Last Train to Transcendental
Here again we return to infinity (if it can be said possible to return to something which one can’t leave). If you hadn’t heard of it before: there are different types of infinity – different sizes of infinity. This area of mathematics has fascinated and fixated mathematicians for literally millennia, though perhaps one of the most famous and prolific mathematicians to contribute to the area was a man named Georg Cantor. Cantor was very likely bipolar and spent a large chunk of his adult life feeling somewhat insane and persecuted by his peers. This last part wasn’t entirely due to the neurochemical roller coaster as some mathematicians were truly unsettled by his work and lashed out, like that schmuck Poincaré who said that Cantor’s set theory work was a ‘perverse illness from which someday mathematics would be cured.’
To be fair, what Cantor exposed for us does, on first blush, seem to make no sense, at best, and be contradictory, at worst. In this article we’re going to look at just a small sliver and, in that, find something bigger than the universe in which we live.
We’ll be using something you first saw in elementary school: the number line. Here’s one i’m referencing from the Wikimedia Commons:
For the moment, this has more information than we need to bother ourselves with, so just pay attention to the numbers -3, -2, -1, 0, 1, 2, 3. In math-ese, these are examples of what are frequently called the integers – but you may have also heard “whole numbers”. So how many of these integers can we imagine? Well, yes, an ‘infinite’ amount of them.1 The better questions are things like How big is this infinity? and Can this infinity be compared to the infinity of some other group of infinite stuff?
To improve the former ‘better’ question, we’ll start using a funky word for ‘size’ which is the math-ese word “cardinality”; so we could more formally phrase the former2 as What is the cardinality of this infinity? Now, to answer the latter ‘better’ question, and thereby the former, first we need to round up some other groups of infinite stuff. To start to do this, we’ll define a group of numbers that you grew up with knowing as “fractions”; in math-ese, we call these “rationals”. All of the numbers in this group are a number gotten by dividing any integer by any other integer that is not 0.
Cantor came up with an ingenious way to lay out all fractions in a 2 dimensional matrix such that for each integer, there is one listed fraction. Because we can do this “one to one” pairing3 between the integers and the rationals, this means that the integers and the rationals have the same cardinality, or size. This will be where we find the first bump in the weird world of the sizes of infinities, because the astute reader will notice that all of the integers are included in this list of rationals (2/1 = 2, 12/4 = 3, etc.) — and compounding that: more than once (6/3 = 12/6 = 5200/2600 = … = 2) — and yet the infinity which captures the rationals is the same size as the infinity which captures the integers.
While we casually refer to an infinity of this size as “countably infinite”, Cantor decided to start labelling the cardinality of different infinites using the first letter of the Hebrew alphabet, aleph — — and he labelled the cardinality of the infinity which captures the integers, and the rationals, as
– spoken as “aleph null”. The grand scheme being that subsequently bigger sizes of infinity would have an increasing subscript — so the next biggest infinity would be labelled as
(“aleph one”) and so forth.
Now let’s talk about the “real numbers”; the real numbers can be roughly thought of as any number which can be written as a decimal representation (like 3.500…)4. How big is the set of real numbers? Well, if just the integers are infinite, then the real numbers (which contain all of the integers, and rationals, and more) must also be infinite; but how infinite? It turns out that there is no way to do our ‘one-to-one’ trick from the integers to the reals. Cantor (again) presented a robust proof of this which is known as “Cantor’s diagonal argument” – which ultimately means that the real numbers must be of a greater infinity than the integers, et al.
This size of infinity is often referred to as “uncountably infinite”; Cantor, who was unsure whether there was some size of infinity that existed between the infinity of the integers and the infinity of the reals, could not label this infinity and so adopted a convention utilizing the second letter of the Hebrew alphabet (‘beth’) and called this infinity
(“beth one”).
