Phineas and Ferb hoverboarding. Image source: LA Times. License: Disney. Fair Use.

My seven-year-old son understands infinity and how easily it is reached. This post reports his discovery.

We watch the television show, Phineas and Ferb, whose main characters are two brothers named, funnily enough, Phineas and Ferb. It’s a great show and I strongly recommend it. In almost every episode, the step-brothers, aged ten and eleven¹ respectively, invent outlandish things like time machines, space ships, and animal-mind-reading machines. If you think they’re too young for such things, yes, yes they are.

What does all this have to do with infinity, you ask? We’re getting there.

In the episode “Bowl-R-Ama Drama”, the boys participate in “The World’s Most Pointless Awards” and enroll in two categories: ⑴ the world’s largest bowling ball and ⑵ the world’s largest game of pinball. They win and the award adjudicator remarks, “Wait! Two records in one day? That’s another world record!” and gives them a third record. Here, the show’s writers missed something that my son immediately picked up.

He said, “If they won a third record, then that’s also a record so they get a fourth record, and then they get a fifth record and then they get infinity awards!”

Meta awards

My son’s insight shows why meta awards—awards for winning awards—are a bad idea. Once you give them then it’s impossible not to reach infinity.

**The non-argument
**Wait, you argue, it isn’t necessary that Phineas and Ferb win a fourth award for being the first to win three awards in a day because it could be that someone else already won three in a day. Leaving aside the problem of infinite awards for this hypothetical awardee, this still doesn’t defeat my son’s argument. But it won’t do because were this the case, they would now be the first to win two awards in a day and not win three in a day thereby winning a fourth and then carrying onwards to infinity.

The problem most closely related to the meta awards case is the proof that all numbers are special. It goes something like this:
✏ 1 is special. It is unity, the first number, the only number that when raised to itself is itself.
✏ 2 is special. It is the first and only even prime number.
✏ 3 is special. It is the only prime number that’s one greater than another prime, the first odd prime.
✏ 4 is special. It is the first non-prime even number, the first square number. It is the first number that can be summed up in two different ways (1+3, 2+2) and we’ve established that two is special.
…
Now, suppose we come across a number that’s not special; how special is that! an unspecial number!
…

The list continues until infinity. A concept my son understands.

Children…, to my mind are the best philosophers.
From “5000 BC and other Philosophical Fantasies” by Raymond Smullyan, 1983

Footnotes

¹ Phineas and Ferb’s ages are never discussed in the show so I asked the best authority on the subject—my son. He decided their ages.