In mathematics, classical calculus depends on the idea of ‘infinity’, and in the late 19th century mathematicians became quite concerned with nailing down that elusive concept.

Georg Cantor (1845 – 1918) proposed a solution, which can be explained using only two simple math concepts you learned in grade school.

The first is the idea of a ‘proper subset’. Suppose you have two sets: ** I** = { A B C } and

**= {A B C D E }.**

*II***is a subset of**

*I***because every element in**

*II***is also in**

*I***. However**

*II**contains the elements D and E where are not in*

**II***so*

**I,***is not a subset of*

**II***.*

**I**We also have the special case of * I *= { A B C D } and

*= { A B C D } — that is, they’re identical. Strictly speaking they are both ‘subsets’ of each other because every element in*

**II***is also in*

**I***, and vice versa. But this special case isn’t very useful, so this is called an ‘improper subset’, and in the example above*

**II****is a ‘proper’ subset of**

*I***because the two sets are not identical.**

*II*The second simple idea Cantor used to tackle infinity involves “do two sets have the same number of elements?” For finite sets we can just count the number of elements and compare the results: “17 in * I* and 17 in

*— they have the same number of elements.” But we can’t count infinite sets. Instead, Cantor observed, we can determine if two sets have the ‘same number of elements’ by pairing up the elements and see if there are any elements left over.*

**II**For example, suppose we have ** I** = { A B C D } and

*= { w x y z }. Then we can pair A with w, B with x, C with y and D with z, and we have no elements left over in either set:*

**II**When we can pair up their elements with none left over like this, we say that the sets are ‘equivalent’.

Now let’s consider two infinite sets: the set of all counting numbers ** ALL** = { 1 2 3 4 … } and the set of all even numbers

**= { 2 4 6 8 …}. We can pair 1 with 2, 2 with 4, 3 with 6 — in general every counting number with its double:**

*EVEN*So the set of all counting numbers (* ALL*), and the set of all even numbers (

*) are*

**EVEN****equivalent**because we can pair their elements with none left over.

But notice: EVEN is also a **proper subset **of ALL! So we come to Cantor’s definition: *a set is ‘infinite’ if it is equivalent to a proper subset of itself! *

When a set is equivalent to the counting numbers, it is called ‘countably infinite’. It turns out that the set of all possible fractions (1/2, 736/100236, etc.) is also countably infinite (which I won’t prove here). Cantor used the symbol ℵ_{0} (“aleph-null”) to represent ‘countable infinity’.

So the question arises, are there other types of mathematical ‘infinities’? It turns out there are. The set of all ‘real numbers’ (which includes all square roots, all transcendental numbers like π) is larger than ℵ_{0}. Cantor called the next larger infinity ℵ_{1}, and ℵ_{0} is a proper subset of it. It turns out there are an infinite number of infnitities: ℵ_{0}, ℵ_{1}, ℵ_{2} … (Of course there are!)

Cantor’s ideas remind me of the ouroboros^{1} — the ancient symbol of the snake or dragon eating its own tail:

Pressing on, John Wheeler (1911 – 2008) — one of the great quantum physicists of the 20th century — proposed a solution to a thorny problem assuming there is only one electron in the entire universe, it’s just very busy. Wheeler’s idea didn’t quite work out, but Richard Feynman (1918 – 1988), one of greatest physicists of the 20th century, realized that Wheeler was on to something, solved an important quantum physics problem with a variation of Wheeler’s ideas, and earned himself a Nobel Prize for his insight. ^{2}

So, along similar lines, let’s consider a thought experiment in which there is only one Soul, including G-d and everybody in all of space time, and this Soul is just ‘very busy’.

Why might G-d want to create the Cosmos this way? Well, suppose being infinite wasn’t enough for G-d, perhaps S/He also wanted to experience finiteness from the *inside* — in the same way that it’s one thing to watch someone swimming, and quite another to jump in the pool. In a sense this might be a way for an infinite G-d to be even more infinite! Something like Cantor’s ideas of ever larger infinities.

This idea has interesting implications. For one thing, of course I should love my neighbor as myself, because my neighbor is ME!

Or when Jesus said, “I and the Father are One”^{3} He meant that quite literally.

It also embodies a certain perfect justice: everyone whom I have treated badly — I get to experience being on the receiving end of that ill-treatment myself. Hitler gets to experience what it is like to be a Jew who was killed by the Nazi’s — 6 million times. And likewise, for every act of compassion and kindness I do, I will experience the receiving end of that as well.