What if a digit (say, 9) only appears in pi up to a certain decimal point? If your sequence contains that digit and doesn't appear by that point, pi won't contain your sequence.

If pi is indeed normal (whether proven or not), that would never happen. A normal number, by definition, contains an even distribution of digits. There would be no "last 9" because there would have to be more after to satisfy the distribution. Basically, the count of all nines must be the same as the nine other numbers, as the number of digits goes to infinity.

Even if the first million digits of some unnamed number were all nines (with no more afterward), it would not be normal because the distribution of nines, as the number of digits goes to infinity, would go down.

Right, the point is that we don’t know whether pi is normal.

No. Encoding is not a clever way around infinity.

You can encode orange in RGB and HSL too, but the set of all integers still does not contain the concept of an orange. You’ve just assigned meaning to an integer.

In the same way you can’t count to orange, you can never start at 1 and count up to -1. There are an infinite number of different integers greater than 1, and that infinity does not contain -1.

SciFi likes abusing this. Just because there are an infinite number of universes doesn’t mean there is a universe that contains anything you can dream up Rick and Morty style.

> You can encode orange in RGB and HSL too, but the set of all integers still does not contain the concept of an orange.

Nobody claimed "the concept of an orange" can be found. The claim that the word "orange" can't be found, however, is provably false.

If we're going by this logic, then nothing but 0 and 1 can ever be stored on any medium, because a hard drive can't possibly store "the concept of an orange" either.

> Just because there are an infinite number of universes doesn’t mean there is a universe that contains anything you can dream up Rick and Morty style.

And you know this with certainty.... how? With enough branches in a truly infinite timeline, the likelihood of there being a combination which produces a specific outcome is quite high.

> And you know this with certainty.... how? With enough branches in a truly infinite timeline, the likelihood of there being a combination which produces a specific outcome is quite high.

It is not. It is quite low. The number of possible states grows at a faster rate than the rate necessary to maintain “infinity.” It’s only true if you assert that all possible states at a branch are added to the set, which is a tautology.

A fun puzzle. Let’s say you have an empty set whose members are sets. You add a set to it of infinite size (like the set of all positive integers). Next, you add another set to it whose size is also infinite but whose members are not contained in the first set. Now you repeat this process an infinite number of times.

You have an infinite set of infinite sets. The question is: is such a set possible and, if so, does your infinite set of infinite sets necessarily contain all possible infinite sets?

> It’s only true if you assert that all possible states at a branch are added to the set

That is generally how "multiverses"/branching timelines are handled, yes.

Most pitches for Everett branches (which are a specific enough concept that it's silly to get all technical, but whatever) hold that there is an amplitude distribution over the entire configuration space of the universe. While the proposed mechanism of decoherence implies a sort of discreteness, every single possible configuration would have non-zero amplitude. Of course there's no specific evidence for this over, say, collapse theories (or something stranger, like Bohmian mechanics), but there's certainly no philosophical issue. Not sure what you think ZFC has to do with physics, but an empty set with members is a tad nonexistent.

What if I created a new system of encoding words, where the digit "1" represents the word "orange"? Does that disprove the claim, just as encoding it with bytes does?

> the set of all integers still does not contain the concept of an orange.

Nor can any file system store the "concept of an orange".

File systems are really just pointers to numbers, if you think about it.

Did you reply to the wrong person? I actually said just as much in my comment.

> If we're going by this logic, then nothing but 0 and 1 can ever be stored on any medium, because a hard drive can't possibly store "the concept of an orange" either.

Ah yes, sorry! This should have gone one level up.

> the set of all integers still does not contain the concept of an orange. You’ve just assigned meaning to an integer.

We all do, all day, every day. Isn’t that exactly what computer science and mathematics is?

> Just because there are an infinite number of universes doesn’t mean there is a universe that contains anything you can dream up Rick and Morty style.

Sure, there are many different kinds of infinities, but what does that have to do with pi and file systems?