When I was in **kindergarten**, we would argue about whose dad made the most money. I can’t fathom the reason. I guess it’s like arguing about who’s taller? Or who’s older? Or who has a later bedtime. I don’t know why we did it.

**Josh Lenaigne:**My Dad makes one million dollars a year.**Me:**Oh yeah? Well, my Dad makesmillion dollars a year.*two***Josh Lenaigne:**Oh yeah?! Well My Dad makes*five, hundred, BILLION dollars*a year!! He makes adollars a year.*jillion**(um, nevermind that we were obviously lying by this point, having already claimed a much lower figure … the rhetoric continued …)***Me:**Nut-uh! Well, my Dad makes, um,*Infinity*Dollars per year!*(I seriously thought I had won the argument by this tactic. You know what they say: Go Ugly Early.)***Josh Lenaigne:**Well, my Dad makes**Infinity Plus One**dollars a year.

** I felt so out-gunned.** It was like I had pulled out a bazooka during a kickball game and then my opponent said “Oh, I got one-a those too”.

Sigh.

Now many years later, I find out that **transfinite arithmetic** actually justifies Josh Lenaigne’s cheap shot. Josh, if you’re reading this, I was always a bit afraid of you because you wore a camouflage T-shirt and talked about wrestling moves.

**Georg Cantor** took the idea of **∞ + 1** and developed a logically sound way of actually doing that infinitary arithmetic.

**¿¿¿****¿¿¿ INFINITY PLUS ??????**

You might object that if you add a finite amount to infinity, you are still left with infinity.

- 3 + ∞ = ∞
- 555 + ∞ = ∞
- 3^3^3^3^3 + ∞ = ∞

and Georg Cantor would agree with you. But he was so clever — he came up with a way to preserve that intuition (**finite + infinite = infinite**) while at the same time giving force to 5-year-old Josh Lenaigne’s idea of **infinity, plus one**.

Nearly a century before **C++**, Cantor overloaded the plus operator. *Plus on the left means something different than plus on the right.*

- ∞ + 1
- ∞ + 2
- ∞ + 3
- ∞ + 936

That’s his way of counting **“to infinity, then one more.”** If you define the **+** symbol **noncommutatively**, the maths logically work out just fine. So **transfinite arithmetic** works like this:

- 1 +
**∞ + 3**=**∞ + 3** - 418 +
**∞ + 3**=**∞ + 3** - 1729 ×
**∞ + 3**=**∞ + 3** - 43252003274489856000 × 287 × 1.4142135623730954 + 3→3→64→2 ×
**∞ + 3**=**∞ + 3**

All those big numbers on the left don’t matter a tad. But **∞+3** on the right still holds … because we ”went to infinity, then counted three more”.

By the way, Josh Lenaigne, if you’re still reading: you’ve got something on your shirt. No, over there. Yeah, look down. Now, flick yourself in the nose. That’s from me. Special delivery.

**##### ORDINAL NUMBERS #####**

W******ia’s articles on ordinal arithmetic, ordinal numbers, and cardinality flesh out Cantor’s transfinite arithmetic in more detail (at least at the time of this writing, they did). If you know what a “well-ordering” is, then you’ll be able to understand even the technical parts. They answer **questions** like:

- What about
**∞****× 2**? - What about
**∞ +****∞**?*(They should be the same, right? And they are.)* - Does the entire second infinity come after the first one?
*(Yes, it does. In a < sense.)* - What’s the deal with parentheses, since we’re using that differently defined plus sign?
*Transfinite arithmetic is associative, but as stated above, not commutative.*So**(****∞ + 19) +****∞ =****∞ + (19 +****∞)** - What about
**∞****×****∞****×****∞****×****∞****×****∞****×****∞****×****∞**?*Cantor made sense of that, too.* - What about
**∞ ^****∞**?*Yep. Also that.* - OK, what about
**∞ ^****∞ ^****∞ ^****∞ ^****∞ ^****∞ ^****∞ ^****∞**?*Push a little further.*

I cease to comprehend the infinitary arithmetic when the ordinals reach up to the **∞** limit of the above expression, i.e. **∞** taken to the exponent of **∞**, **∞** times:

It’s called **ε₀**, short for “*epsilon nought* gonna understand what you are talking about anymore”. More comes after **ε₀** but Peano arithmetic ceases to function at that point. Or should I say, 1-arithmetic ceases to function and you have to move up to 2-arithmetic.

**===== SO … WHAT COMES AFTER INFINITY? =====**

You remember the tens place, the hundreds place, the thousands place from third grade. Well after infinity there’s a ∞ place, a ∞2 place, a ∞3 place, and so on. **To keep counting after infinity** you go:

- 1, 2, 3, … 100, …, 10^99, … , 3→3→64→2 , … ,
**∞**,**∞**+ 1,**∞**+ 2, …,**∞**+ 43252003274489856000 ,**∞×2**,**∞×2**+ 1,**∞×2**+ 2, … ,**∞×84,**+ 1**∞×84**… ,**,****∞^∞**,**∞^∞**+ 1, …,**∞^∞^**… ,**∞^****∞^****∞^****∞**,**ε**,_{0}**ε**+ 1, …_{0 }

Man, infinity just got a lot bigger.

**PS** Hey Josh: Cobra Kai sucks. Can’t catch me!