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.

  • From your description in second paragraph, the number of jewels is countably infinite. – Wojowu Dec 21 '15 at 17:57
  • Please clarify. If you are asking if you've laid out a clear and effective approach to "realizing" the reflections of Indra's Net as rational numbers, thus showing that a countably infinite number of reflections suffices, I'd say the exposition leaves room for improvement. Bear in mind that a countable union of countable sets is countable. However you say, "are there enough real numbers for this?", which suggests you do not believe the results are countable. – hardmath Dec 21 '15 at 18:04
  • This question was answered in the first comment to a previous version. – vadim123 Dec 21 '15 at 18:11

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.

  • Why must the strings be finite? – vadim123 Dec 21 '15 at 18:10
  • I have assumed that the number of reflections can only be finite, because no real-life-like scenario which I can imagine allows "seeing" an infinitely nested "reflection". If you are somehow allowing this, then your question requires further clarifications. – Wojowu Dec 21 '15 at 18:13
  • There is nothing about this question that is a real-life scenario. If the removed portions of a Cantor set correspond to finite strings (i.e. reflected images), the Cantor set itself corresponds to infinite strings. – vadim123 Dec 21 '15 at 18:16
  • I don't see what Cantor set has to do at all with reflections and/or this question. – Wojowu Dec 21 '15 at 19:00

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.

  • Why must the sequence be finite? – vadim123 Dec 21 '15 at 18:12
  • @vadim123: How could it not be finite? If there's a certain minimal distance between the jewels, then a reflection path containing more than finitely many reflections would need to be infinitely long, which means that (a) light cannot yet have propagated along this path since the net was made, and (b) no finite magnification of the image we see can possibly show the jewel at the end of such an infinite path (whatever that is taken to mean) -- and there's not such a thing as "infinite magnification" (how would you even define that?). So no actually occurring image has an infinite sequence. – Henning Makholm Dec 21 '15 at 18:17
  • @vadim123: Certainly there will be uncountably many points in our field of view that correspond to infinite sequences of reflections -- but it doesn't sound like a reasonable interpretation to say that there is "a jewel" to be seen at such a point. – Henning Makholm Dec 21 '15 at 18:20
  • If we assume that the entire surface of each jewel consists of reflected jewel, then how else would you describe a point that corresponds to an infinite sequence of reflecions? – vadim123 Dec 21 '15 at 18:20
  • @vadim123: "A point that corresponds to an inifnite sequence of reflections". Since such a point does not correspond to any particular reflected jewel (which one would that be if the infinite sequence contains all different points of reflections, for example?), your assumption that "the entire surface of each jewel consists of reflected jewel" leads to a contradiction and cannot be upheld. – Henning Makholm Dec 21 '15 at 18:23

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

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