Let's Take The Gun Issue To Its Extremes

I think we can all agree that we have a societal problem with gun violence here in America.  Compared to all the other developed, industrialized, advanced nations in the world, the level of gun violence in the U.S. is just ridiculously out of whack, and everyone is in favor of somehow reducing the bloodshed..

The question is how to do it.

(Let's eliminate law enforcement and the military from the discussion, and let's also agree to set aside Second Amendment interpretations.)

One school of thought says the only way to decrease gun violence is with more guns for more people; that as gun possession becomes easier, the number of armed citizens increases, and so does our relative safety.

The other side says the only way to cut gun violence is to cut the number of guns via legislation, registration, backgound checks, buy-backs, etc.; that as guns and ammo gradually become harder to obtain, rather than easier, fewer people will have access to them, and less guns means less gun violence.

Now let's take these two diametrically opposed viewpoints and carry each to its "logical" endpoint.  Strictly hypothetical, you understand.

On one extreme, you'd have a world where everybody has a gun (or guns) on their person all the time wherever they may be.

On the other extreme, you'd have a place where no one (except cops and military) ever has a gun, anywhere.

Which hypothetical world would be safer?