Something I've been wondering for a while now: if concepts like "natural number" and "set" can't be adequately pinned down using first-order logic, how the heck do we know what those words mean? Take "natural number" as a given. The phrase "set of natural numbers" seems perfectly meaningful, and I feel like I can clearly imagine its meaning, but I can't see how to define it.

The best approach that comes to my mind: for all n, it's easy enough to define the concept "set of natural numbers less than n", so you simply need to take the limit of this concept as n approaches infinity. But the "limit of a concept" is not obviously a well-defined notion.