Question: Instead of discovering that earth was round, we find that we live on a infinite, flat "earth".

What are some cultural implications if we were to live in a world like that (disregarding any physical impossibilities, and that gravity, biological/ecological systems remain roughly the same as reality).

One interesting idea is that cities could be tens of thousands of kilometers from another and that one can only explore so much of the world in their lifetime.

EDIT: I don't mean flat as in geometrically flat, but flat in general. There would still be mountain ranges and oceans and forests and stuff. But there would be different and exotic ecological systems as you travel.

Answer: 
A lot of people ignored the question and delved into the rabbit hole of physical impossibilities given the physics of our known universe that were ruled out in the question. Given an infinite, planar type world with similar characteristics and physics to earth (though I am not sure how different temperate zones would come about) I agree that with others that life would develop separately in different locations. With infinity there is the possibility that completely separate forms of life (e.g., non-DNA based) would come into existence. I think the question is also interesting starting with a planar non-infinite landscape.

But in general, I think life forms would spread gradually over the landscape as they do on earth (though probably from different pockets of life on the infinite landscape). I do not see why, as one suggested, that we would continually (as a whole) be nomadic just because the world was infinitely flat. The answerer's premise was that we would run out of resources in one area and because of the infiniteness of the world, land and resources would not be valued and we would pull up stakes and move on. But just as happens here on earth (according to the premise), the land would replenish itself, and I think people would spread out naturally as population/resource pressures arose. The whole population would not pick up and move as nomads. And unless there was a scientific way to determine it, I do not think the population would realize the world was infinite so population migration would probably proceed much like it did no earth. They probably just wouldn't do the type of exploration we did to find shorter routes to places based on the curvature of the world.
Not good (or at least, not much better than our history).

The issue is not only as much "real estate" you have, but how much of it is usable. For example, most of the Earth surface has no human population (the oceans, Antartica) or just a bare minimum (deserts, Amazonian).

In that sense, if your area becomes overcrowded, maybe the nearest free soil is too far away (or has to pass through too many of other tribes'territory) to be of any actual interest.

So you would begin with isolated human groups, which would grown and expand. At the beginning, most of them would be able to send excess population out of the frontier, but with time the "frontier" tribes will find that the free soil is too far away (think how once, the USA frontier run through the Appalachian Mountains).

As technology progresses, the "frontier tribes" would become bigger and bigger (because with better technology is easier to keep the political unity). There would be cycles of frontier nations becoming too big, desintegrating in several nations, and the new "inner" nations using their power to push back the weaker nations of the privous cycle. Probably, sometimes population pressure in the older (inner) nations would lead to massive migrations/waves of conquests, like those of the Germanic peoples or the Mongols.