Is it possible to eliminate induced drag completely?
The infinite span has nothing at all to do with having no ends.

The ratio of lift to induced drag increases with span. If you fix the lift, and vary span, the induced drag will tend to zero as the span tends to infinity. That means in limit a wing of infinite span generating finite lift (and therefore flying at zero coefficient of lift!) will have zero induced drag.

This is a highly abstract approach to things and I don't think it provides any useful insight.

Anyway, forget about the vortices. They only happen behind the wing anyway and therefore are obviously irrelevant! Instead, call upon the good old laws of motion. The lift is upward force of the air on the wing. By principle of action and reaction (Newton's 3. law) the wing must apply force of equal magnitude and opposite direction, that is downward, on the air. Now if the ambient pressure above and below the wing is the same, this force will accelerate the air downward, which will increase its kinetic energy. And by law of conservation of energy, this energy has to come from somewhere and the only way work can be done here is by drag.

This suggests the way to reduce it. If the pressure can be increased below the wing without accelerating the air, there won't be induced drag. Well, that's exactly how ground effect reduces drag. Normally it does not eliminate it, because the air can still move outward, but a wing flying in a trench could perhaps get quite close (though not to zero, because the air still needs to move at least a bit to maintain the pressure difference).