|Complex: a complex is composed of components that can be distinguished from each other and are relatively autonomous. Complex behavior refers to systems that consist of several components. The relative independence of the components is manifested in their behavior. Relative autonomy of the components is determined by the description of the complex as a whole.|
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|Barrow I 78
Complexity/Decidability/Paradox/Chaitin/Barrow: Order: "Print a sequence whose complexity can be proved to be greater than the length of this program!".
The computer cannot respond to this. Each sequence that it generates must be of lesser complexity than the length of the sequence itself (and also of its program).
(> Neumann: a machine can only build another machine if it is one degree less complex than this one itself,> Coursebook 8, 139 ff)
In the above case, the computer cannot decide whether the number R is random or not. Thus the Goedel theorem is proved.
In the late 1980s, even simpler evidence was found for the Goedel theorem, with which it was transformed into statements about information and randomness.
Information content/Barrow: You can assign a certain amount of information to a system of axioms and rules by defining their information content as the size of the computer program that checks all the possible concluding chains.
If one attempts to extend the bounds of provability by new axioms, there are still larger numbers, or sequences of numbers, whose randomness remains unprovable.
Chaitin: he has proved with the Diophantic equation:
X + y² = q
If we look for solutions with positive integers for x and y, Chaitin asked,...
...whether such an equation is typically finite or has infinitely many integral solutions if we let q pass through all possible values q = 1,2,3,4 ....
At first sight it hardly deviates from the original question, whether the equation for
Q = 1,2,3 .. has an integer solution.
However, Chaitin's question is infinitely more difficult to answer. The answer is random in the sense that it requires more information than is given in the problem.
There is no way to a solution. Write for q 0 if the equation has only finitely many solutions, and 1, if there are infinitely many. (> Kronecker symbol). The result is a series of ones and zeros representing a real number. (> Putnam)
Their value cannot be calculated by any computer.
The individual spots are logically completely independent of each other.
omega = 0010010101001011010 ...
Then Chaitin transformed this number into a decimal number...
...omega = 0.0010010101001011010 ...
and thus had the degree of probability that a randomly chosen computer program would eventually stop after a finite number of steps. It is always not equal to 0 and 1.
Still another important consequence: if we choose any very large number for q, there is no way to decide whether the qth binary digit of the number omega is a zero or a one. Human thinking has no access to an answer to this question.
The inevitable undecidability of some statements follows from the low complexity of the computer program, which is based on arithmetic, however.
John D. Barrow
Warum die Welt mathematisch ist Frankfurt/M. 1996
John D. Barrow
Die Natur der Natur: Die philosophischen Ansätze der modernen Kosmologie Heidelberg 1993
John D. Barrow
Die Entdeckung des Unmöglichen. Forschung an den Grenzen des Wissens Heidelberg 2001