it cannot be represented by a Boolean circuit which accepts at least some input if we use the definition which appears in the paper.

Definition: A representation of a set S with n positive integers is a Boolean circuit C, such that C accepts the binary representation of a bit integer j if and only if j is in S.

The above circuit can accept the binary representation of a bit integer. If there is no number in the register then the value stored in memory by the adder will be all zeros. You can then use multiple of same and nand gates to do exactly what you say can't be done.