GÖDEL'S INCOMPLETENESS THEOREM AND THE PHILOSOPHY OF OPEN SYSTEMS
Carlo Cellucci · 1992
, whose main feature is that they are always subject to unanticipated outcomesin their operation and can receive new information from outside at any time [cf. Hewitt 1991]. WhileGodel's incompleteness theorem has been widely used to refute the main contentions of Hilbert'sprogram, it does not seem to have been generally used to point out the inadequacy of a basic ingredientof that program - the concept of formal system as a closed system - and to stress the need to replace itby the concept of formal system as an open system.A partial exception seem to be provided by van Heijenoort who states:The notion of formal system, introduced by Frege in 1879, had become by then the acceptedstandard of precision in the foundations of mathematics. It seemed to embody the Aristotelian ideal of aperfect deduction from first principles. Godel's results, by showing that mathematics cannot becompletely and consistently formalized in one system, shattered this ideal. The bounds of mathematicscannot be those of one formal system. Since mathematics has often been regarded as the standard ofrational knowledge that other sciences should strive to attain, Godel's theorems seem to acquiresignificance for the whole body of human knowledge; they certainly establish that the old ideal of adeductive system cannot be maintained [van Heijenoort 1967: 356].Here van Heijenoort rightly points out the impact of Godel's result on the concept of formalsystem as a closed system but does not suggest any alternative to such a concept.In this paper, on the one hand, I want to stress the role of Godel's incompleteness theorem inshowing the inadequacy of the concept of formal system as a closed system, and, on the other hand, Iwant to point out the interest of the concept of formal system as an open system, which is essential bothfor current developments in artificial intelligence and for the emergence of a new paradigm of logic,alternative to mathematical logic: