This paper constructively proves the existence of an effective procedure generating a computable (total) function that is not contained in any given effectively enumerable set of such functions. The proof implies the existence of machines that process informal concepts such as computable (total) functions beyond the limits of any given Turing machine or formal system, that is, these machines can, in a certain sense, "compute" function values beyond these limits. We call these machines creative. We argue that any "intelligent" machine should be capable of processing informal concepts such as computable (total) functions, that is, it should be creative. Finally, we introduce hypotheses on creative machines which were developed on the basis of theoretical investigations and experiments with computer programs. The hypotheses say that machine intelligence is the execution of a self-developing procedure starting from any universal programming language and any input.
Deep Dive into Informal Concepts in Machines.
This paper constructively proves the existence of an effective procedure generating a computable (total) function that is not contained in any given effectively enumerable set of such functions. The proof implies the existence of machines that process informal concepts such as computable (total) functions beyond the limits of any given Turing machine or formal system, that is, these machines can, in a certain sense, “compute” function values beyond these limits. We call these machines creative. We argue that any “intelligent” machine should be capable of processing informal concepts such as computable (total) functions, that is, it should be creative. Finally, we introduce hypotheses on creative machines which were developed on the basis of theoretical investigations and experiments with computer programs. The hypotheses say that machine intelligence is the execution of a self-developing procedure starting from any universal programming language and any input.
Hilbert's program aimed to reduce mathematics to a formal system in order to avoid inconsistencies in mathematics. In particular, Hilbert's Entscheidungsproblem (decision problem) aimed to find "a procedure that allows one to decide on the validity, respectively satisfiability, of a given logical expression by a finite number of operations". 1 In order to prove that Hilbert's Entscheidungsproblem is unsolvable, Turing [ 1936 ] introduced his "computing machines" which are a formalization of a procedure in Hilbert's sense. Gödel [ 1965, p. 72 ] writes:
Turing’s work gives an analysis of the concept of “mechanical procedure” (alias “algorithm” …) … This concept is shown to be equivalent with that of a “Turing machine”. A formal system can simply be defined to be any mechanical procedure for producing formulas, called provable formulas. Hopcroft and Ullman [ 1979, p. 147 ] write that “the Turing machine is equivalent in computing power to the digital computer as we know it today”. They implicitly assume that the computer is used for executing a given procedure or program that was developed manually. Turing [ 1986 ] asks whether a computer can be used in another way:
It has been said that computing machines can only carry out the processes that they are instructed to do. … Up till the present machines have only be been used in this way. But is it necessary that they should always be used in such a manner? Turing [ 1969 ] discusses the development of intelligence in man and in machines:
If the untrained infant’s mind is to become an intelligent one, it must acquire both discipline and initiative. So far we have been considering only discipline. To convert a brain or machine into a universal machine is the extremest form of discipline. But discipline is certainly not enough in itself to produce intelligence. That which is required in addition we call initiative. … Our task is to discover the nature of this residue as it occurs in man, and to try and copy it in machines.
We investigate “the nature of this residue” called “initiative” (see Sieg, 1994, Section 5, Final remarks). Section 2 proves the existence of an effective procedure generating a computable (total) function that is not contained in any given effective enumeration of such functions. This procedure can be regarded as a bridge to the informal concept of computable (total) functions, that is, to Turing’s uncomputable residue. On the basis of this procedure Section 3 defines creative machines which can, in a certain sense, “compute” function values beyond the limits of any given Turing machine. Section 4 introduces hypotheses on creative machines which say that Turing’s uncomputable residue is the execution of a self-developing procedure starting from any universal programming language and any input. This process, which produces formally irreducible experience, can be regarded as a new use of computers. The remaining sections discuss our proof, creative machines and related work.
In this and the following sections we simply write computable function for an effectively computable total function of natural numbers which is defined for all natural numbers.
Theorem 1 There is an effective procedure generating a computable function that is not contained in any given effective enumeration of such functions.
Proof. Let f 1 , f 2 , … be an effective enumeration of computable functions. We define a new function g by
for all natural numbers n. Obviously, g(n) is defined for all natural numbers n because f i (n) is defined for all natural numbers i and all natural numbers n. Furthermore, g is computable because f 1 , f 2 , … is an effective enumeration of computable functions according to our original assumption. The expression f n (n) + 1 in the definition (1) of g can be regarded as a functional pseudocode, that is, as a computer program, say R, that computes the function g for all natural numbers n. There is an effective procedure that generates the program R. This procedure can be represented as a computer program whose input is the effective enumeration f 1 , f 2 , …, that is, a program, say E, generating the functions f 1 , f 2 , …, and whose output is the program R.
In order to generate g(n) from any natural number n, the program R thus generates the function f n by applying E to n and then adds 1 to the result of applying f n to n. Because of definition (1), g(n) is different from f n (n) for all natural numbers n. This implies that the computable function g is different from all functions f n , where n is any natural number. Therefore, there is an effective procedure generating a computable function g that is not contained in any given effective enumeration of such functions.
Theorem 2 There is an effective procedure generating a computable function that is not contained in any given formal system with a predicate for such functions.
Proof. Let S be a formal system with a predicate Q for computable functions. Because a formal sy
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