There is a song that has been stuck in my head since early childhood. It plays during the end credits of a 90s children’s show called “Lamb Chop’s Play-Along”. It goes something like this.

This is the song that doesn’t end.
Yes it goes on and on my friend.
Some people started singing it not knowing what it was.
And they’ll continue singing it forever just because
this is the song that doesn’t end…

The song plays along with a skit that depicts kids singing it to the torment of the single nearby adult. The adult’s only relief comes when the kids march off and eternally sing their song somewhere else. As a child, I was, and still am, interested in complete experiences. Finishing a thing was just as important as starting that thing. So a song that cannot be completed does stand out, and occasionally I, my siblings, or my friends would attempt to sing it to completion as well. However, there is something I learned very quickly that the skit doesn’t portray. After leaving, the kids in the skit are assumed to be still singing their song. However, in my world the point of the song was to annoy the adults. I could use it as a tool to elicit a response, preferably a new response or at least change the status quo in an interesting way; however, once there was nobody left to hear it, all enjoyment of the song disappeared. Other looping songs, like “99 bottles of beer on the wall”, at least have endings. If you sing them long enough, your effort is rewarded by a final verse that changes in some subtle but important way. However, the same is not true for the song that never ends. It only takes a few cycles before there is nothing left to get out of the experience and the game becomes boring.

Even as a child, this song taught me that the reality of infinity is far less interesting than its concept. It’s just the same thing repeated ad nauseam without an end and without change.

Set Theory

Even though infinity is one of my favorite topics, I still find it awkward to write about it. Infinity absolutely falls into the category of “weird science” that is interesting to a general audience and generates mountains of explainer articles and videos. As well, I’ve already written extensively about infinity before as a quarter of my master’s dissertation is dedicated to it. However, there are subtleties to infinity that generally don’t make it into explanations geared towards a general audience, because such subtleties rely on much deeper understanding of the foundations of mathematics. Understandably, this doesn’t make for light reading and is often skipped over. However, these subtleties are exactly what I want to talk about today, even though I will freely admit I don’t fully understand them myself.

Every couple of years, I pull out my copy of “Set Theory and the Continuum Problem1” and try to understand the proof the book builds toward. The continuum problem is one of the more interesting mathematical theorems that I encountered during my undergraduate degree, and I’ve always wanted to understand it better. The Continuum Problem is an open problem in mathematics that speaks to our understanding of infinity. Mathematicians have discovered, and described, at least two types of infinity: countable infinity, the infinity that is created when you have a process of discrete steps, like counting, with a beginning but not an end. And uncountable infinity, the type of infinity that comes into being when you can infinitely divide an object into immeasurable pieces2. We can show that uncountable infinity is larger than countable infinity, but we do not yet know what, if anything, lies between these two values. Gregory Cantor, the mathematician who normalized using infinity as a valid mathematical number, hypothesized that no such value could exist but could not prove it. His hypothesis has come to be known as the Continuum Hypothesis, and the larger problem called the Continuum Problem. The issue is that the Continuum Problem does have an answer, although that answer is far more complicated than just simply proving or disproving the Continuum Hypothesis. This complexity fascinates me, but also eludes me, as I need to come back to the book year after year.

Part of the problem rests with set theory itself. It is the language within which the continuum problem is commonly defined, yet it resists metaphor in ways that make all other mathematical frameworks jealous. Sets are not, as the choice of word might lead someone to believe, a collection of objects. Sets are instead defined recursively, they are collections of other sets which are themselves collections of other sets. Likewise all of their properties are simply references to possible ways they could reference themselves. Even seemingly simple things like ‘order’ are properties of sets that need to be proven. Some sets have order, but not all of them.

For example, the set theoretic representation of the number 5 is the set that contains all previous numbers. So 4 = {3, 2, 1, 0} where zero is the set that contains nothing; however, all of those other numbers are themselves sets that contain all previous numbers. Expanding this out we arrive in the bizarre reality where the number five is an object that contains the empty set sixteen times at various depths which looks something like this.

