With "total randomness" I mean that every number generated has exactly the same probability of appearing and there is no way to predict which number will come next in a set. With order I mean that a pattern appears in a sequence of totally random numbers, and that pattern is consistent with every set of totally random numbers. What would be the implications of finding that? Is that a paradox?
There isn't a final test of total randomness, some tests have been developed and possibly more will come. So it might be possible using a given process to be developed, that we find there is a pattern in total randomness. This is just a mental experiment but still possible. If we find such pattern we could create an algorithm that cancels the pattern, then we would have a natural predictable randomness and an artificial unpredictable randomness.
Example of possible pattern: http://www.slate.com/articles/health_and_science/do_the_math/2013/05/yitang_zhang_twin_primes_conjecture_a_huge_discovery_about_prime_numbers.html (thanks to @Chris Degnen who answered below)