My four-year-old has recently started asking me questions about infinity (“is there a biggest number?”, “what’s infinity plus one”, etc.) and he and my three-year-old often get into competitions to name the biggest number.

The correct answers to these questions depend on how you interpret “number” and “infinity”; I’ve decided it’s best to pick one interpretation of these terms and answer their questions consistently based on that interpretation. Clearly, when they are older they will remember the answers their dad gave them when they were three and four and be grateful that all the answers turned out to be consistent.

But what interpretation should I give to these terms? Here are some possibilities:

Cardinal number	$\aleph_0$	no	yes	yes	no
Ordinal number	$\omega$	yes	no	no	n/a
Ordinal numbers with “natural” operations	$\omega$	yes	yes	no	n/a
Surreal number	$\omega$	yes	yes	yes	yes

Let me go through some of the tradeoffs:

If “number” means cardinal number, then I’d have to explain to my kids that “infinity plus one” is the same as “infinity”. I don’t think they’d like that. Also, when they get into a “name the highest number competition”, they often say things like “infinity plus infinity” and “infinity plus two”. I’d have to explain that those are all the same, defeating the point of the game. That’s no good.

“number” meaning ordinal number is better, but then if my kids ever ask about “one plus infinity” instead of “infinity plus one”, I’d have to explain that “one plus infinity” equals “infinity” but “infinity plus one” is bigger. Huh? What four-year-old wants to hear that?

If I keep “number” as meaning ordinal number, but interpret “plus” to be the so-called “natural” plus that’s commutative, then that’s better. But even better still is surreal numbers. Much like a Stokke high chair, it will grow with them as they age. When they start asking about “infinity minus one” I can explain that that’s an actual number that’s one less than infinity as expected. Similarly with more advanced things like “sqrt(infinity)” and “log(infinity)”. If they ever ask about infinitely small things, I can explain that as “1/infinity”.

And yet, this essential parenting tool was only invented ~50 years ago. Just another of the modern parenting conveniences that weren’t available to our ancestors.