How Far Is Up?

My Dad does this schtick where sticks his pointer finger in the air to arrest his audiences' attention and in his best professorly voice poses the question "How Far Is Up"? This is usually met with sighs since everyone in the family has heard this a thousand times. He then produces the punchline-like answer: "Half way between up and down" followed by a "eh eh?" For years, I dismissed it as nonsense. My Dad is an inquisitive guy, but not a physicist. I even thought about trying to come up with a mathematical proof to dissuade him from saying this. Something something recursive definitions. However, I only realize now, this schtick actually touches on one of my favorite areas of mathematics: countability. 

Countability

Loosely defined, countability is a property of a set of numbers to have an injective (one-to-one) mapping of each element in the set to some element in the set of natural numbers [1, 2, 3, 4, ...]. This is sort of obvious with finite things (i.e. enumerable). You can count the number of fingers on your hand. You can count the number of leaves on a tree. Heck if you had the patience, you can count all the grains of sand on a beach or all the molecules of water on the planet. All of these things are finite.

To Infinity and Beyond

Things get interesting when you start talking about infinite sets of numbers however (i.e. denumerable). The set of even numbers [2, 4, 6, 8, ... ] is technically countable even though they go on into infinity. This is true because you can take the position of each number and align it to the set of Natural Numbers. The even number 2 is in the first position, 4 is in the second position, 6 is in the third position, and so on all the way to infinity. This may not make intuitive sense, but smarter folks than I have proven that basic operations on a set do not alter their countability. Adding 1 to an infinite set is still infinite. As with the set of even numbers, removing half of the elements is also infinite.

Half of Infinity

This brings me to why my dad might be on to something. If we define UP as maximum distance we can move from a center of mass (such as the center of the Earth) and that the universe is infinite (or at least converges towards a shell-like boundary where spacetime is infinite), then the distance you can go UP is infinite. Using Natural Numbers, we can discreetly map the distance in meters away from DOWN all the way to infinity - i.e an infinite countable set. If we divide that in half (re: "Halfway between up and down") we still have an infinite distance. 

So whether or not my dad knew it, his schtick touches on a fairly advanced mathematical concept. I only realized this while binge watching the video "Which way is Down?" from my new favorite Youtube Channel VSauce. 
 

One More Thing

Countability has a lot of interesting concepts. I first touched on it in my finite field theory class at the U of M. One of my favorite topics we briefly discussed was Hilbert's Paradox of the Grand Hotel. I won't even attempt to try and explain it, but watch the TED video below for an amazing illustration of the concept. Enjoy. Love ya Dad.

The first part of this video is a good intro to countability, math with infinite sets, etc. 

Bonus: What really is down?

Further Reading:

• Mathematical proof that infinity divided by infinity does not equal 1