1. Every person will trade with every other person in the world. This is clearly false, although it would seem to follow from some presentations of the concept.
2. In a world of only two people, they will probably trade with each other. This is very likely true (especially if they are both "hot"), though not certain at the theoretical level.
3. Everyone will trade with at least someone. Whoop-dee-doo.
Worse, #3 is not even true.
The key counterexample is animals. They have well-defined preferences, downward-sloping demand curves, demand for multiple commodities (if only both food and water, plus of course sex), and differential abilities. Yet most animals don’t trade with any other animals.
Why not? There are high fixed costs to trading at all. Most animals can’t overcome those costs. They aren’t smart enough. Lack of opposable thumbs, or lack of extended long-term trust, are other obstacles, not to mention "fear of being eaten." There is nothing in a Walrasian model to rule out q = 0 no matter what kind of critters are walking around.
In human communities there will be much more trade than we find amongst the eagles. But the theory of comparative advantage deflects our attention away from the fixed costs of trading. As a result, many people overestimate the benefits that free trade (which, I might add, I fully favor) will bring to the developing world. And they underestimate the importance of those fixed costs of trading in holding nations — and people — back from a better future.