Computation&Technology

What Turing disregards completely is the fact that the mind is not static, but
constantly developing. This is seen, e.g., from the infinite series of ever
stronger axioms of infinity in set theory, each of which expresses a new idea
or insight. A similar process takes place with regard to the primitive terms.
E.g., the iterative concept of set became clear only in the past few decades.
Several more primitive ideas now appear on the horizon, e.g., the selfreflexive
concept of proper class. Therefore, although at each stage of the mind's
development the number of its possible states is finite, there is no reason why
this number should not converge to infinity in the course of its development.
Now there may exist systematic methods of accelerating, specializing, and
uniquely determining this development, e.g., by asking the right questions on
the basis of a mechanical procedure. But it must be admitted that the precise
definition of a procedure of this kind would require a substantial deepening of
our understanding of the basic operations of the mind. Vaguely defined
procedures of this kind, however, are known, e.g., the process of defining
wellorderings of integers representing larger and larger ordinals or the
process of forming stronger and stronger axioms of infinity in set theory. 
- Kurt Godel