The cardinality of Indra's net?

Question

This question has been asked before, but the title of the post was so general that it received no sufficient answer.

What is the cardinality of the set of jewels and reflected jewels in Indra's Net?

For those who don't know, Indra's Net is a poetic image used in Hindu and Buddhist cosmology to represent the universe. Imagine a grid-like net with a countably infinite amount of intersections and at each intersection there is a jewel. Now, on the surface of each jewel there are reflected images of every other jewel in the net. And since each of these reflected jewels is a copy of the actual jewels, they also have an infinite amount of reflected jewels within them. And within that second level of infinite reflected jewels, each of them also have infinite reflections... and so on, and so on, forever.

Basically, Indra's Net is like the scenario of two ideal mirrors facing each other, only the number of mirrors is countably infinite.

I've been trying to picture this with the number line. The integers could represent the original jewels, and the non-integer rational numbers could represent the first level of reflected jewels. Just pair each integer n with every rational number greater than n but smaller than n+1.

But then, each non-integer rational number would then have to be paired with its own distinct countably infinite group of numbers to represent the second level of reflection, and then every number representing that next level of reflection would in turn need its own distinct countably infinite group of numbers to be paired with, and so on, and so on.

Are there enough real numbers for this? The continuum always surprises me. Thank you.

EDIT: Reading the answers to the first version of this question made me realize how I underestimated the power of the old "1,2,3...".

Now, I have a similar question to my previous one, but I want to change the image of Indra's Net a bit. Imagine the standard image mentioned above, but then project the starting position infinitely "deep down" the levels of reflection so that not only do you see the standard Indra's Net before you, but as you look "outwards" you realize that your position is just one of infinitely many reflections within a larger meta-reflection, which itself is just a one reflection within an even larger meta-meta-reflection, and so on and so on, there being no "actual" jewels to begin with. You can "zoom-in" and "zoom-out" forever in both directions.

Is the cardinality of reflected jewels in this altered Indra's Net also the same as the set of integers?

Thank you.

Answer 1

First enumerate the jewels $1,2,3,...$.

Every jewel, every reflection of a jewel, every reflection of a reflection of a jewel etc. can be described using a finite string of natural numbers. To not go into boring details of this encoding, let me instead give a few examples:

Jewel $k$ will be coded by a one-element string $(k)$.

Reflection of jewel $k$ in jewel $l$ will be coded by a string $(k,l)$.

Reflection of the above reflection in jewel $m$ will be coded by $(k,l,m)$.

And so on.

We see that the total number of reflections is at most the number of finite strings of natural numbers. I claim that there are countably many such strings.

There are obviously countably many one-element strings. Using some kind of pairing function (or using argument similar to proof that $\Bbb Q$ is countable) the number of two-element strings is countable. Similarly the number of three-element strings, or more generally $n$-element strings for fixed $n$, is countable.

Now we can finish by recalling the fact that union of countably many countable sets is countable.

Answer 2

A useful characterization to keep in mind here is:

If you have a set and you can label each element of the set with a finite bit string (or a finite string of symbols in some other finite alphabet), such that every element gets its own label, then the set is at most countably infinite.

In your case, each of the multiply-reflected images of jewels you see will have been reflected through finitely many jewels in some particular order. If you write down that chain of reflections in this order (as a list of intersection addresses, which we already assume are drawn from a countable set), then you get a finite description for exactly that jewel image.

Thus the set of jewel images is countable.

Answer 3

If you allow an infinite number of reflections, the number of reflected images is $\mathfrak c=2^{\aleph_0}$, the cardinality of real numbers. As with the other answers that only allowed a finite number of reflections (and got a countable cardinality) you can identify each reflection with an infinite string of natural numbers, so we are asking the cardinality of $\aleph_0^{\aleph_0}$. But $\mathfrak c=2^{\aleph_0}\le\aleph_0^{\aleph_0}\le (2^{\aleph_0})^{\aleph_0}=2^{\aleph_0\cdot \aleph_0}=\mathfrak c$ as shown here and here on this site