4 = {{},{{}},{{},{{}}},{{},{{}},{{},{{}}}}}

However, this representation is not correct as the left to right nature of our writing implies an order to the above elements that simply does not exist. The following representation is just as correct as I can reorder any portion of if without changing its meaning.3

4 = {{{{}},{}},{{}},{{{}},{{{}},{}},{}},{}}

Order only enters the picture when we specifically construct sets that have order. In the example, the number five contains all previous numbers. So we can define the relationship of being “less than” as the relationship between a set and other sets that contain the first set. So if 4 is contained in 5, then 4 is less than 5. However, we can just as easily talk about two sets that are not contained in each other and therefore cannot be considered “less than” one another. This, along with countless other definitions, can eventually be used to create the numbers as we know them, but at the cost of eliminating all the intuition that we would normally use to understand numbers. It only gets worse from there.

Theorem 2.3: if M is a g-tower then M is a nest, and moreover for any elements x and y of M, either g(x) < y or y = g(x) 4

Literally every word in that sentence is defined in the book, and is a very specific way sets can contain other sets. Any prior intuition about what those words mean will likely act as a barrier to your understanding of the actual theorem. Worse, the translation into language might actually make it harder for the layperson to understand what is happening here because these words contain meaning beyond what they are explicitly intended to convey. All this theorem is trying to say is that if a set adheres to a definition X, defined earlier in the text, then it also gains property Y, defined earlier in the text, and all elements in it are comparable under relationship Z, defined earlier in the text. Importantly, this theorem tells us nothing about sets as a whole, all it says is that if we build a set using a specific method then it gains properties from that method. Likewise, the only things we can know about an individual set are the things that they inherit from the processes that create them. If we do not understand these processes, then we cannot understand the resulting set. Set theory is the mathematical embodiment of reductionism.

The Power Axiom

The earliest success of set theory is that it gave us a way to talk about infinity in a precise manner. Sure, no human can count to infinity, but through set theory we can talk about objects like the “set that contains all natural numbers.” This set necessarily contains an infinite number of elements, and can be meaningfully compared to other sets that also contain an infinite number of elements.

Set theory states that two sets are the same size, or have the same cardinality, if all elements in one set can be put into 1 to 1 correspondence with all elements in the other set. In other words, is there a transformation that can match every object in one set to a unique object in the other, and vice versa. Say we wanted to compare a bag of three oranges with a bag of two apples. We can pull out a random fruit from each bag and set them down on the table side by side. We will repeat as necessary until one of the bags has run out of fruit. Once we are done, both apples will be paired with an orange, but one of the oranges will not have an apple friend. Thus, we say that there are more oranges than apples because the oranges cannot be put into 1 to 1 correspondence with the apples; all of the apples have friends, but one of the oranges does not. Importantly, this same logic can be used to compare infinite sets as well.

Say we want to know which set is bigger, the set of all natural numbers, or the set of all even numbers. Well, the set of all even numbers is just the set of all natural numbers multiplied by two. So ‘multiplication by two’ can act as a function that pairs values in the two sets.

1 <--> 2  
2 <--> 4  
3 <--> 6  
4 <--> 8

Since all numbers can be multiplied by two, and all even numbers can be divided by two we know that every element in both sets will have a pair and therefore both sets are the same size.5

Once we accept that infinity has a size, it’s natural to ask if there are other larger infinities? Gregory Cantor proved that there are indeed larger infinities. He tried to put the natural numbers into 1 to 1 correspondence with another set of mathematical objects, the real numbers, and found something truly unexpected. Through his famous ‘diagonal argument’6 he showed that the real numbers cannot be put into one to one correspondence with the natural numbers, and that any attempt to do so will necessarily leave some out. Thus he showed that the real numbers are larger than the natural numbers, and that infinity comes in at least two flavors.

