Q:

Can a salt water solution conduct electricity forever?

We know that very pure water does not conduct electricity, but salt water is a decent conductor. This is commonly explained by saying that "the ions carry the current through the solution", an explanation that does not really make sense because it is not clear what will happen when all of the ions have migrated to the electrodes. 
Better explanations of conduction through a salty solution (like this one or this one) explain the conduction of electricity in terms of a reduction reaction that takes place at the anode and an oxidation reaction that takes place at the cathode. In the case of salt water, chlorine gas (Cl2) is formed at the anode and hydrogen gas (H2) is formed at the cathode. 
This explanation makes sense to me, but it implies that the conduction of electricity through a solution is fundamentally different than the conduction of electricity through a wire. A copper wire does not change at all after we pass electricity through it. In contrast, in salt water, we are driving a two chemical reactions (one at each electrode), which fundamentally change the composition material.
This implies that it is not possible for a aqueous solution to conduct electricity forever. Since we are driving a chemical reaction, we are either consuming our salt by forming gas (or plating it onto the electrodes) or we are consuming the water by forming H2 or O2 gas. 
This is surprising to me (for some reason). So, I'm asking if my thinking is correct: is it possible for a salt solution to conduct electricity forever, or it will it always eventually consume the reactants and stop as I have surmised?
EDIT: See this discussion of the Chemistry SE: https://chemistry.stackexchange.com/questions/7571/can-an-aqueous-solution-conduct-electricity-forever/7610#7610

A:

I believe the answer is no simply because the electric current breaks the $\mathrm{H_2 O}$ bonds and forms $\mathrm{H_2}$ and $\mathrm{O_2}$ gas.  Eventually you run out of water due to this process.
Also, in a $\mathrm{H_2 O}$ + $\mathrm{NaCl}$ solution, $\mathrm{Cl_2}$ gas is produced which would eventually affect ion concentrations.
Wikipedia has a pretty detailed description of the electrolysis of water.

