Would these 2 infinite sets be equal, if so why?

Question

My friends and I were asking the following question: If Minecraft worlds were to be infinite, does that mean that every Minecraft world is identical? My friends and I are adding this constraint to say they are "identical": $(x_1,y_1,z_1)$ in world1 doesn’t necessarily need to be equal to $(x_2,y_2,z_2)$ in world2 for the worlds to be identical

1. These Minecraft world have a fixed amount of possible blocks that can occupy one space block, let this fixed amount be $x$.
2. There is a height limit from $0$ to $255$ in the $z$-axis, but there would be no limit on the $x$ and $y$ axes. So if we were to focus only on the $z$ axis and one of the other axes, we would have $255$ infinite sequence of numbers, one set on top of another.

I was thinking the following:

1. If we grab 2 series from point 2 above, let's say $s_1$ and $s_2$. Since they are infinite, any "subsequence" (so a small portion of the sequence) in $s_1$ is bound to happen in $s_2$. Is this true? If so, why?
2. I was thinking that if 1 is true, then by extension, all the 255 infinite sequences would be equal, and this further extends to all sequences in the 3d world. Is this true? If so, why?
3. Finally, do 1 and 2 here imply that every sequence mentioned above is identical? If so, why?
4. Does 3 imply that the worlds would be identical? If so, why?

Answer 1

So it is not true that all Minecraft worlds would be identical, so our intuition should hold true here. Your logic breaks down in step 2, though step 1 is still shaky. In step 1, you assert that any finite list of numbers (I don't use the word "subsequence," since that refers to an infinite list of numbers inside the sequence) in $S_1$ would also be found in $S_2$ (with probability 1). This may or may not be true. For more reading, check out this post.

However, even if you can find any finite list of numbers in both $S_1$ and $S_2$, this does not mean that the two sequences are equal. The numbers $\pi\approx 3.142$ and $e\approx 2.718$ are nonterminating, nonrepeating real numbers, but they are not equal. So the infinite sequences you describe in your question also are not equal in general.

Answer 2

Certainly not.

Although there are multiple ways of disproving that all Minecraft worlds are a subset of an infinite one, one method relies purely on the math behind terrain generation. Take, for instance how some worlds such as this one, are "broken" in a way that some of their features repeat indefinitely, which is not the case in some worlds. This is because the code that generates some features relies on modular arithmetic, which is inherently periodic.

Even if we omit these "broken" worlds, we know that the periods of generation of some structures are different in different worlds, and thus cannot be part of the same infinite world.

Another hurdle is is the fact that bedrock formation in each world is the exact same, as discussed here. This makes for the argument of a coordinate transform very difficult, as then bedrock formation would have to conform to a very precise and complicated symmetry, which would be a miracle if true.

Lastly, each world is expected to generate exactly three strongholds, even if the world border were infinite. Hence, if all worlds were apart of an infinite one, it would have to have infinite strongholds, which is prohibited by standard generation.

From a mathematical standpoint, asking if all possible subsequences occur in an infinite sequence has essentially zero probability, but not impossible. However, such a case only occurs under a very specific set of conditions, which Minecraft worlds do not satisfy.

Answer 3

To try to extract a simple mathematical question from your question, you may ask: if one has two doubly-infinite strings over a finite alphabet $A$ (such a string really is a map $\Bbb Z\to A$ from positions to letters), is there always a shift that transforms one string into the other? The intuition why one should think that is so, appears to be that there are infinitely many shifts possible, yet locally one can only have finitely many states.

Nonetheless this intuition is wrong, and the answer is a resounding "no", no matter how you turn the question. Taking $A=\{0,1\}$ for simplicity it is clear that the strings $...0000000000000000...$ and $...11111111111111111...$ are not shifts of one another. If you consider always taking the same letter to be cheating (I wouldn't know why) then note that for instance $...0101010101010101010...$ and $...010010010010010010010...$ are not shifts of one another either. If you don't like periodic sequences, you can easily find quasi-periodic examples, or even random sequences that differ fundamentally from one another (for the latter it suffices to use different probability distributions for choosing the letters). It is true that for a suitable random sequence any finite substring occurs with probability $1$, but the fact that two sequences have this property by no way makes one of them a shift of the other.