Psychology of Mathematical Reasoning

Recently, I've found myself needing to explain what it is that mathematicians do. Sometimes I say, "we add really big numbers". You'd think people would laugh (or at least give an incredulous look) -- but how many times have you heard a layman casually comment how "math people" deal with "numbers"?

Actually, number theory is a nice vehicle for giving layfolk a taste of what math is about. Everybody knows about naturals and primes (and if they don't, and you're a mathematician who's been put on the spot, it's something you can explain in under a minute). So I tell people, look: there are obviously infinitely many naturals (for any number there's always a bigger one) and there are also infinitely many primes -- but this latter fact is less obvious and requires proof.

Here is where I've run into unexpected troubles. People have no problem accepting the infinitude of the naturals, but what they have trouble appreciating is that it's not obvious tha! t the primes are infinite. "C'mon -- there are infinitely many numbers, so of course there are infinitely many primes!" I've heard this response from more than one person. "Now wait a minute" I protest. The primes are a subset of the naturals, so a priori, they have every right to be a smaller subset. "OK, gimme an example of a finite subset of the naturals". I'm happy to provide the example {1,2,3,4,5}. "Yeah, but you've constructed it as a finite set, so it doesn't count" is the sort of reply I get.

What seems to be happening is that to an untrained intuition, any subset of the naturals defined by a property without explicit bounds appears to be obviously infinite. Has anyone else encountered this phenomenon? Alexandre? Can my mathematician readers try this out on some non-math friends (no need to obtain signed consent forms) and let me know what you find?
Finally, does anyone have a simple example of a "nontrivia! lly fini te" subset of the naturals? That is, a set defined by a (simple!) property P that makes no reference to explicit bounds, yet P is provably finite?

Mathematical Reasoning