Consistency, Turing Computability and Gödel’s First Incompleteness Theorem

Abstract

It is well understood and appreciated that Gödel’s Incompleteness Theorems apply to sufficiently strong, formal deductive systems. In particular, the theorems apply to systems which are adequate for conventional number theory. Less well known is that there exist algorithms which can be applied to such a system to generate a gödel-sentence for that system. Although the generation of a sentence is not equivalent to proving its truth, the present paper argues that the existence of these algorithms, when conjoined with Gödel’s results and accepted theorems of recursion theory, does provide the basis for an apparent paradox. The difficulty arises when such an algorithm is embedded within a computer program of sufficient arithmetic power. The required computer program (an AI system) is described herein, and the paradox is derived. A solution to the paradox is proposed, which, it is argued, illuminates the truth status of axioms in formal models of programs and Turing machines.

Appendix A

For those not accustomed to dealing with axiomatic models of Turing machines, the following remarks should be helpful. It is widely recognized within the field of computer science that programs presented in modern programming languages can always be written so as to accommodate a previously specified input format. Moreover, it is common knowledge among researchers in the logic programming community that, corresponding to any working computer program, there is an equivalent program written in Prolog (which is the most widely used, logic programming language). Prolog, like all other modern high-level programming languages has the computational power of a universal Turing machine.

Now, any Prolog program consists of a series of axioms in the form of horn-clauses, most of which are syntactic variants of axioms written in a first-order predicate calculus. When input data is to be supplied to a Prolog program, it must first be converted, either by humans or by supplementary program code, into axioms that have the form of atomic sentences. In order for the Prolog program to be successful, it must be designed in a fashion that is adapted to some previously determined, axiomatic representation of the input data. Happily, this is always possible, because if need be, it is always possible to augment the main program’s axioms with special axioms that perform a bridging function, so that the entire axiom set, including the “input data axioms”, is deductively harmonized.

Now, admittedly, the axioms of Prolog programs do not always perfectly correspond to sentences written in classical first-order logic. However, there are programming environments, based upon first-order theorem provers, which do permit the required horn-clause axioms to be written entirely within classical first-order logic. Examples of such programming environments are Otter and PTTP (Prolog Technology Theorem Prover), both of which include powerful first-order theorem provers. ⁶ Otter, in particular, is a very powerful, inferentially complete theorem prover that permits the full range of predicate calculus syntax to be employed in the programs it executes. A remarkable characteristic of the programs which Otter and PTTP accept is that the axiom set that comprises each such program can serve as the program’s own equivalent formal deductive model.

It is also noteworthy that, for any Turing machine, there exists a corresponding set of first-order axioms that is functionally equivalent to that machine (see Davis, Chapters 1 and 6, 1958, for a discussion of Post’s method, whereby any Turing machine can be presented axiomatically). Moreover, the axiom sets that constitute such programs can be designed to deductively harmonize with their “input data axioms” in a fashion that is strongly analogous to the fashion in which an actual Turing machine is designed to accommodate a predetermined presentation of input data upon its machine tape.