Yet Cantor doesn’t stop there. Using a slight variation of the diagonal argument, Cantor proved that the ‘power set’ of a set is larger. The ‘power set’ of a set is the set that contains all possible subsets of a set. So the power set of the number three is all combinations of numbers below three.

P(3) = {{},{0},{1},{2},{0,1},{0,2},{1,2},{0,1,2}}

The power set is functionally the best definition of ‘everything’ that set theory is capable of. A power set is the set that contains all possible ways to organize its child elements. As well, the power set is a general operation meaning it can always be applied to a set, even an infinite one. So if we apply it to an infinite set we necessarily create a larger infinity. We can show that the cardinality of the power set of countable infinity is the same as uncountable infinity, and the power set of uncountable infinity is larger still. Iteratively taking the power set of each new infinity produces a countably infinite sequence of infinities: a process that never has to end.

Both the natural and real numbers have analogues outside of pure mathematics and have been the subject of discussion, mathematical, scientific, and philosophical, for thousands of years. Every infinity we knew about, prior to set theory, falls into one of these two categories. Yet, a power set is something new, and also something that exists only within set theory. The only property a power set has is a set theoretic sense of completeness. It is a ‘multiverse’ of possibilities because it is built with the fundamental assumption that everything that can exist within it does exist. However, it is also something that the axioms of set theory cannot construct, so in order to reason about a power set we first must force it to exist through an axiom.

A6 [Power set axiom] For any set x, P(x) is a set7.

Unfortunately, axioms are not things we can really understand; they are instead the building blocks we use to understand other things. The existence of the power set is an assumption of set theory, and as a trivial result of that assumption there are now an infinite number of infinities. However, as mentioned above, the properties of these infinities are entirely determined by the process that creates them, and the process that creates them is an assumption of set theory. Unfortunately, for anyone seeking a greater understanding of infinity, the power axiom is a dead end. I cannot say anything more about these objects because it cannot say anything more about itself.

The continuum problem speaks to this gap in knowledge. The power axiom can demonstrate that this sequence of infinities can exist, but that is all it can do. To move forward and see between these values we need a new perspective. Thankfully, set theory does have this. We can generate a second sequence of infinite values using different axioms.

The Axiom of Choice

Imagine a pair of shoes. They are a set that contains two objects: a left shoe and a right shoe. Now imagine a pair of socks. They are also a set that contains two objects; however, individually those objects do not have names they are just two socks. There is no left sock or right sock, and the only way to tell them apart is by their physical properties. Maybe there is a darker sock, or an older sock. Perhaps you grab the nearest sock, or the sock on the top of the pile, but in all cases you are always choosing based on some physical property beyond its simple existence as a sock. However, in a mathematically pure setting, like set theory, two socks may not have any physical properties. If I ask you to hand me a sock, and there is no way to tell them apart, how can you select one?

What is the smallest real number between 0 and 1? This question has no answer. A fundamental property of the real numbers is that there are always an infinite number of numbers between any two of them. So if we select a number, let’s say 0.1, then we can identify a number smaller than it, 0.01, that is still greater than 0. Thus we cannot use a statement like “the smallest number” as a way to choose a single number from the set of numbers between 0 and 1. Thus even having properties does not necessarily solve the problem of choice. In order to ‘choose’ an object that object first must have properties and those properties must make it stand out in some way. Put another way, in order to ‘choose’ an object we must first be able to assign it a name.

The problem with the real numbers is that most of them are ‘transcendental’ numbers, which means that I cannot use algebraic notation to single them out. In some cases, like e or pi, there might be other mathematical theorems that we can use to define them. Other numbers, like \(\sin(5)\), can be defined by their relationships to other numbers that we can assign names to. However, these are exceptions. Most transcendental numbers cannot be assigned names, and thus cannot be selected from a set.

