The concepts of creativity and incompleteness are tightly connected. In a nutshell, a system is creative if every description of it is incomplete.
Creativity can be regarded as the ability of a system to produce something new. A “system” in this sense might be an organism, a human being, a culture, an organization, a physical system like the solar system, a machine or any other dynamic structure in which some processes are happening, even the whole universe. Being creative in this sense can be viewed as the ability of a system to always generate or incorporate new information.
Newness, however, is not an absolute property. Something is new only relative to some description. But we would not call a system creative just because it did something that a certain given description of it could not predict. We would instead say that that particular description was incomplete. And since we can always produce incomplete descriptions of any system, doing so does not make those systems creative.
To be really creative, the system must instead have the ability to move out of the scope of every description we can give of it. In other words: every description of the system would have to be incomplete. A complete description of such a system would not exist. So no matter what we know or think we know about the system, it would be able to surprise us. And if we learn something new about the system, the resulting bunch of knowledge will again be incomplete. All knowledge we can have about such a system would contained gaps or mistakes or would be partially vague or would contain expressions that we don’t know how to calculate. We could never come up with a description of the system that is complete and exact at the same time.
An exact description of a system can be thought of as a formal theory from which statements about properties of the system or processes within it can be derived. When the description is incomplete, there are true statements about the system that cannot be derived within the formal theory. So for a creative system, there is no complete description in terms of a formal theory.
Non-creative systems, on the other hand, can be described completely in terms of exact descriptions from which all of their properties can be derived. Such descriptions can be viewed as formal theories describing the laws according to which the system is working.
Creative systems, then, don’t have fixed, laws describing how they function. The way they work can change over time, i.e. they develop and have a history.
I think that biological systems like ecosystems and species are creative in this sense. The creative process here is the process of evolution. Human beings are creative as well and our cognitive processes are creative. Systems consisting of human beings, like societies, cultures, organizations and institutions are creative as well. The historical processes taking place in these systems are creative processes in the sense described here. I also think that artificial (technical) creative systems are possible. However, showing that these types of systems are indeed creative in the sense defined here is a topic beyond the scope of this article.
(The picture, a painting by artist Yvan Richeux-Rey, is from http://commons.wikimedia.org/wiki/File:Bandon%C3%A9on_-_Yvan_Richeux-Rey.jpg.
I like this painting a lot. Itself a product of creativity, I take this picture here as a symbol of the fractured nature of our knowledge that is the result of being creative but also our means of dealing with our creative surrounding. The world around us is a creative system in the sense described here. To cope with it, we are continually producing and extending descriptions of it, but these are always incomplete. As a result of this process of producing new knowledge, every description of ourselves is itself incomplete, so we ourselves are creative in the sense defined here. Art is one result of this our creative nature.)