Not only is it still unknown whether (a problem which is known as the “Continuum hypothesis”), it appears logically evident that we can never prove it nor disprove it.5
If you’ve become somewhat settled with the idea that the rationals, which contain all of the integers, and the integers themselves are groups of the same size infinity, then it shouldn’t be too much of a stunner to learn that the size of infinity of the group of all real numbers is actually the same size of infinity of a group for any continuous range of real numbers. What i mean by ‘continuous range’ is any continuous flow of real numbers between two points; using the number line above as reference, we can consider the group of every real number between 0 and 1 to be a continuous range [between 0 and 1]. Since this equivalence of infinity size is true for all continuous ranges of real numbers, we could then grab a continuous range within that 0-1 range and it, too, would be a group of the same size infinity as all of the real numbers.
This doesn’t translate at all well to what we experience in our physical world and so is often difficult to digest. For example, it would be pure Alice-world were you to operate a power shovel; dig your shovel into the ground and remove a basket full of dirt; rotate and dump that dirt into a pile; dig your shovel into just the center of the pile removing that; find that even though some of the pile is still on the ground, your basket is completely full again.
Ok. So if the real numbers are ‘bigger’ than the rationals (which contain the integers) what else is that stuff hanging out in the real numbers – that stuff which is so numerous as to make the real numbers uncountable? This is the stuff which we call the ‘irrationals’; the irrationals can be thought of as roughly being composed of ‘algebraic’ numbers and ‘transcendental’ numbers. It turns out that our ‘one-to-one’ tool works with the algebraic numbers and not with the transcendental numbers – so here is our real stuff.
Now that we’ve finally arrived at our station, what are transcendentals? Well, we know what some of them are – there are the famous ones shown on the above number line image, like ,
and
– but, truth be told, it can be quite hard to prove that a given number is transcendental and only a small zoo exists. I suppose you could think of them sort of as the ‘dark matter’ of the real numbers.
As an aside, when was finally proven to be irrational in 1882, it sounded the official death knell for a problem more than 2,000 years old called, ‘Squaring the Circle’. The earliest attempts to solve this was first described by the famously named Plutarch 500 years after it was done by Anaxagoras (who was cooling his heels in jail for one thing or another6 ). It was a welcome toll as squaring the circle had become such the past-time of math-crackpots that the Paris Academy (Académie des Sciences) had to take an official stand in 1775 and declare that none of its officials would be allowed to review papers claiming to address the solution of this (this, and two other favourites of the time: the trisection of the angle and the duplication of the cube).7
We can now finish here with pondering what all of this means when we talk about an “infinite” universe. In what we experience in the physical world and in terms of reality, the concept of an uncountable infinity definitely doesn’t correspond to our idea of physical space as we drill down. From every indication, there is indeed a smallest possible unit of physical space (quantum foam, for example, bumps down to an eensy weensy length of space referred to as the “Planck length“); since there is a smallest definable unit, and if we assume that the universe does expand out “forever”, we can pull out our “one-to-one” tool and see that the group of every smallest definable unit of physical space within the existing universe is countably infinite – its cardinality is .
Rephrased more loose and fast: what this means, in a final twist a weirdness, is that on our above number line the group which contains the continuous range of real numbers between only 0 and 1 has a greater cardinality than the group which is the entire physical space of the universe.
- A sloppy quick proof would be to pick the largest possible integer you can think of; now just add 1 to it and you have one even bigger. Repeat endlessly. [↩]
- mmmm, consonance [↩]
- math-ese: ‘mapping’ [↩]
- where, technically, there are an infinite number of digits after the decimal point; it could be an infinite number of 0s, in which case 3.00… would represent the same item as the integer ‘3’ [↩]
- This statement is too general, there are specifications which the reader can discover reading up on the CH… [↩]
- where the ‘one thing or another’ appears to have been that he attempted to explain to people that the Sun was not a deity but rather some physical entity [↩]
- “L’Académie prend en 1775 la resolution de ne plus examiner aucune solution des Problémes de la duplication du cube, de la trifection de l’angle, & de la quadrature du cercle…” — The last entry on page 2 in Volume 9 of “Mémoires de l’Académie royale des sciences”, published 1786, freely available here (archive.org rocks) [↩]
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Loki, you are most qualified to speak on this topic. But is there an idiot’s guide to problematic arithmetic? Love to own one! 😉