1 + 1/2 + 1/3 + 1/4 + ... = Infinity
what we mean is that no matter now big of an N you pick, I can always find enough terms in that series whose sum will exceed N. In this case, infinite really is shorthand for "increasing without bound" or becoming "arbitrarily large".
When dealing with cardinalities -- as opposed to magnitudes -- all bets are off. That was the crux of the closure under star question. [BTW, I think I've found the bug in Ivan's proof that closure under concatenation implies closure under star. The set U need not be a complete lattice. Otherwise, your argument could be used to show that the function f:Z->Z defined on the integers by f(x)=x+1 has a fixed point.] Set theory is rife with examples where some property P holds for arbitrary finite collections but not infinite ones:
- in point-set topology, a finite (but not necessarily countable) intersection of open sets is open
- in analysis, there are finitely (but not countably) additive measures
There are techniques, most (all?) of them based on transfinite induction, for proving that P holds for infinite collections given that it holds for arbitrary finite ones. Knaster-Tarski is one; Hausdorff maximality principle and Zorn's lemma are other favorites. All are equivalent to the axiom of choice.