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 makes *two* million dollars a year.
**Josh Lenaigne:** Oh yeah?! Well My Dad makes *five, hundred, BILLION dollars* a year!! He makes a *jillion* dollars a year.

*(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.

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:

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, ****∞×84 **+ 1**,** … , **∞^∞**, **∞^∞ **+ 1, …, **∞^∞^****∞^****∞^****∞^****∞**, … , **ε**_{0}, **ε**_{0 }+ 1, …

Man, infinity just got a lot bigger.

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