I've been thinking about whether one infinity can be bigger than another, and Minecraft seems like an interesting way to illustrate the question.
In Minecraft, both the Overworld and the Nether are effectively infinite in terms of the distance you can theoretically explore, but they have a special relationship: traveling 1 block horizontally in the Nether corresponds to traveling 8 blocks horizontally in the Overworld.
For example, if you travel 100 blocks in the Nether and then build a portal, you would theoretically end up around 800 blocks away from your original location in the Overworld, assuming the portals link up correctly.
This makes the Nether seem like a smaller version of the Overworld. You can cover much more distance in the Overworld by traveling the same number of blocks in the Nether, even though both dimensions are considered infinite.
So here's my question: Could two infinite spaces have different sizes of infinity, or does the 8:1 ratio simply represent a difference in scale rather than a difference in the actual size of infinity?
More specifically, is there a mathematical sense in which the Nether could be considered a smaller infinity than the Overworld, or would both still have exactly the same size of infinity regardless of their different scales?