I have been told many times by people professing to be Traditional Catholics that one of my main “problems” is the fact that, as a convert, I take Catholicism way too seriously. The argument goes that as not having been raised in an ethnically Catholic home or community, I simply don’t understand that Catholicism is not supposed to be taken… as seriously as I take it. I’ll never forget the famous Trad priest telling me to “wait five years… you’ll calm right down.”
I was always puzzled by this critique, as I perceive myself to be lukewarm at best. I don’t do even remotely what I could, and often don’t do what the Church teaches should be the bare minimum for a person in my position.
Well, here is a math lesson that I hope will scare the absolute crap out of every person that reads it. If love of God won’t motivate someone, so be it. Let’s use abject terror.
And that abject terror comes in pondering eternity. Infinity. Arithmetic.
To save time, I’m going to assume that everyone knows what exponents are. The power of an exponent is the number of times you multiply the number by itself.
3 ↑ 4 = 81
Remember how we talked about the number of angels and made a lowball guesstimate that there were SEPTILLIONS of angels (and thus demons as well?) We arrived at this number by looking at the number of stars scientists now guesstimate to be in the observable universe: 200 SEPTILLION or so. Well, one SEPTILLION is notated as:
10 ↑ 24
For reference, the earth weighs 6 SEPTILLION kilograms.
10 ↑ 80 is the number of atoms in the universe.
10 ↑ 90 is the number of grains of sand to fill the entire universe.
10 ↑ 100 is called a “googol” – note the spelling, different from the search engine. A googol is the number of grains of sand to fill the entire universe multiplied by TEN BILLLION. To make things easier as we move forward, comprehend now the distinction between a number, and the number of DIGITS in a number. A googol is a 100 DIGIT number.
10 ↑ 122 is the number of protons you could pack into the universe.
10 ↑ 185 is the number of Planck Volumes (the smalles volume discussed in science) in the entire universe.
A GOOGOLPLEX is 10 raised to the power of one googol: 10 ↑ googol
Now, remember the difference between a number itself, and its NOTATION, and read on. Fill the universe with sand. On each grain of sand write TEN BILLION ZEROES. This would accomplish merely writing out the NOTATION for the number googolplex – not googolplex itself. Not even remotely, remotely close.
Now, folks, THIS IS NOTHING. Absolutely NOTHING.
Let’s start talking about EXPONENT TOWERS. With EXPONENT TOWERS, we stack exponents on top of each other.
So 2 raised to the power of two, raised to the power of two:
2 ↑ 2 ↑ 2
Or, in exponent tower notation:
2 ↑↑ 3
With exponent towers, the second number after the arrows indicates the HEIGHT of the tower. You start at the top, and work down to the base, so:
2 ↑ (2 x 2)
= 2 ↑ 4
“Meh,” you might say. But let’s look at 3 ↑↑ 4
Remember, three cubed is 27, so do that operation at the top of the tower first:
3 ↑ 3 ↑ 27
Now, three raised to the POWER of 27 is… 7,625,597,484,987. That’s 7.6 TRILLION.
So, 3 raised to the POWER of 7.6 TRILLION is going to yield a number that has has 3.7 TRILLION DIGITS. Remember, that isn’t the number itself – just the number of digits in its notation, written out.
If we simply increase from 3 ↑↑ 4, to 3 ↑↑ 5, so adding one more story to the exponent tower, that gets us to 3 raised to the power of a 3.7 trillion digit number. 3 multiplied by 3, 3.7 TRILLION times – not 3.7 TRILLION times 3, no, multiply three times three, and each successive PRODUCT times three…
3 x 3 = 9
9 x 3 = 27
27 x 3 = 81
81 x 3 = 243
and so on like that, over and over again, 3.7 TRILLION times. We only did four.
We have now DWARFED a googolplex.
But we’re just getting started. We are still dealing in numbers so small that they could be described as “basically nothing”.
If you go to the next level, using three up arrows instead of two, what you are doing is saying that the product of the topmost expansion is NOT merely the power that the number below it is raised to, but rather the HEIGHT OF THE EXPONENT TOWER ITSELF. So instead of 3 x 3 x 3… 3.7 TRILLION times as immediately above, what
3 ↑↑↑ 4
is saying is writing 3 raised to 3, raised to 3, raised to 3 (not multiplied by, RAISED TO) 3.7 TRILLION times. Remember, the product of the topmost tower is the HEIGHT of the next tower below it, and so on. If you wrote this out in normal handwriting, with each “raised to the power of 3” in superscript (this blog won’t do superscript, but superscripting exponents is standard notation), the mere NOTATION of the number would reach from the earth to the sun. So, we can call this number the “Sun Tower”. Remember, as we learned before in
3 ↑↑ 3
the top few CENTIMETERS of this number, whose notation reaches from the earth to the surface of the sun, would be a 3.6 TRILLION DIGIT EXPONENT. And the mere NOTATION of
3 ↑↑↑ 4
would be 150 million kilometers tall.
But we’re just getting warmed up. There is a number called g1, which is
3 ↑↑↑↑ 3
This is getting to be almost impossible to keep up with. Just imagine the Sun Tower number raised Sun Tower number of times, and that product, let’s call it, “I’m getting scared”, being the HEIGHT of the next tower.
But we’re just getting started.
The difference between g1 and g2 is that g1 is merely the number of ARROWS in g2. Not the height of the exponent tower – that’s child’s play – the number of ARROWS.
So if g1 is 3 ↑↑↑↑ 3, four arrows,
then g2 is
3 (with Sun Tower Number of arrows) 3
Multiply g2 all the way out, and that is the number of arrows in g3.
Mathematics stops at what is called “Graham’s Number”, which is g64.
Now, to the point of this, and I truly hope this puts terror in your heart, and that this haunts you for the rest of your life on this earth. I hope it makes you cry.
Count to g64.
Congratulations. You have just passed your first NANOSECOND in hell.
Nah, I’m just kidding.
You’re still AN ETERNITY AWAY from completing your first nanosecond in hell.
Now, tell me more about how I take Catholicism and the fate of the human soul (both my own and other people’s) too seriously. Tell me more about how I just need to “calm down”. Go ahead. I’m all ears.
You don’t want to go to hell. You don’t want your kids to go to hell. You don’t want your worst enemy to go to hell.
Antipope Bergoglio is driving people into hell by the millions.
Have mercy on me, O Lord, a sinner.
(The source of the mathematics in this essay, with clearer notation, but considerably more detail, is HERE.)