The choice of impedance used to shunt a Josephson junction determines if the charge transferred through the circuit is quantized: a capacitive shunt renders the charge discrete, whereas an inductive shunt leads to continuous charge. This discrepancy
leads to a paradox in the limit of large inductances L. We show that while the energy spectra of the capacitively and inductively shunted junction are vastly different, their high-frequency responses become identical for large L. Inductive shunting thus opens the possibility to observe charging effects unimpeded by charge noise.
