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
- These Minecraft world have a fixed amount of possible blocks that can occupy one space block, let this fixed amount be $x$.
- 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:
- 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?
- 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?
- Finally, do 1 and 2 here imply that every sequence mentioned above is identical? If so, why?
- Does 3 imply that the worlds would be identical? If so, why?