This is actually a greater property of uncountable infinity. In order for something to be uncountable, we must first lose the ability to assign names to all of the elements8. Indeed, this is a huge part of why the diagonal argument works. Pairing elements in two sets is essentially using one set to assign names to another set; however, if one set, the countable numbers, can be assigned names and the other set, the uncountable numbers, cannot, clearly the attempt to pair them will fail. Inevitably there will always be some that slip through the cracks.

Put another way, in order to ‘choose’ an object we must first be able to assign it a name.

The problem of choice in set theory speaks to this issue. The axiom of choice states that a choice function will always exist. So while there might not be a least element between 0 and 1, there will always be a least X that a choice function can pull out. Then if we iterate X it will eventually become a sequence of numbers that has a first element. This sequence is called a ‘well-ordering’. Notably, the real numbers cannot be well-ordered without the axiom of choice. However, if we assume this axiom, then the well-orderings that suddenly appear cannot be expressed because such an expression would rely upon our ability to assign names to all the elements. So, any well-ordering of the real numbers that appears because we are assuming the axiom of choice are simply logical consequences of that assumption.

I bring this up because we can use well-ordering to generate another sequence of infinite numbers. As referenced in this numberphile video. The number of ways to well order a set, produces a larger set similar to how the power set of a set produces a larger set. Yet, instead of relying on the power set axiom, this new sequence relies on the axiom of choice. In theory, comparing these sequences should give us deeper insight into the logical structure of infinity, and by extension the continuum problem.

But how would we do this? Can we put any of the elements of these two sequences into 1 to 1 correspondence with each other? Well, unfortunately, no. Both sequences are entirely defined by the axiom that generates them, and they share no properties that would allow us to meaningfully compare them. We can’t even use the diagonal argument because the diagonal argument makes use of the fact that at least one of the sets being compared has a choice function that we understand. What even is a subset of an infinite set of transcendental numbers? What would a choice function of a power set of a power set even look like? At some point we would have to construct a countable infinite sequence of number without any names. How can we meaningfully categorize a type of well-orderings when the choice function that generated them cannot be expressed using language?

The issue with the continuum problem is that it relies on the assumption that mathematical truth is discovered and not created. Cantor begins with the assumption that set theory has allowed him to see into a space that already existed and thus questions like, “Is there anything between these two values?” have meaning. However, the continuum problem does have a proof, we have an answer. It’s just not a very exciting one.

In 1938 Kurt Godel proved, using logical arguments that are beyond my understanding, that the continuum hypothesis was consistent with the axioms of set theory meaning that set theory could never disprove Cantor’s hypothesis. Then in 1963 Paul Cohen finished the proof by showing that set theory could not prove it either. So in the end, the accepted solution to the continuum problem is that set theory cannot say one way or the other. The continuum hypothesis is independent of set theory, and infinities generated in different ways say nothing about each other. Essentially, we can, and we do, create any infinity we want at any size and assign it all the properties we need it to have; however, doing so tells us nothing about other infinities. It’s like assuming that ‘aliens’ exist and separately assuming that ‘unicorns’ exist. We can make those statements, but the fact that those statements were made tells us nothing about how these aliens and unicorns will interact.

Beyond Infinity

When I first learned about the ‘solution’ to the continuum problem I was disappointment, but I have since come to believe that it is, in fact, the most interesting way the problem could have been resolved. It should come as no surprise that the study of one mathematical subject can, at times, fail to produce results in another mathematical subject. Nobody studies chemistry to tease out deeper meaning in political science. Yet, if someone could come up with a chemical reaction that demonstrated, with mathematical rigor, that the chemical properties of hydrogen can tell us nothing about the nature of democratic governments, then that would be interesting. Why? Because it disqualifies chemistry as a valid reduction of political science. Superficially, all chemical elements come from fusing hydrogen molecules to other hydrogen molecules. This chemical reaction produces energy which powers our sun. Biology is a chemical reaction that uses the sun’s energy to produce other more complicated chemicals and organisms. Those compounds produce even more complicated chemical reactions that evolve slowly over time to produce animals and eventually humans and human society. Politics could be, in a reductionist sense, just a higher order byproduct of the chemical properties of hydrogen. Thus, a proof that hydrogen alone doesn’t explain human behavior is a proof that something else exists out there in the universe, something we can’t see if we are only looking at hydrogen. Something that could never be generated so long as we are wholly engrossed in the ‘limitless’ potential of hydrogen.

