Wikipedia page about randomness says that "complete disorder" and "true randomness" are impossible according to Ramsey Theory and Cristian S. Calude. I don't understand it.
https://en.wikipedia.org/wiki/Randomness
So if I randomly select numbers from an infinite set of numbers e.g. from a set of complex numbers, will there always be some formula or pattern or a way to predict successive numbers from the previous ones?
There are no nondeterministic theories, patterns, methods or elements in mathematics? What with quantum mechanics in Copenhagen way of understanding it? What with probability theory and stochastic process?
Thanks!
You're essentially right. Ramsey theory says that, in any sufficiently big system, there will be some small (often very small compared to the whole system) parts that look orderly. That doesn't contradict randomness or disorder in the system as a whole.
For example, if I flip a coin 100 times to produce a sequence of "heads" and "tails", you can pick out 50 of those terms that are all heads or all tails. Those 50 terms look very orderly, but that doesn't contradict the randomness of the whole sequence of 100 terms. It doesn't mean that the coin was biased or that I cheated.
In nontrivial theorems of Ramsey theory (as opposed to the pigeonhole principle used in the preceding paragraph), the orderly part of the system is generally much smaller, not half of the whole system (50 out of 100 coin flips) but maybe the logarithm (or log log, or even smaller) of the system size.