The Continuum Hypothesis: You Can't Count for Shit

Warning: TL;DR

In an effort to write more, and to watch SportsCenter reruns less, I’m beginning a series of blogs which will (hopefully) present an understanding of The Continuum Hypothesis and its “solution” to the lay person. The Continuum Hypothesis is suitable for a number of reasons. Most importantly it’s a result primarily in logic, so it requires little formalized mathematics, which can be stated fairly easily. It’s interesting to me because it does not resemble math and has absolutely no application to anything in the corporeal world.

Mathematics’ ability to force one to reexamine, and frequently discard, long held assumptions about the nature of the world interests me immensely. It’s one of the reasons I find the discipline so fascinating. For example consider the following, if you have two groups of things, group A and group B, such that group A contains everything in group B and more, then you would say that group A must be bigger than group B. As young children we deduce this, and it serves us well throughout our lives. If I draw two circles, one surrounding the other, then the surrounding circle must be bigger. If you have every single comic book that I do, plus a collector’s edition that I don’t have, then you have more comic books than me. The logic seems to check out. In general though, this assumption is unfounded.

The give us some minimal technical jargon, mathematicians like to call any group of stuff a set. If something is in a set, we simply say it is an element of the set. If set A has everything that set B has in addition to something that set B doesn’t have, we say that B is a proper subset of A or dually that A is a proper superset of B. To be concrete we could say set A is the group of numbers {1,3,5,8} and set B is the group {1,5,8}. B here is a proper subset of A. You don’t care about what grey bearded aloof professors call things though. I just dropped a bombshell on you. You want to hear more about that.

Where did the logic go wrong? Well as children we did something that makes every mathematics professor cringe and every beginning mathematics student bomb a test he thought he passed, we made a tacit assumption. The assumption probably seems good as children. It probably seems pretty good to most adults. We assumed that we can count everything and give it a number. In the above I can say that set A has 4 elements. It seems like a good bet that we can always do this, but unfortunately we can’t. When we can, our childhood logic holds true. The “proof” goes like this: Set A has everything that set B has. I can count the things in set B. Set B has x elements. Set A has things that set B doesn’t have. I can count those things. Set A has y things that set B doesn’t have. Therefore set A has x+y things. x+y things is more than x things. So set A is bigger than set B. So sets are always bigger than their proper subsets. Sometimes though, this breaks down.

A common breakdown in logic occurs when we attempt to jump from familiar finite cases, which we test through our daily experiences, to the world of the infinite, of which it seems we have a poor understanding. Suppose we had the set of numbers, called the counting numbers (N), {1,2,3, …} and the set of numbers, called the integers (Z), {…, -3, -2, -1, 0 , 1, 2, 3}. Which one is bigger? By our faulty logic, Z contains all of N, plus a bunch of stuff N doesn’t have, so that Z is bigger. But we can’t say this. We don’t really know how big either of them are. How many things are in N? The way we usually count doesn’t work here. We can’t assign a usually number to the size of N. We can’t physically count all of them up. The same applies to Z. The “proof” used earlier to verify our logic breaks down here. How many elements are in N and Z? How many more elements does Z have? These questions can’t be sufficiently answered. There’s no number to assign to the size of each.

Let’s bypass this problem. Instead of worrying about giving a name to the size of each, let us just find a way to say if two things are the same size. Consider the following thought experiment. We have a classroom of students. There’s some number of kids in the classroom. There’s some number of desks in the classroom. I want to know if I have the same number of students as desks. If I could count each up, this would be an easy exercise. However, we want this method to work for things infinitely big so we can’t do that. I propose a different method. I’ll take each kid and assign him to a desk. I’ll assign every single kid to one desk and only one desk. I will assign no two kids to the same desk. If at the end, I see that every desk is filled with no kids left, I can say that there is the same number of kids as desks. Simple right? Take a moment to make sure you agree that this method captures the essence of two things being the same size. Make sure you see the importance of the fact that every kid gets only one desk, every kid gets his own desk, and every desk is filled.

Here’s some more jargon to impress the ladies at the bar. A function is a way of assigning things from a set A to exactly one thing in another set B. In the above, me assigning kids to desks was a function from the set of kids to the set of desks. If no two kids share a desk, that is if a function sends everything to a different place, it is called injective. If every desk is filled by a kid, that is if a function ensures that everything in set B has something from set A sent to it, the function is called surjective. If we have an injective and surjective function from what set to another, we can say that the two sets are the same size. Now back to the narrative.

Now that we have a way to compare sizes, what can we say about those two sets that previously eluded our examination? Can we say that Z is bigger than N? The shocking answer is no. These two things are actually the same size. Cool huh? To see how, we’ll start assigning students to desks. The elements of N will be our students, and the desks here will be the elements of Z. I’m an idiosyncratic teacher so here is how I tell the students to get in their desk. I’m going to tell student 1 to sit in desk 0. Now for every other student who has an odd number, I’m going to have him sit in the desk that is half than one less of his number ( [x -1]/2). In this way student 3 goes to desk 1, student 5 goes to desk 2, and so on. Now for all those students with even numbers, I’ll have them sit in the desk which half of his number and negative (-x/2). In this manner, student 2 goes to desk -1, student 4 goes to desk -2, and so on. With this method of assigning seats, I’ve given every student a place to sit and only one place to sit. I’ve also done it in such a way that every desk will be filled. Therefore, contrary to popular belief, the two sets are the same size even though Z has infinitely many things in it that N does not. Behold the power of arcane mathematics.

