balderdash9, 1 year ago

To establish whether one set is of a larger cardinality, we try to establish a one-to-one correspondence between the members of the set.

For example, I have a very large dinner party and I don’t want to count up all the forks and spoons that I’ll need for the guests. So, instead of counting, everytime I place a fork on the table I also place a spoon. If I can match the two, they must be an equal number (whatever that number is).

So let’s start with one $1 bill. We’ll match it with one $100 bill. Let’s add a second $1 bill and match it with another $100 bill. Ad infinitum. For each $1 bill there is a corresponding $100 bill. So there is the same number of bills (the two infinite sets have the same cardinality).

You likely can see the point I’m making now; there are just as many $1 bills as there are $100 bills, but each $100 bill is worth more.