Understanding Arrow’s Impossibility Theorem

http://www.whydomath.org/node/voting/Arrow's_Impossibility_Theorem.html
Understanding Arrow’s Impossibility Theorem
Hodge and Klima (p.79, Mathematics of Voting and Elections: A Hands-On Approach) include the following number-theoretic example to develop intuition about Arrow’s theorem.  They ask the following question:

• Is it possible for a positive integer to be divisible by 2, 11, and 23 and to be less than 500?

Notice how each of the characteristics (from the end of the list to the front) reduces the set of possible positive integers.  The set of all positive integers less than 500 consists of 499 integers, S1 = {1, 2, 3, … , 499}.  The 21 integers in S1 that are multiples of 23 form the set S2:
{23, 46, 69, 92, 115, 138, 161, 184, 207, 230, 253, 276, 299, 322, 345, 368, 391, 414, 437, 460, 483}.
Let S3 be the set of integers in S2 that are evenly divisible by 11.  Then, S3 = {253}.  Because there are no even integers in S3, there is no positive integer less than 500 that is evenly divisible by 2, 11, and 23.
Arrow’s theorem is referred to as Arrow’s Impossibility Theorem because the inclusion of a fifth axiom (No Dictatorship) ensures that there is no procedure that satisfies all five axioms.  Hence, it is impossible for a procedure to satisfy all five axioms.  As such, Arrow’s theorem is in the spirit of the above example.