dure according to Theorem 1 these considerations also apply to humans who can thus extend their knowledge beyond any formal limits. Post's **[1944, p.** 295] conclusion from G¨odel's theorem is that "**mathematical thinking is ...**
essentially creative**" (see Shapiro 1998, pp. 291-292, "creative step").**

## 7 Conclusion

Turing points out that intelligence requires a "residue" called "initiative" that is not captured by his machines. We introduced the concept of **a creative machine by requiring that it can evaluate uncomputable functions in**
a certain sense. It is capable of extending informal concepts such **as (total)**
computable functions beyond any given formal description which is restricted to incomplete enumerable parts of these concepts. We argue that **"intelligent" machines should be capable of processing such concepts. Hypotheses** on creative machines say that Turing's uncomputable "residue" is the execution of a self-developing procedure that starts from any programming language and any input. This process, which produces formally irreducible experience, can be regarded as a new use of computers. Acknowledgments**. The author wishes to thank colleagues for valuable** comments.

## References

[Ammon, 1988] **Kurt Ammon. The automatic acquisition of proof methods.**
In National Conference on Artificial Intelligence, St. Paul**, San Mateo,**
Calif., 1988. Morgan Kaufmann.

[G¨odel, 1965] **K. G¨odel. On undecidable propositions of formal mathematical**
systems - POSTSCRIPTUM. In M. Davis, editor, The Undecidable**. Raven**
Press, New York, 1965.

[G¨odel, 1990a] K. G¨odel. G¨odel 1972: On an extension of finitary mathematics which has not yet been used. In S. Feferman et al., editors, **Collected**
Works: Publications 1938-1974**, volume 2. Oxford University Press, New**
York, 1990.