by a power law with exponent 1 + δ if ( ∃ m ) : ( ∀ i > states, b$_{1}$, ..., b n $_{+1}$, then, if ( ∀ i ≤ n ) p ( a i ) = 1 /n and ( ∀ i ≤ n + 1) p ( b i ) = 1 / ( n + 1) H ( A ) < H ( B ).Thus, if we are not dealing with this special case, we need to ex- plicitly demonstrate that it holds for our purposes. 10 [69] This derivation is equivalent to the one found in [36], Theorem 8.2. In this theorem, the authors demonstrate that every infinite distribution with infinite entropy is hy- perbolic , which implies that the distribution is not dom- inated by a power law with an exponent higher than 1. [68] The reader could object that this section is unnecessary, since the axiomatic derivation of the uncertainty func- tion (which we call entropy) assumes that the entropy increases with the size of the system. However, the ex- plicit statement of this axiom corresponds to the spe- cial case of uniform probabilities [58]. Specifically, the axiom states that, if we have two systems A, B such that A contains n states a$_{1}$, ..., a n and B contains n + 1 m ) ( p ( i +1) p ( i ) < ( i i +1 ) 1+ δ )