is capable of constructing a computable function g **on the set of natural numbers not produced by**
any given Turing machine M**. Figure 1 illustrates the corresponding proof** which shows that there is no predefined Turing machine M **producing all**
computable functions f1, f2, ... on the set of natural numbers C **can construct.**
A difference between C and a Turing machine is that C **is not regarded as**
a static predefined object isolated from its environment. Rather, C **and the** Turing machine M **in the proof are regarded as separate interacting entities**
in the sense that C assumes the existence of a Turing machine M **producing** all computable functions f1, f2**, ... on the set of natural numbers in order to** construct a computable function g not produced by M**. Roughly speaking,**
C **can change itself by interacting with its environment, that is, the Turing**