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!

Tags: arithmetic, bazooka, cardinal number, cardinality, cardinals, children, elementary school, generalised number, generalised numbers, Georg Cantor, gradeschool, infinite, infinity, kickball, kids, math, mathematics, maths, money, noncommutative, noncommutative operator theory, noncommutative operators, noncommutativity, numbers, object-oriented programming, ontology, operator overloading, ordinal number

## Leave a Reply