There's a problem you can't circumvene though. Population pressure will eventually break down human society. Humans(like any species) can't survive with a constant high population density
...already addressed multiple times: we're building in three dimensions.
Even then, there's only a limited amount of people you can support before society collapses, if only because nobody has anything useful to do.
The most important problem will be the weather though. Cities tend to absorb heat
That's not a problem. That's beneficial, because the waste heat would be reclaimed.
This is, incidentally, a significant part of the solution for the energy consumption issue. One doesn't actually need to collect all those exojoules every year, any more than one needs to "acquire new dirt" to plant new seeds each season.
Waste heat is waste heat. You can't recuperate the energy. You can use it to warm things up, or exploit temperature differences to produce power, but the Chaos theory will remain in effect. Can't infinitively reuse the same energy.
Also heat tends to form more of a problem when the average temperature lies between 40 and 60 degrees, or higher if you build superhigh megacities.
If you are talking about a population of trillions of people then you are not talking about "anytime soon". Many centuries if not a few millennia would be the better estimate.
...I admit that does skew the naure of the debate somewhat. If 6.3 trillion people materialized overnight, that would be very different than if it were gradual growth over a few hundred years. And looking at even the fastest predicted growth rates, we would have those centuries.
Honestly I think a few decades would be enough time to finish up all technological requirements of supprting trillions of people. We're almost there. The majority of everything discussed so far could be done with today's technology. But the infrastructure would take quite a lot longer. And if we do have centuries to work with...I think that the technological advances in that time will make a lot of our accounting now pretty much irrlevant. I've been avoiding futuristic Star Trek style solutions here, but we've gone from our very first flight to sending an object outside the solar system in roughly 100 years. So...looking to what we'll be capable of 100 from now...stuff like interplantary teleportation and matter replication seems pretty reasonable to me.
In any case, the planet herself would have no problem accomodating trillions of humans. It might not might not actually happen...but the people saying it's "not possible" aren't any different from those who thought it impossible to fly, to travel in space, etc. Of course it's possible. In fact, even saying it now...I suspect that "trillions" is probably a pessimistic figure by several orders of magnitude. If we simply colonized down like ants, the available space is so large that I don't have an intuitive grasp of it. And each level we build up has more surface area that the last. Space is totally not an issue. It's just a matter of providing proper input and output in the form of food, water, climate, plumbing, entertainment, communication, etc. And those are all simply engineering problems.
Trillions is no problem. It just a question of what technology is required to fill in the empty gaps in what we can do.
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Almost everything is possible with the right tech level. Hell, with enough energy investement you could probably make a discworld or something.
Then again, none of this will help you against a shortage of resources because popualtion growth and resource usage exceeds technology and infrastructure investements. As might happen in a relatively short time.