Is encryption possible without secure random numbers? All cryptographic algorithms I know of that satisfy Kerckhoffs's principle rely upon secure random number generation. Assuming the existence of a 'numerical oracle' capable of perfectly predicting the outcome of any random number generator, is a Kerckhoff-compliant algorithm possible?
All secure cryptography requires a secrete key, it can be a symmetric shared secret or an asymmetric key pair with secret private key. 
The question is how do you get a secret if there are no secure random number sources. 
Note there are natural sources of random numbers which don't rely on cryptography for their security, but the question said the oracle can predict any RNG including predicting radioactive decay and cosmic rays, and the number I'm thinking of. That is an impressive oracle you have there. 
If there is no initial secret, and anything we do to try to create an initial secret is detectable by the oracle then anything the legitimate user can do so can the attacker. 
A secret is how we differentiate between legitimate users and attacker, the attacker is assumed to not know the secret. 
With all secrets given up by the Oracle there is no difference between legitimate and attacker and no secure cryptography.