A lot of interesting discussion has sprung up! Its good to see
So I thought it was worth commenting on a couple of themes pop up:
1) What constitutes "ruling the world"?
People have asked if the Wizard has to be iron fisted or maniacal, can ruling by proxies work, etc etc. Ultimately, I would say the Wizard can "rule" in any way you see fit. My thought would be, a wizard that rules the world can effectively leverage any resource of that world at his whim, and can inflict his whim on the world in a large scale. For example, if the Wizard needs an army, he can have it mustered. If he needs a particular magic item or treasure the world possesses, people will give it to him. If he needs to sacrifice a young maiden to fulfill a dark ambition, people will (even if reluctantly) provide them one. If he declares a new edict in a country, it becomes law. Now whether the world knows they are providing said resource to the "Wizard" or whether they knew new edicts are coming from the "Wizard" is up to the poster, its fine if the wizard works through a proxy or people do it because of beliefs, threats, mental influence, whatever.
Now that doesn't mean the control is absolute. I could absolutely see periodic rebellions and underground insurgencies. But we are assuming a level of control where these are more "annoyances" than true threats to the wizard's power. If a rebellion has grown to truly challenge the wizards authority, than the Wizard can no longer be considered to rule the world.
2) How to win the argument.
So I think there are two key factors you have to prove in order to say a wizard can rule the world.
a) Step 1: Rule the World
Aka how does the Wizard exist such considerable influence on the world? Is it a hammer, meteor swarming any who oppose you and taking an iron fist. Is it slower, more influence, domination, and politics. Or going super slow, literally changing cultural paradigms. Or something inbetween.
b) Step 2: Beat the Resistance
At some point, its assumed that the other powerful people of the world would attempt to counter the Wizard, or the Wizard would look to systematically take out the powerful people. So the question becomes, how effective would the Wizard be at countering these threats?
c) General Beats Specific
So a lot of discussion has been on the Aberrant Mind Sorc vs the Wizard. I think its an interesting discussion. My main counter would be it assumes that a 10th level character (already established to be pretty rare) is a sorceror (even rarer) and then is a specific subclass (rarer still). So the likelihood that one would exist in a world is pretty low (though clearly pretty effective)
Lets contrast that to the 10th level cleric argument. While 10th level clerics are still pretty rare based on our assumptions, any subclass will work....so the chance that one of these exist in the world is a lot more likely.
And of course, strategies that work with groups of lower level spellcasters are even a better argument, as those are common enough that we could safely assume a world would have them.
So ultimately the best arguments to me are the ones that rely on a less and less specific circumstances.