What Comes After Infinity?

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.

  • 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.

to infinity, and beyond

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

1 + infty   =   infty  <   infty + 1

  • ∞ + 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:

lim_{i to infty}  underbrace{{{{{{{{ infty ^ infty  } ^ { ^ infty} } ^ {^ infty}} ^ {^ infty}} ^ { ^ infty  }} ^ {^ infty }} ^ {^ infty}   } ^ {^ ldots  }   }_i

It’s called ε, short for “epsilon nought gonna understand what you are talking about anymore”. More comes after ε 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,  ε+ 1, …

Man, infinity just got a lot bigger.

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

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About isomorphismes

Argonaut: someone engaged in a dangerous but potentially rewarding adventure.
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