Not all possible functions from states to {0,1} (or to some larger discrete set) are implementable as some possible state, for cardinality reasons
All cardinalities here are finite. The set of generically realizable states is a finite set because they each have a finite and bounded information content description (a list of instructions to realize that state, which is not greater in bits than the number of neurons in all the human brains on Earth).
Yes, I knew the cardinalities in question were finite. The point applies regardless though. For any set X, there is no injection from 2^X to X. In the finite case, this is 2^n > n for all natural numbers n.
If there are N possible states, then the number of functions from possible states to {0,1} is 2^N , which is more than N, so there is some function from the set of possible states to {0,1} which is not implemented by any state.
All cardinalities here are finite. The set of generically realizable states is a finite set because they each have a finite and bounded information content description (a list of instructions to realize that state, which is not greater in bits than the number of neurons in all the human brains on Earth).
Yes, I knew the cardinalities in question were finite. The point applies regardless though. For any set X, there is no injection from 2^X to X. In the finite case, this is 2^n > n for all natural numbers n.
If there are N possible states, then the number of functions from possible states to {0,1} is 2^N , which is more than N, so there is some function from the set of possible states to {0,1} which is not implemented by any state.
I never said it had to be implemented by a state. That is not the claim: the claim is merely that such a function exists.