Ammon (2016) gives proofs for the existence of physical reasoning processes that cannot be described by a formal system, that is, they cannot be computed by a Turing machine. The proofs can be outlined as follows:
The generality of Gödel’s first incompleteness theorem implies its applicability to a formalization of its proof. Because this formalization is incomplete as well the formalization cannot be completely represented in the incomplete formalization. Therefore, there are physical processes that correspond to the application of the theorem to a formalization of its proof which cannot completely be represented in the formalization.
Gödel’s first incompleteness theorem states that any formal system, say S, satisfying some simple requirements contains an undecidable formula, that is, neither this formula nor its negation can be proved in S. If we add a formal symbol S for S to S we get an another formal system which is called the observing system of S. The theorems and proofs in the observing system correspond to theorems and proofs in S but contain the formal symbol S which refers to S. The observing system is more powerful than S because it contains all proofs in S and additionally a proof of a formula that corresponds to the undecidable formula in S. Therefore, there exist physical processes that correspond to the construction of observing systems although this construction cannot completely be represented by a formal system because this system is incomplete as well.
Physical systems that can apply Gödel’s theorem to any formal system satisfying some simple requirements and can construct an observing system from any such system and thus prove a formula in the observing system that corresponds to the undecidable formula are called creative systems.
Computer experiments yield the following principle for creative systems: A creative system is a self-developing process which starts from any universal programming language and any input. This process cannot be reduced to a formal system, that is, a Turing machine, but to the language and the input from which it starts.