Spacetime is Uncountable: A Proof
A proof showing that spacetime is uncountable.
Spacetime
We’re all familiar with the concept of spacetime at this point. It is simply three dimensional space (x, y, z) and the dimension of time. Visually, a point can move through spacetime like the below. This is called a reference frame.
The amalgamation of all these reference frames creates space and whatever causes time to move makes those frames move through time. Space and time are not separate. The faster you move, the slower time is for you relative to someone else standing still. Furthermore, when you move in one direction, then you contract in that direction! When space contracts, time slows down to compensate it. Together, we have spacetime. What we want to show is that spacetime is uncountable. Let’s see if we can.
Numbers
What does uncountable even mean? For this we have to go back to our numbers.
Natural Numbers include numbers 1, 2, 3, 4, …
Integers include numbers … , -2, -1, 0, 1, 2, …
Rational Numbers include numbers 2/3, 6/8, 1/2, ect… (basically any integer over another integer, as long as the bottom integer isn’t zero)
Real Numbers include every number on the number line (without real numbers, π or e would have no home)
Functions
Now that we have numbers, let’s look at functions. A function is an imaginary machine that takes something from one set, or space, and transforms it into something else in another set, or space. This concept is illustrated below.
There are three main types of function that we want to look at: injective (one-to-one), surjective (onto), and bijective (one-to-one & onto). Injective means that every element in A is mapped to an element in B only once. Surjective means that A is mapped to all of B. Bijective means that A is mapped to all of B & each element in A is mapped to an element in B only once. Visually, it looks like the below:
It would make sense that if two sets have the same amount of elements, or equal cardinality, then there would be a bijective map between the two. For instance, look at this adorable picture below. There are 3 dogs on the left and 3 cats on the right. If we map one dog to every cat only once, we get a function that is one-to-one. Furthermore, we’ve capture all three cats and the function is therefore onto. Since it’s both one-to-one and onto, it’s bijective. This allows us to state the following proposition.
Proposition: If and only if there is a bijection between two sets, then those two sets have the same number of elements (i.e. equal cardinality). We remember that this is an “if and only if” statement, which means that if the first part is true then so is the second (and vise versa). In english, if two sets have equal cardinality, then there is a bijection between the two sets.
Countability
With this we’re equipped with everything that we need to prove spacetime is uncountable — almost everything actually. First, we have to define what countability even means. Let’s assume we have some arbitrary set. Let’s call it, S. Don’t overthink this. Picture an empty circle labeled S with nothing in it — for now.
Our set, S, is:
Finite if it is empty or for some n in the Natural Numbers (1, 2, 3, …)
Infinite if it is not finite
Furthermore, our set, S, is:
Denumerable if S has the same cardinality, or number of elements, as the Natural Numbers.
Countable if it is finite or denumerable.
Uncountable if it is not countable.
Note: infinity and countability are not mutually exclusive since the Natural Numbers can go to infinity! An example will help illustrate. Let’s bring back our good friend, set S. This time we’ve filled it with things. How many oranges are in this set?
If you got 10 — congrats, you did it. This set of oranges was countable, because, well, you and I just counted it. Put another way, in our mind, we probably did something like the following:
It looks pretty similar — we just assigned a Natural Number to each orange. What you’ll notice is that this is a bijection (i.e. one-to-one & onto) mapping between the Natural Numbers and our oranges. So, if there is a bijection between the Natural Numbers and any arbitrary set, then that arbitrary set is countable. That is what we mean when we say whether something is countable or not.
Spacetime is Uncountable
Surprisingly that’s all we need to prove that spacetime is uncountable. Before we get into it, let’s recap what we have. We have different types of numbers: natural, integers, rational, and real. We have three types of functions: injective (one-to-one), surjective (onto), and bijective (one-to-one & onto). We also know something is countable if there is a bijective map between it and the Natural Numbers. Lastly, we have our reference frames in spacetime reposted below for convenience.
Let’s start by proving space is uncountable — you’ll see that we’re really going to do something once and then it’ll automatically show the whole shebang. Specifically, let’s start with only one dimension of space. Looking back at the reference frames, let’s pick the x-spatial dimension. What this means is this. Lift your head up — look straight. That is x. To your right is y. Look up to the sky and you have z.
If you want to measure something in the x-spatial dimension — what would you do? I’d get out a ruler and get to measuring. Since this is a thought experiment, let’s imagine we have a ruler that can measure any measurement imaginable. 2.543 meters? We got it. 1,203,021 meters — no problem. 1.6x10^-35 meters? Sure, why not. Essentially, we’re saying that everything we can measure in the x direction can be measured using the Real Numbers. Think of a giant, straight, number line extending outward in front of you. Wherever you land, you’ll find a point to measure.
If I wanted to count every point, I could do something like the table below. Here, it’s worth commenting on the notation. All the Z’s are integers (…, -2, -1, 0, 1, 2, …) and all the x’s are numbers between 0 and 9. You’ll agree that if we agree to never terminate this sequence to the right and downward we’ll get every real number imaginable. For instance, 2.94872839000000032300 is on there. So is 312,412,50,910.2419334923941. So is 0.3333333… You get the point.
