One of the most intellectually astounding ideas I’ve ever encountered is that there are different types of infinity. Not only that — some infinites are larger than others, and there are an infinite number of infinities. The tool used to create these ever larger levels of infinity is know as a power set.

One of the most existentially relevant ideas I’ve ever encountered is the Buddhist notion of emptiness. In a sense, calling it an idea devalues it —as it is nothing short of an epistemic tool leading to liberation from suffering.It’s function is to eradicate the erroneous ways in which we look at the world.

In this piece, I want to explore the structural similarities between these distant concepts — and demonstrate that power sets and emptiness are closer than you think.

The generation of infinitely bigger sets is analogous to the way our mind generates unceasing thoughts and elaborations.

Sets, Subsets, Power Sets

A set is simply a collection of elements. If we consider the set {1,2,3} — we see that it has three elements: 1, 2 and 3. The curly brackets {} denote that it’s a set. A subset is a set that is contained within our original set. More precisely, if we make A={1,2,3} and want B to be a subset, then every element in B must be an element in A.

Now things become interesting. The power set is the set of all subsets. In the branching diagram below, I’ve shown all the possible subsets of the set {1,2,3}. You’ll notice that the original set is a subset, since it qualifies based on our earlier definition. We also see the empty set, { }, the set containing no elements, is counted as a subset.

As such, the power set of {1,2,3} will contain, as elements, all of the above sets. The size of a power set is determined by the number of elements in the original set, so in this case we can know in advance that the power set will contain 2³= elements. (A power set contains 2^n elements, with n being the number of elements in the original set).

Infinity and Beyond

The power set of a set is always larger than the original set. This applies to infinite sets as well. The size (number of elements), of the Natural Numbers (1,2,3,4,5,…) is denoted by the Hebrew letter Aleph, with a subscript zero — referred to as Aleph Null. It represents the smallest level of infinity. (To learn more about infinite sets, read this: https://bit.ly/36hRiV9 and watch this: https://bit.ly/34nQext)

However, to obtain a larger infinity, we need only take the power set — then we are guaranteed a new set with more elements.

We can then take the power set of our new set and continue this process forever. There is no end to this infinite landscape. Which is what prompted mathematician David Hilbert to say:

“No one shall expel us from the paradise Cantor has created for us”

Georg Cantor was the first mathematician/logician to schematize the landscape of the transfinite. The above facts about power sets are contained in a theorem bearing his name: Cantor’s Theorem.

The crux is this: with the tool of the power set, we can generate infinite infinities.

Introducing Emptiness

In Mahayana Buddhism, some of the most profound teachings are those on Emptiness. Legend has it that the philosopher Nagarjuna obtained them by visiting the Nagas — snake like demons. We can summarize emptiness as such:

All phenomena are seen as empty of inherent existence.

This is not a declaration of voidness or nothingness — but a claim of interconnection. Emptiness is the statement that upon final analysis — everything is empty of any essential qualities. Everything is a sum of its parts — there is no permanent or stable essence of anything. If we break down a racecar and try to identify its racecar-ness, we’re going to have a hard time. Is it the color of the car? The parts? The shape? It’s function? In fact, it’s all of these things — and nothing more. Each of those aggregate components can themselves be broken down in a similar way.

Okay. How can this help us in meditation? Well, we can utilize emptiness as a tool to see the interconnection and lack of substantiality of thoughts. Particularly — we can use it to undercut our habit of living in our heads and in our thoughts. Consider this scenario of metacognition (thinking about thinking). This process of iterating on thoughts is endless — we can remain in thought loops indefinitely, continually elaborating on an initial observation.

Putting an ending to our intellectual explorations can be done with emptiness.

“The victors have taught emptiness

To definitely eliminate all views,

Those who have a view of emptiness

Are said to be incurable” — Nagarjuna, MMK 13

Using this tool — at any moment during a series of thoughts, if we can have insight into their insubstantiality and lack of essence — we can immediately release them. We can let go of them as we recognize there is neither, 1) any reason, nor 2) anything substantial to hold onto in the first place.

This is not a rejection of the beauty of the thinking mind. This is a tool to release oneself from the grip of an overactive mind. We are learning to see the reality of things as they truly are. One of the epistemological changes in perspective that has to occur is from content to quality. We must disengaged with the subject matter of our thoughts and recognize its emptiness nature of impermanence and interdependence.

Instead of focusing on what the words in your head are about — focus on the fact that they arise for a moment, abide, and pass away. If you’re too caught up in the content, just remember the emptiness-nature of the thought in question. As some Zen masters recommend:

Watch attentively for the beginning of your next thought.

Emptiness is a vast topic in Buddhist philosophy, and I have barely scratched the surface. However, it is worth clarifying the last line of the above poem — “Those who have a view of emptiness are said to be incurable”. This is a warning. Emptiness is a way to get rid of views — if you take it as a view itself, then you’re really stuck with it! Accusations that Buddhism is Nihilistic often stem from this point — yet this is based on the misconception that emptiness is a viewpoint.

Infinity and Emptiness

Bringing it all together — we see that the power set is a way to endlessly generate bigger and bigger sets. Similarly, our mind can continually metacognize without end. Both of these ‘endless’ processes can be stopped if we undercut them on first principles.

How to stop the infinite generation of sets? We simply reject the axiom of infinity. Our initial notion of infinity is axiomatic — meaning it is postulated and assumed to be true by definition. If we rid ourselves of this axiom, no matter how much we apply to power sets to finite sets, we will never reach Aleph Null (the smallest infinity) again.

How to stop the infinte generation of thoughts? Stop the process by recognizing the empty nature of any thought — change from focusing on the content to the quality.

I’ll leave you with a quote on the emptiness-nature of the mind.

“Although mind is the source of the entire spectrum of experience, if we look for mind itself we cannot find it. Mind’s actual nature is emptiness. We cannot point to any substantial entity and say, “That is mind.” At the same time, we cannot deny the phenomena that arise in the mind, the continual experience of emotions, thoughts, and perceptions. Mind’s nature remains beyond limiting concepts of existence and nonexistence” -Dudjom Rinpoche

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