putable partial function of natural numbers is a total function is uncomputable. The preceding paragraphs in this section imply that a creative system can determine an unlimited number of values of the function d **beyond**
its "knowledge", that is, it can repeatedly generate computable functions beyond the limits of its "knowledge", which is finitely describable, and the limits of a given Turing machine or formal system. The Church-Turing **thesis** states that every effectively calculable function is computable by a Turing machine (see Kleene 1952, pp. 317-323, 376-381). A creative machine can determine values of the function d beyond its finitely describable "knowledge" but it cannot calculate the value of the function d(x**) "for each value**
of x for which it is defined" (see Kleene 1987, p. 493), that is, it cannot determine for each computable partial function x whether x **is a total function.**
Therefore, a creative system can compute values of uncomputable functions beyond the limits of given Turing machines or formal systems, but this has no bearing on what number-theoretic functions are effectively calculable (see Kleene 1987, p. 493). Furthermore, the computable functions generated by a creative machine cannot be specified in advance **but must be generated**
by repeated applications of an effective procedure according to Theorem 1. This process involves the generation of a sequence of more and more powerful effective enumerations of computable functions, equivalent Turing **machines,**
or equivalent formal systems each of which forms the basis for the **generation of its successor. Roughly speaking, this process can be regarded as a**
self-developing procedure which must be generated step by step and cannot be reduced to a finite description given in advance (see Theorems 3 and 4 in Section 3). Kleene [1987, p. 493] **requires that an "effective calculation procedure" or "algorithm" is "fixed in advance for all calculations" and that it**
is "possible to convey a complete finite description of the effective procedure or algorithm by a finite communication, in advance of performing computations in accordance with it". For any computable (total) function **that a**
creative system can generate it can give a complete finite description of its generation but there is no complete finite description for the generation of all computable functions it can generate (see Theorems 3 and 4).

Hypothesis 1 in Section 4 states that the knowledge in a creative machine is developed in a formally irreducible empirical process that starts from a programming language and any input. As described above, the development of knowledge is formally irreducible because it must be generated step by step and cannot be reduced to a finite description given in advance. For example, this process can involve the generation of a sequence of more and more