Now, are there a countable or uncountable number of Real Numbers? Well, if there was a countable amount of Real Numbers then there would be a bijection (i.e. one-to-one and onto) between the Natural Numbers and the Real Numbers. Let’s see if we can draw it out:
Ok — so what are we saying here? We are saying that the above table we made maps every Natural Number imaginable to every Real Number imaginable. Therefore the Real Numbers are countable. I beg you the following question:
Can you think of a number that is not represented in Real Numbers table above?
Don’t worry if you can’t, the answer is forthcoming. Imagine that we have a Real Number, a. And, similarly to above, let’s write a as a decimal expansion.
Now, we will say this: If the first decimal in our Real Number’s table is equal to say 5, then the first decimal in a can’t equal 5. Moreover, if the second decimal in our Real Number’s table is equal to say 7, then the second decimal in a can’t equal 7. Repeat ad nauseam and you get something like the below.
Notice earlier that we said we listed all the Real Numbers in our handy dandy table, so let’s see if we can find a in our table. Is it in the first row? Nope — the first decimal point in that table can’t be equal to the first decimal point in a. Is it in the second row? Nope —that second decimal point can’t be equal to the second decimal point in a. Third row? Nope — third decimal point. Fourth? Nope. 254,901th? Nah. This concept is illustrated below and is known as Cantor’s Diagonal after Georg Cantor (1845–1918).
So — where is our a? We said that that table held all the Real Numbers, but we just found one that wasn’t there — where did we go wrong? We went wrong by assuming that we could count all the Real Numbers. We can’t. Put another way, there is no bijection between the Natural Numbers and the Real Numbers. And if there is no bijection, then they have a different amount of elements in them — think back to the dogs and cats.
You may ask, “but the Natural Numbers and the Real Numbers are both infinitely big — how can infinite be bigger than infinite?” Great question. I don’t know what to tell you there other than it can.
So, let’s go back to our spacetime. Look straight. If I were to measure everything in my x direction — I simply couldn’t. There are an uncountable number of points there. Put another way, every number that you could pick in that direction, I can always choose a number in there that you did not pick. Look right. This holds true there too. Look up — still check out. Now, take a deep breath. Try to focus on your breath and feel the passage of time . Yet again, that is uncountable — I can always choose a time that you did not pick.
If we do the same thing that we just did for all the dimensions of spacetime, we get a little something like the below:
The above graphic is the climax of our story. We have proved that spacetime is uncountable.
Meaning of It
I’m not sure myself — but it does beg the ominous question:
Are there countable or uncountable objects in our universe?
You may ask, “Well, what’s an object?”. Let’s look at a few. Are there a countable number of galaxies? Stars? Black holes? How about atoms? Quarks? Ideas?
In another article, I wrote about the idea space. For a too long; didn’t read (tl;dr), consciousness is the light one shines onto an idea space. Furthermore, an idea space simply consists of thoughts, emotions, sensations, and perceptions. So, are there a countable or uncountable number of ideas in consciousness?
Let us assume that an idea has this special property that whenever it is discovered that another idea appears that was not previously in the users’ idea space (i.e. consciousness). This thought can be expressed in common terms as: A new discovery creates more questions than answers. Don’t take my word for it, test this idea out yourself. When was the last time you discovered something new? Did it just end right there? Or did the discovery lead to a million more questions about what you just discovered? For instance, the last time you read a Medium article, did you learn something new? If so, did you then learn something new about that something new? Probably.
If that’s the logic, then we can use Cantor’s Diagonal argument one last time to prove that there are an uncountable number of ideas in the idea space. This phenomena is represented below. If each row represents an idea, then we can always pick an idea that was not represented in the table. Just like spacetime, the idea space is uncountable.
The Identity of this particular idea space (i.e. article) is as follows: The world has an uncountable number of ideas.
Look up in your x direction again. What are you thinking? Everything that you’re thinking and have ever thought is uncountable. It is not possible to count all the thoughts you had, the thoughts you are currently having, and the thoughts you will have. It’s just not. That is the power of imagination. That is pure bliss. That is a new way of looking at the idea space, or consciousness, or the mind.
On that note, go out and make the most out of your day and life with this new shift in mindset. Remember, you can always go back to where you started with a S.T.O.P.P (Stop, Take a Breath, Observe, Purpose, Proceed) — added below for those that actually want to do it.
Thank you for reading this — cheers.
If you want to learn more about The Idea Space, I’m currently writing a book on the subject. The newsletter/waitlist can be found here: https://www.ideaspacebook.com/ .
STOP. Stop what you’re doing. Stop thinking. Stop getting distracted. Just stop.
TAKE A BREATH. 3 of them even. Slowly inhale into your body, feeling the breath circulate inward. And slowly exhale out. 2 more.
OBSERVE. What do you notice? Your breath? A thought? Gravity? Simply notice.
PURPOSE. Now, what do you want to get out of this? Learn something new? Up to you.
PROCEED. And now that you’ve achieved beginner’s mind, we can proceed.
