BusyBeaver(6) Is Quite Large

> So I said, imagine you had 10,000,000sub10 grains of sand. Then you could … well, uh … you could fill about 10,000,000sub10 copies of the observable universe with that sand.

I don't get this part. Is it really rounding away the volume of the observable universe divided by the average volume of a grain of sand? That is many more orders of magnitude than the amount of mass in the universe, which is a more usual comparison.

Yes, that's only some normal number amount of orders of magnitude. Even 10,000,000^10,000,000 is already so large that it doesnt matter, let alone after exponentiating _the exponent_ nine times more.

With tetration you're not dealing with orders of magnitude anymore, but orders of magnitude of orders of magnitude.

Exactly. This number is so so much bigger than 10^100000 or however many grains of sand would fit, that dividing by that amount doesn’t really change it, certainly not enough to bring it down closer to 9,999,999sub10

Yes, that's right, dividing by that ratio essentially barely affects the number in a sense that 'adjacent' numbers in that notation give a much bigger change.

10↑↑10,000,000 / (sand grains per universe) is vastly larger than, say, 10↑↑9,999,999

So on system we're using to write these numbers, there's really no better way to write (very big)/ (only universally big) than by writing exactly that, and then in the notation for very big, it pretty much rounds to just (very big).

Here's a more common example of this sort of comparison:

In significant figures, 1.0 billion minus 1.0 million equals 1.0 billion.

True but this is a ratio.

However many universes in question, there is a qualitative difference between that many empty universes (with 1 grain), and that many completely packed with grain.

Ask anybody who lives in one!

It's the other way around: we're talking about 10^(10^(10^(10^…))) (which is vastly bigger).

At very large numbers, even ratios don't really matter.

For instance, if you personally owed $100 trillion, you wouldn't be much relieved by a court order that reduced your liability by 99%. Or, if you're looking at numbers in scientific notation, you don't much care about the difference between 2e40 and 5e40.

In this case, the ratio is around 10^200. An incomprehensibly vast number, to be sure.

But because tetration is the next operator up from exponentiation (the way exponents are from multiplication), any fixed divisor ceases to "matter" very quickly. The difference between 10^^10,000,000 and 10^^10,000,001 is (10^^10,000,000 to the tenth power), if my understanding is right.

There's basically no way to get it into comprehensible territory even with repeated divisions. 10^^1 = 10, 10^^2 = 10^10 (ten billion), and 10^^3 is 10^(10^10) = 10^10,000,000. Already, dividing by 10^200 isn't going to meaningfully affect your number (10^99,999,800).

10^^10,000,000 is that kind of incomprehensible growth that we just saw from 1 to 2 to 3, repeated 10 million times.

> For instance, if you personally owed $100 trillion, you wouldn't be much relieved by a court order that reduced your liability by 99%.

It is *never• true that differences don’t matter. Only true that in some respects the difference matters, others it does not.

You manufactured a reasonable situation for differences not mattering.

But if I had $1 trillion, 99% off $100 trillion would matter.

As I noted, from the perspective of anyone in those universes, a 1 grain universe, or a solid grain universe would each be a spotty context to make a living.

But in very different ways!

So in this case, the ratio between two incomprehensibly large numbers, happens to be highly comprehensible under the circumstances in which they were described. I.e. universes and grains.

One can imagine that one of unexplained constants of nature might be a result of differences between unimaginably large numbers. Which again shows, that there is no such things as numbers so large differences don’t matter. Only cases where they don’t matter, or do. As with all approximations.