Infinity
Infinity
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.
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.
But the monetary value of the bills in each stack still adds up to infinity for both. It’s like having an uncapped Internet connection at 56 KBit/s versus 100 MBit/s: You can download all the things with both, but that doesn’t make them equal in practice.