Now, if every n-binary sequence can, in principle, be obtained by this procedure, then all n-binary sequences are random, including such sequences as 01010101010...01 which obey a precise rule.

But on the other hand, if certain sequences can never be so obtained, then it's not true that the probability of obtaining any one of the 2^n sequences is 1/2^n--which contradicts a basic assumption of theoretical probability.

In either case, we seem to reach a conclusion that must be rejected. Can you explain a "way out" of this paradox?

Yours sincerely,

--Arturo

1. A sequence is random if it has been created by a random process, or might have been.

2. A sequence is random if it looks disorderly, i.e. random.

Any sequence is random in the first sense whether it looks random (in the second sense) or not. Even of a sequence appears to follow a certain rule, it does not mean that it might not be the result of a stochastic process.

The apparent paradox brings to mind Solomon Golomb's syllogism:

All governments are unjust.

To prove the assertion for all governments, it is sufficient to prove it for an arbitrary government. If a government is arbitrary, it is obviously unjust. And since this is true for an arbitrary government, it is true for all governments.