Counting, counting...

This is a continuation of the post of 31st January.  What we were doing with our people and chairs experiment was putting two sets of things (people and chairs) into one-to-one correspondence - that is, pairing off each item in one set with an item in the other set, and this will tell us if the two sets are equal, and if not, which set is bigger.  And this can give rise to some seriously spooky results.  We all know that the whole is greater than its parts - right?  Well...  consider the set of positive numbers (1,2,3,4...and so on).  Now consider the set of even numbers (2,4,6,8... and so on). Common sense says that there are only half as many even numbers as numbers, but put them into one-to one correspondence, so 1 pairs up with 2, 2 pairs up with 4, 3 pairs up with 6, 4 pairs up with 8 and so on. And that's the point, "and so on" goes on for ever.  You never reach an end, and so are forced to the conclusion that the two sets are equal - there are as many even numbers as there are numbers!  More to come.