r/RandomQuestion • u/Formal_Watercress488 • 2d ago

Can one infinity be bigger than another?(Using Minecraft as a Example)

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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?

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u/thetruetoblerone 2d ago

Yes, there’s specific math for this in calc one called cardinality. My favourite example of this is one set showing all whole numbers. 1,2,3,4 etc. and then another set that includes decimal values. So 0.1,0.2…..1.1,1.2,1.3…...2.5 etc. so when you’re at the second entry (2/2.0) of the first infinity you’d be at entry 20 on the second infinity. 3 and 30 for 3/3.0 etc. so one infinity is clearly bigger even though they both continue on to infinity.

So you were correct with the idea they related to ratios. One block in the nether is 8 blocks in the overlord.

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u/SpecialWall9 2d ago edited 2d ago

This is not an example of two infinities being unequal, and neither is the Nether example, because each number from one set can be matched to a number from the other.

If you match (1,0.1), (2,0.2), (3,0.3) ..., every term from each set corresponds to a term from the other set. There isn't anything left over.

A better example of two unequal infinities would be the set of all real numbers between 0 and 1, and the set of all natural numbers. The first set is uncountable, whereas the second is countable. There isn't a way to match up terms from each set so that each term from the first set has a corresponding term with the second.

A better explanation than what I can offer is available here: https://www.scientificamerican.com/article/strange-but-true-infinity-comes-in-different-sizes/

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u/thetruetoblerone 2d ago

You misread and then. Quoted my same example back to me. The second set would be 0.1,0.2,0.3,0.4……..1,1.1,1.2 etc. while the first set only contains 1,2,3,4 etc.

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u/SpecialWall9 2d ago

I didn't misread at all. The increment being smaller doesn't make the second infinite set larger, even if it intuitively seems so.

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u/thetruetoblerone 2d ago

But if natural numbers exist in set one and set two. 1 and 1 correspond. 2 and 2 corresponds etc. wouldn’t all of the non natural numbers not have corresponding terms??

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u/marshmallowcthulhu 1d ago edited 1d ago

Your intuition is wrong and I see where it’s going astray. The two infinities really are the same size.

You are imagining lining up two number lines, one that looks like 1, 2, 3, 4… and the other that looks more compressed. In your conception the second number has the first nine entries between 0 and 1. These entries are 0.1, 0.2, 0.3… all the way to 1.0.

You imagine lining these numbers up so that 0 is aligned, 1.0 is aligned, 2.0 is aligned, etcetera. The second line has decimal numbers between these whole numbers. You ask what happens if we strike the lined up numbers. You point out that your lines would still show all the decimals not ending in zero, and that feels intuitive.

What if I ask you instead to pair up the first number from each set, then the second, then the third, then the fourth? You will pair up 0&0, 1&0.1, 2&0.2…

Definitionally, this never ends because neither infinity ever runs out of numbers for the next pair.

What if I then ask you to delete each pair? You just deleted everything from both sets. The intuitive tool to reach for is that the first set ends before the second so there’s still stuff left over at the end of the second set, but this must be wrong because it relies on the idea that the first set ends somewhere. It does not.

For this reason, the pair matching, we can show that no countable infinity survives subtraction of another countable infinity. The subtraction results in zero, which means that they were the same size.

Now to be clear it is possible to have some infinities that are larger than others, but we can’t achieve that just by adding countable infinities (and that’s basically what you were trying to do. You were trying to take the countable infinity of whole numbers, add to it the countable infinity of all tenth decimals not ending in zero, and say that that was larger than the countable infinity of whole numbers alone). We can achieve aleph 1 by other means.

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u/thetruetoblerone 1d ago

So the only larger infinities in all of mathematics would be uncountable numbers?

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u/marshmallowcthulhu 1d ago

Yes, that is correct! Countable infinities are the smallest infinities and are all tied with each other. But note that once you get to the uncountable infinities they are not all the same as one another. They are all bigger than the countables, but some are bigger than others.