> That doesn't make any sense.

Oh yes it does, it's the basis for almost all electronic circuit design.

> More precisely, if we assume that V and I are not necessarily in phase, 
> this immediately allows me to derive a contradiction.
>
> Apparently, all I have to do is generate a sufficiently low-frequency 
> wave, with V and I sufficiently far out of phase, and we arrive at an 
> impossible situation. I could have a system with an arbitrarily large 
> current passing through it, for an arbitrarily long time, despite the 
> entire system having zero potential difference.

Note that the complex version of Ohm's Law only holds for steady-state 
continuous sinusoidal operation, it won't explain what happens when you 
first turn on a circuit or suddenly introduce voltage or current 
spikes/steps.

In your example, if you have previously charged up a capacitor an arbitrary 
amount, then yes, you can then get an arbitrary amount of current out with 
no voltage.  Just charge up a capacitor with a constant voltage, then short 
the terminals ;-)  But the voltage must be in a sine wave cycle though for 
Ohm's law to hold.

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