# Who knows this approach to cardinality?

Where did I take this analogy from?

My algebra classes have detoured into three days of set theory. Once upon a time (when I was in school) the detritus from New Math was scattered across the land, and 2nd and 3rd and 4th graders knew about sets and elements, subsets, union and intersection, the empty set. But no more.

So instead of walking my kids through 20 minutes of stuff that might show on the new state exam, I went for an ever-so-slightly deeper 3 days of sets. (seTS, with TS, not X, an easy slip you don’t want to make in front of 25 giggly, gangly teenagers).

On the last day, I get them to remind me, when sets are finite, . And then I tell them that things go all goofy when the sets are not finite. And I digress.

Tell me if you recognize this:

Let’s walk into a huge auditorium, and notice that every audience member is seated, only one per seat. And let’s also notice that every seat is filled. No empty seats. No standing audience members. What conclusion do we draw?

I’ll save the fun stuff for a follow-up.

“Where did I take this analogy from?”

I’m not sure where you got the seats and kids analogy, but it seems very close to the hotel infinity analogy which I first read about in Martin Gardner’s book Aha Gotcha!, but I have recently seen ascribed to Hilbert (which seems likely–didn’t Hilbert do half of everthing?)

Actually, this looks more to me like the pigeonhole principle, with audience instead of pigeons. Of course, for infinite sets this is close to Hilbert’s Hotel as lsquared noted.

Didn’t Hilbert do half of everything Euler did? Just kidding..

“Didn’t Hilbert do half of everything Euler did?”

Yes, but he did it with more axioms! (also, just kidding)

I think it is a demo of 1-1’ness rather than pigeonhole.

In fact, a few days later, a kid tried to relate it to . She had focused entirely on the 1-1 aspect.

This is the only formal approach to cardinality I really know: two sets have the same cardinality iff there is a 1-1 and onto from one to the other.

I was pretty thrilled to see that the Singapore Math curriculum uses this extensively in their kindergarten books to introduce the concepts of “more”, “less”, and “the same number”. They have the kids draw lines to make maps between sets (give each cat a ball, give each bird a nest, etc.), then after you make the best pairing you can, you say whether one set is larger. This worked really well for my kids. A bit of the new math may be lingering…

I’m not sure where this particular analogy originated. The first time I was introduced to the concept of equal cardinality, my instructor started out with a shepherd with a large flock of sheep and couldn’t count to more than 5; the shepherd used stones in 1-1 correspondence with the sheep to know whether any of them got lost while they were out grazing.

There’s a similar analogy using stadium seats at http://www.infiniteproduct.info/struppma.htm.