International Encyclopedia of Systems and Cybernetics

Formal logic version: "For every consistent formalization of arithmetic, there exist arithmetic truths unprovable within that formal system".

Complexity version: "There exist numbers having complexity greater than any theory of mathematics can prove".

Both version are given by J. CASTI (1994, p.139/146).

At the beginning, the theorem seemed to be merely a formal logical one about arithmetic truth, in correspondance with the formal logic version.

Various different versions have appeared through time, and slowly, it dawned that "… we'll never get at all the truth by following rules; there is always something out there in the real world that resists being fenced in by a deductive argument" (p.150).

G. CHAITIN has been one who most insistently inquired into the matter.

"If a theorem contains more information than a given set of axioms, then it is impossible for the theorem to be derived from those axioms" (Cited by Joseph FORD, 1989, p.609)

"…in a random sequence each sequential digit carries positive information since it cannot be predicted from knowledge of its predecessors. In consequence an infinite random sequence contains more information than all our finite human logical systems combined, hence, verifying its randomness lies beyond constructive proof" (Ibid).

"…in all truth, GÖDEL permeates the physical world even more than he does the mathematical. Indeed we now have no choice; we must come to grips with GÖDEL, with the paradox of solving the unsolvable, predicting the unpredictable" (Ibid).

(as stated by F. HEYLIGHEN)

"…in each formal system (containing the axioms of number theory) there exist propositions such that it is impossible to distinguish between the truth (probability) and falsity of the proposition within the system itself" (1990a, p441 ).

Although this seems a purely negative statement, it "can also be interpreted in a positive way by moving to a higher level of representation, where (it) expresses a certain high-order closure.

"The theorem of GÖDEL is proven by means of a construction allowing the formal system to represent propositions of the system by numbers so that it is possible to interpret propositions about numbers to be propositions about propositions of the same system. This self-reference is an example of cyclical closure. The incompleteness result can be understood as an application of the more general principle of the impossibility of complete self-reference (LÖFGREN)" (p.441).

GÖDEL's theorem is related to A. CHURCH's work. As stated by R. SCHOENFELD: "CHURCH 's thesis (is) that decidability is equivalent to recursiveness, which is equivalent to computability" (1994, p.88).

The practical systemic meaning of GÖDEL's incompleteness theorem is thus resumed by J. CASTI: "A universal feature of knowledge is that one must get outside of a system in order to really understand it" (1994, p.17).