Objection! Objection! says the weary observer. I claimed that every “desk” needed to be filled. How can that possibly happen here? It is true that in physical reality we can never have every single desk filled in this manner. However, in reality the sets themselves couldn’t exist. It should be remember that infinity is really just a concept. You could never actually have every number in N. Instead I would say that for any number you can think of, N contains a number that is bigger. In the same way this desk assignment process, think of any number of desks that you wished to have filled. The process detailed above will fill more than that if you give it enough time. It we imagine that this process could carry on forever, in this imaginary land of infinity I would actually fill up all the desks.

Let’s briefly go back to my method of assigning desks. What did I actually do? Since I was assigning the number {1,2,3, …} to each desk labeled by Z, in reality what I was actually doing was counting the desks. Imagine the Z desks were lined up in order. You are standing at the zero desk with the positive desks extending forever to your right and the negative desks extending forever to your left. What I described was a way to count these. My method told you to first point at the desk right in front of you and count it as one. Then you were to go to the first negative desk and call it two. Next we go to the first positive to call it 3. To finish counting them all, you just need to keep alternating between positive and negative as you count. For this reason, sets that are the same size as N are frequently called countable sets. Notice that the way that I counted was very important. If I just attempted to first count all the positive desks, and then count all the negatives, I would never have been able to count everything. When calling something countable, or when showing two sets are the same size, we only need to find one way to do the assigning. If there are billions of ways that we can’t count the set, but just a single way that we can, the set is still countable.

The German mathematician Georg Cantor introduced these concepts in 1874. The mathematical community at the time violently reacted to his work. Unfortunately, scientists conduct themselves no differently than the average man. They are biased, selfish, and blinded by their beliefs. His work gradually gained acceptance after a period of ridicule. Georg Cantor spent the final days of his life driven to the brink of insanity pondering the extensions of this simple idea of counting. He died in a sanatorium consumed by what became to be known as the Continuum Hypothesis.

To fully introduce this idea, next time we will ask whether there are sets that cannot be counted. Intuition begs us to call this question absurd. As we have seen though, intuition has no place in the realm of the infinite.

The Solution to our Education System : There is none.

This post begins a series that I will pen (in theory only) to reveal the superficialities of my theory of pedagogy. I’ll point out what is wrong with the American Education System, as I now find myself to be firmly a part of it. My overall belief is that the system will never be fixed. To do so would require that many rich and powerful people losing their money and prestige, common folk accept extremely discomforting realities and the education system attempts to educate children rather than adhere to its historical purposes.

Education progresses cyclically. Every two decades or so a new revolution in reaching occurs which quickly claims to eradicate our education woes. The zealots of the movement cite some loosely related, over-reaching articles drawn from the social sciences to give credence to the unstoppable force that is this new theory. The fact that this approach to education mimics a previous failed approach never gets mentioned. Millions of dollars are spent in training, curriculum reform, textbook changes, and new school projects to accommodate the “new” untested solution which has won over a select group of bureaucrats and politicians, forced to make some change as inaction spells political suicide. Another massive failure reveals itself as false hope subsides, and another faux solution rises from the ashes of previous ineptitude.

Humans have been able to accomplish an awful lot. We’ve tackled seemingly more difficult problems than the question of how to properly educate our children. To invoke the mantra of every complaining S.O.B. on the face of the Earth, “We can send a man to the moon, but we can teach Johnny to add fractions?” In the face of such monumental failure to solve a problem well within our means to solve, it helps to examine the underlying axioms that theorists use. For centuries, astronomers were unable to explain the precession of the perihelion of Mercury despite the extraordinary success of celestial mechanics. The problem, as Einstein eventually revealed, was that everyone operated with the innocuous and friendly assumption that time is an unyielding constant. Bad assumptions give you junk results every time. The educator’s assumption is just as seemingly benign as the physicist’s. Educational theorists began with the axiom, “There is a solution.”

Unfortunately, there isn’t.

Malcolm Gladwell touches on this same point in a TED talk (here if you’re interested). He talks about sauces, but the overarching principle remains valid. Social scientists have a nasty habit of treating people like physical scientists treat control populations: heterogeneous and indifferent to the act of observation and experimentation. However, an individual’s subjective evaluations vary remarkably from person to person. The population of people to be educated gives a myriad of differing curiosities, strengths, weaknesses, educational levels, social pressures, desires, career paths, learning styles, and reactions to stimuli. We’ve been operating as if we’ve failed because we’re trying to jam a round peg into a square hole. We could succeed if only we could find the right dimensions to carve. In reality, we’ve been trying to jam thousands of fractal images into neat polygons that we carve one at a time.

With this idea in mind, centralized academic control is a cancer to society. This isn’t a Republican talking point. I’m not suggesting that we just need to abolish the Department of Education, offer a voucher system, or let kids pray in the classroom. The issue is much, much deeper. This is an admonishment of private schools as much as it is of public schools. Returning education to State control would allow for some heterogeneous products. However, State’s are still behemoth sized beings. Most families do not have the fluidity necessary to move across State lines just for a set of educational standards. Add to it the fact that these would still be at the whim of feeble minded, reactionary bureaucrats and this is even less of a feasible solution.

To truly provide the diversity of instruction methods to meet the abound diversity of the individual, all forms of regulation on the delivery of knowledge must be eradicated. Education cannot exist alongside academic standards. All forms of (governmental) accreditation and certification must be thrown into the dustbin of history. Along with them must follow the laws which dictate mandatory attendance in institutions of learning.

This is the first step towards a coherent system of education with this one being more of an organization than a pedagogical nature. Nobody has the correct answer for how to educate the nation. The answer doesn’t exist. However, methods can be devised to educate the person, but the mechanics which will do so cannot operate bound to a small group of individuals’ preconceived notions of what is right or proper. This is not a free market argument. The nature of the individual dictates this regardless of what economic system is actually ideal.