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: 
While abiogenesis might have a singular origin (with descendants of that singular event outracing any "new" events) as some of the answers have described, various other "steps" [multicellular life, intelligent life, industrial civilization] may not be so rare, and you could have multiple societies of different species and even entirely different anatomical basis within the same "life island" with the same basic chemistry and backdrop of oxygen-producing single-celled life. You wouldn't have to go so far as the "edge of life" to find "aliens". The edge of hominids, or of land vertebrates, would suffice. The further out, the more alien.

The first "aliens" would probably be human, or very nearly so (almost certainly a species descended from the same *Homo/Australopithecus* line.), which had independently developed to the point of inventing rapid transportation and radio.

On the subject of rapid transportation, in the book "Missile Gap" (in which Earth's continents are transplanted to the surface of a very large but finite flat megastructure), a large [Ground-effect vehicle](http://en.wikipedia.org/wiki/Ground_effect_vehicle) was used for exploration. (The *name* of the book, incidentally, comes from the fact that intercontinental ballistic missiles are not a viable technology in the uniform gravity field caused by such a 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.