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”.
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:
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, …, ∞^∞^∞^∞^∞^∞, … , ε0, ε0 + 1, …
Man, infinity just got a lot bigger.
PS Hey Josh: Cobra Kai sucks. Can’t catch me!