Bigger Than Time Itself

I've just been down a very interesting if mind-boggling rabbit-hole, courtesy of the podcast 'The Rest is Science',  which is co-presented by Professor Hannah Fry and Micheal Stevens; talking about very large finite numbers. Discounting the various flavours of infinity that theoretically exist, albeit mostly conceptually, very large finite numbers are numbers that, given enough time, could actually be counted, but are in any remotely practical sense infinite to the human mind's conception. Even one of the 'smaller numbers they discussed during the podcast - Fifty-Two Factorial, or 52!, 1x2x3x4x5 ... x52 - is pretty much inconceivable to most people, even given that it 'only' represents the total number of possible orderings in a deck of cards: approximately 8.0658 x 10⁶⁷, or 8-ish followed by 67 zeroes. The method of mentally imaging the scale of this number that was used in the podcast by Stevens ran roughly along these lines:

• Set a timer to count down in seconds from 8.0658*10^67

• Stand on the equator, and take a step forward every billion years

• When you've circled the earth once, take a drop of water from the Pacific Ocean, and repeat

• When the Pacific Ocean is empty, lay a sheet of paper down on dry land, refill the ocean and carry on.

• When your stack of paper reaches the sun, take a look at the timer.

The 3 left-most digits won't have changed. 8.063*10^67 seconds left to go. You have to repeat the whole process 1000 times to get 1/3 of the way through that time. 5.385*10^67 seconds left to go. That's just the number of permutations of a pack of 52 playing cards...

If that alone were not in itself a staggering ask for someone to try and visualise [extremely difficult to nigh-on impossible], they then moved on to Graham's Number, which takes the notions of factorialisation and exponential growth and blows them toward infinity itself in a very short space of time, all the while remaining finite and theoretically countable. Graham's number was devised as the upper bound in an abstruse mathematical exercise in Ramsey Theory: you'll have to look it up, as adequate description is well beyond the scope of a scribble like this and way beyond my pay grade, so to speak.

However, the basis of Graham's Number starts quite simply. It uses a notation invented by a mathematician called Knuth in 1976: 'up-arrow' notation, which has the nicety of managing very large numbers without having to consider decimal places [there's way, way too many of those in Graham's Number to even consider considering]: say, take 3↑3 which is three times itself times three => 27. Now, the next step in the sequence is 3↑↑3, which is 3↑(3↑3) => [three to the power twenty-seven, each arrow to the right equalling the output of the previous step] => 7,625,597,484,987: 7.6 trillion. We've gone from twenty-seven to seven point six trillion in a single step. You can see where this going.

OK, even given that we are rapidly dropping down the rabbit-hole at a decent rate of knots, we ain't even got started. We then move on in just two increments to 3↑↑↑↑, which is given the name grahal (g1). We are currently already inhabiting a number space well beyond our mortal comprehension. The next step is the Graham grahal (g2), which is 3↑↑↑↑... ↑↑↑↑3, where the number of up-arrows is (g1). Onto (g3): 3↑↑↑↑↑ ... ↑↑↑↑↑3, where the number of arrows is (g2). We repeat this process until we reach Graham's Number => 3↑↑↑↑↑ ... ↑↑↑↑↑3, where the number of arrows is (g63). A number so big, that written out in standard notation, even in tiniest possible text, would not fit within the volume of the known universe.

Graham's number is so infeasibly large that if it were even remotely possible for a human mind to conceive a model of it in thought, the mass and concentration of the data would simply create a Black Hole of the host cranium, where it would form its own event horizon, whence nowt could escape: a number of practically [but not] infinite size corralled by the gravitational force created by itself into an infinitely small point. Imagine, your own personal Schwarzschild Radius; a Schwarzkopf, if you like, accreting matter and data alike into your - ahem - singularity. But is Graham's Number the largest (in)conceivable non-infinite number out there?

Is it buggery: there's Rayo's Number, which raises the bar so far beyond into the realms of irreality, whilst still being actually real [but still not an infinity] that it transcends my very humble powers to even try and describe the meta-language that allows its conception. In short, it is a number so vast that it would require a googol [ten to the power of one hundred] discrete symbols simply to to notate it. That the human mind can come up with notions of this scale and grandeur, despite their complete lack of any conceivable real-world utility to its creator is testimony to just how peculiar and singular [sorry!] the human race is. Ponder and wonder, mes amis...