All Infinities Are Big, But Some Infinities Are Bigger Than Others

Someone on EP asked the following question:

If you combine two infinite continuums to make an infinite plane, would the number of points be infinity squared?

Of course, the site’s math nerd HAD to answer this question! And my response got rather long.  Not infinitely long, though!

Any algebraic manipulation (with or without scalar multiples) of infinity is still infinity. This is demonstrated in the Hilbert Hotel Paradox.

Imagine you have a hotel with infinite rooms with room numbers of every integer, and each one of them is occupied.

Infinity plus one: All the manager has to do is ask each guest to move to the room number that’s one greater than theirs. Room number 1 can now accommodate the new guest. Infinity plus one is still infinity.

Infinity plus infinity: This is what you’re asking about. One would think it’d be 2*infinity. It’s not. Now a tour bus full of infinity people want to find a room. The existing guests simply need to move so that the number of spaces moved is greater than one (in essence, leave at least every other room empty) Each guest is told to move to a room number that’s double their existing one. So 1 would move to 2. 2 would move to 4. 3 would move to 6. And so on. Now 1, 2, 3… extra spaces are created, which eventually becomes infinite free rooms. So now the guests can fill in the empty rooms. Infinity plus infinity is still infinity.

Infinity times infinity: THIS is infinity squared. Repeat the above process over and over and over again, since each time you add infinity to infinity, you still have infinity.

Really, any algebraic/arithmetic manipulation of infinity is still infinity. Now you may wonder. Are there any infinities larger than others? YouTuber ViHart provides a great explanation of this in detail. I shall explain one of them, Cantor’s Diagonal Proof.

Remember how I said these doors are integer numbered? Now a tour bus full of every real number between 0 and 1 arrive at the Hilbert Infinite Hotel, demanding rooms. But because if you choose any two numbers on the point, and add an “extra” digit of specificity, you’d always be needing rooms within rooms. The ViHart video does a great job of explaining this mathematically, too.

It turns out that the infinity of the reals is a bigger infinity than the infinity of the integers! Integer infinity, otherwise known as countable infinity, is known as aleph null. An infinite series is known to have the same cardinality if there is a bijection between the numbers of set A and set B. In simpler terms, If we have set A{1, 2, 3, …} and set B{2, 4, 6…}, there is a one-to-one ratio between the elements in each set. Each number can be paired with each other. Cardinality is easier to understand with finite sets. Think of two groups of dancers. Cardinality here is the number of elements in each set (which works differently with infinity, but is a similar concept). If the dancers from one group can pair up with the dancers of the other group without leaving any dancer out, then they have the same cardinality. So set A and set B can pair up like such: {1|2, 2|4, 3|6…}. This also explains the movement in the second scenario. But the cardinality of aleph null is different from the cardinality of the reals (which was proven in Cantor’s Diagonal proof, explained in ViHart’s video).

If you want to learn more, here’s her video:

All Infinities Are Big, But Some Infinities Are Bigger Than Others

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