Set theory operates in the same way. For a period of time it was considered to be important to the foundations of mathematics itself. “Principia Mathematica (PM)”, a three volume treatise written by Alfred North Whitehead and Bertrand Russell, was intended to be a foundational mathematical text that uses set theory to reconstruct the entirety of mathematics. Through PM, set theory would be the mathematical language from which all other mathematical languages would emerge. Under PM, the limits of set theory become the limits of mathematics itself. Any theorem independent of set theory would necessarily exist beyond the limits of mathematics as well. So proving that the continuum hypothesis is independent of PM is also proving that that the continuum hypothesis is equally independent of any mathematical study under PM. Instead of infinity being an underlying constant of the universe, that set theory gives us a glimpse of, it is now just an arbitrary object that is completely defined by the assumptions we make about it.

Now, don’t get me wrong, this does not imply that modern mathematics cannot find another solution to the continuum problem, only that a mathematics defined by PM would not be able to. More specifically, that there is more to mathematics than what PM is capable of describing. It’s like being Yuri Gagarin, the first human to make it to outer space. Before we humans experienced space, our concept of the heavens was really just an extension of our concept of earth. We could argue about what orbited what, and how far apart things are from each other, but ultimately space is just more of the same of what we normally experience. If I travel in the direction of the sun long enough I would eventually reach it, just as traveling towards a mountain will eventually bring me to it. However, to be Yuri, to travel into space and experience the atmosphere thin and the earth shrink, even just a little bit, to witness the boundary between where our experience of space and location disappear and become something else, something alien. An experience that really brings home the concept that space is different from earth and not just more of the same. We could if we wanted to continue Yuri’s trip and travel upwards forever as our current understanding of space does not require that that trip have an end; however, the continuum problem demonstrates why it is naive to think that that transition from atmosphere to outer space is the only time a threshold would be crossed. Infinity, is just our current experience repeated forever, what Yuri experiences is beyond infinity.

Conclusion

Had the continuum hypothesis been proven, or disproven, then infinity itself would become just another studied mathematical sequence. Which would only reinforce the biggest issue I have with infinity as a concept: the idea that it is understandable. I know for me, that if I spend too much time thinking about it, sooner or later things begin to feel normal, comfortable, understandable. I can start counting at one, and continue on for eternity. Each number follows the number before it. One, Two, Three, Four… each number leading into a different number. At no point do I produce something other than a number, and at no point do I either run out of numbers or come to a crossroad that I didn’t expect. Infinity of this type is just more of the same, continuing indefinitely without change. Everything there is to know about the process is summed up in the methodology that creates that process. To claim that something can grow infinitely, is to claim that everything to come is defined by a process we, at least are pretending to, understand. A song with no end still feels like it has an ending because I understand how it will continue.

Set theory is a model, and like all models its theorems are fully defined by the underlying assumptions that create it. However, those assumptions cannot generate infinite knowledge because sooner or later those assumptions lose the ability to speak to each other. If I were to build a weather model I would populate that model with the assumptions that more or less mimic my experience of weather and using these assumptions the model would tell me useful things about the world I live in. This is the scientific method. However, if we don’t continuously correct that model with new experiences of reality, we quickly end up in a world where the predictions of that weather model are completely determined by the model itself and not by our experience of reality.

Models built on today’s assumptions do not necessarily tell us anything interesting about tomorrow’s reality.

