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Qiaochu_Yuan comments on Godel's Completeness and Incompleteness Theorems - Less Wrong
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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.
I don't think "set" has a fixed meaning in modern mathematics. At least one prominent set theorist talks about the set-theoretic multiverse, which roughly speaking means that instead of choosing particular truth values of various statements about sets such as the continuum hypothesis, set theorists study all possible set theories given by all possible (consistent) assignments of truth values to various statements about sets, and look at relationships between these set theories.
In practice, it's not actually a big deal that "set" doesn't have a fixed meaning. Most of what we need out of the notion of "set" is the ability to perform certain operations, e.g. take power sets, that have certain properties. In other words, we need a certain set of axioms, e.g. the ZF axioms, to hold. Whether or not those axioms have a unique model is less important than whether or not they're consistent (that is, have at least one model).
There are also some mathematicians (strict finitists) who reject the existence of the "set of natural numbers"...