Set theory pretends to create an infinity of infinities, but in reality the only two infinities it can tell us anything meaningful about are the two that already existed. Once we ride the power set train through just two iterations, we end up with an object that looks exactly like every other object we have already seen. All we have demonstrated is that these higher infinities can exist, we haven’t actually learned anything interesting about them 9.

Yet our understanding of infinity is not confined to the mathematical world, it is deeply embedded in how we think about the future. In my world the term infinity is used to imply limitless potential. However, I’ve come to see it as the opposite. A world where the possibilities are ‘infinite’ is just a world where we are no longer capable of seeing the edges of what is possible. ‘Infinite’ economic growth is just our current society, but bigger. ‘Eternity’ 10 is just our understanding of time without the possibility of change. Technological ‘progress’ is just change for the sake of change. Each of these ‘infinities’ are completely defined by the methodology that created them, and do not allow for the possibility that we might discover something outside our current understanding. Models built on today’s assumptions do not necessarily tell us anything interesting about tomorrow’s reality.

Yet even this is the most optimistic definition. Infinity can also be a willful rejection of something visible, but unexplainable, inside an infinite model. Infinite economic growth requires infinite economic extraction on a world with only finite materials. However, ‘infinity’ lets us assume that other worlds are just more Earth, and if the universe is infinite then so are the materials within it. But what if they are not? What if Mars is just Mars, and not Earth? What if there are more thresholds between us and other earth-like planets than our current technological progression can cross? What if science has limits? What if human understanding has limits?

It may not be the path that everyone takes to get here. But the lasting philosophical impact of the continuum problem on my own personal philosophy is that infinity itself is not what we think it is. Instead of an invitation to endless knowledge, it is in fact a barrier to it. Set theoretic infinity is just a cardboard facsimile of knowledge11 positioned far enough away that we can see it, but can’t study it and are forced to fill in the details about it ourselves. We can never reach it, and we can never learn more about it. It is the illusion of knowledge because it allows us to believe that if we continue on as we have been going, then we might reach somewhere new.

Knowing that the ‘song with no end’ has an infinite number of versus simply prevents us from realizing that it doesn’t even have a second one.

  1. “Smullyan, R. M., & Fitting, M. (2010). Set theory and the continuum problem.” link ↩

  2. I have an old blog post about immesurability located here. It hasn’t aged as well as I would like because I make a lot of wide sweeping claims about Greek philosophy that I now know is not true. However, \(\sqrt{2}\) is still my favorite number and the proof I discuss in this post is still a fundamental concept in everything I have written since. So it’s still a great read if you can get past the cringe. ↩

  3. There are more, but listing them all out is pointless. ↩

  4. Smullyan, pp 54 ↩

  5. If we have a hotel with an infinite number of rooms, adding five more extra rooms does not increase the number of rooms we have. If new guests arrive at the hotel, even an infinite number of guests, we can always make room in our hotel by asking existing guests to move to rooms with larger numbers. I go into this at length in in the third chapter of my dissertation here. ↩

  6. The wiki article does a good job of explaining this argument for anyone interested. ↩

  7. Smullyan, pp 23 ↩

  8. I’m going to pitch my older article on \(\sqrt{2}\) again here. Once again, the problem of measurement is fundemental to all the wierd ideas that go through my brain. ↩

  9. Insert a giant rant about how useless string theory is here. ↩

  10. I’m referring to “an eternity in heaven with God” here. So this is me complaining about religion. ↩

  11. Part of me wanted to just drop the link to Ahnonay here and call it a day, but that reference might be too subtle. In the Myst games, Ahnonay is an age written to demonstrate the creator’s power over time. However, in reality the age turns out to be a “Truman Show” type universe complete with cardboard islands in the distance that hide the mechanisms that control the dome that rotates to make it appear like time has progressed. It has come to be, for me, a fictional example of the illusion of infinity. ↩