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This isn’t a traditional music video. As part of my new AV show I’ve been working on different ways of visualising the infinite. In collaboration with Martin Krzywinski, we set out to explore some of Georg Cantor’s ground-breaking ideas on different sizes of infinity.
What you see is an authentic numerical rendering of Cantor’s work, and if you’re willing to spend the time reading about what each part shows it should provide real insight into some exotic ideas.
The video begins by counting the natural numbers 1, 2, 3, and so on. This list continues forever, but can be thought of as a single entity: the infinite “set” of natural numbers.
We next look at the set of integers (whole numbers including negative numbers), and pair naturals off with integers.This process is called a bijection. Two sets with a bijection have the same size, or “cardinality”. A set with a bijection to the naturals is considered “countable”.
Even though the cardinalities of the naturals and integers are infinite, they’re the same “kind” of infinity. This first (and smallest) infinity is called Aleph 0 (Aleph numbers and section labels shown top left).
Cantor's diagonal progression between fractions (rationals) and natural numbers is demonstrated next. We build up an infinite table of fractions and then apply his pairing function, which snakes across the table, to match up every fraction with a unique natural. A bijection. So, the cardinality of rationals is the same as naturals, and we see that the rationals are countable.
Our story of infinity now expands in scope to include uncountable infinite sets-those that are infinite but for which there is no bijection with the naturals. Cantor’s diagonal argument is visualized to make this proof by contradiction. First, we assume that there is bijection between the naturals and the “reals” (numbers with decimal expansions), and write a list of reals each assigned a natural number to count them. But we can see that whatever our countable list of real numbers contains, we can always change one digit of each member of the list and work through them all diagonally, to construct a new number which is not on the list. This proves that bijection between the naturals and the reals cannot exist, and the reals, the numbers with (infinite) decimal digits, are uncountably infinite - a bigger type of infinity.
There are many sets that, like the reals, are larger than the naturals. We can use the naturals to construct one such set: the power set, which is the set of all possible combinations of natural numbers. We build up the power set by sampling from the first few naturals-a process that rapidly explodes in complexity.
The size of the set of reals, the so-called cardinality of the continuum is the same as the size of the power set of naturals. But we don’t know if any other sizes of infinity exist between the countable naturals and the uncountable continuum of the reals. This question is settled via the “Continuum Hypothesis”. If it’s true, then the cardinality of the continuum is Aleph 1, which is the next smallest infinity after Aleph 0. But because we don’t know whether the Continuum Hypothesis is true, all we can say is that the cardinality of the continuum is equal to or larger than Aleph 1.
In fact, the Continuum Hypothesis is apparently formally undecidable and our mathematics can work regardless whether it is true or false. Each assumption leads to different and contradictory-but internally consistent-outcomes. There is no doubt that this formal undecidability of the continuum hypothesis led to bouts of anxiety and instability in the minds of its early pioneers. Imagine working hard to prove something is true one day, only to prove that it is false the next.
We go past Aleph 1 and reach the lofty infinite heights of Aleph 2, which we visually show by power sets of reals, whose cardinality is Aleph 2 if the Continuum Hypothesis is true (as we assume it is for the animation). We can keep going to Aleph 3 (power sets of power sets of reals) and beyond, but Aleph 2 seems to capture the basic incomprehensible nature of the whole thing for me, and musically I maxed out my distortion chaos just getting to Aleph 2 so I had to end there!
That may sound a little impenetrable explained so briefly, but the point is that the essence of the techniques which put the infinite onto firm mathematical grounds by Cantor have been visualised. And they form their own equally intense aesthetic for storytelling in the live show context. It’s annoying it came out looking a bit Matrix, but there you go, that’s what it looked like, and the idea had to be shown as clearly as possible.
For a more quantitative explanation of all of this, see Martin’s pages at: mkweb.bcgsc.ca/infinity/math.o...
www.yearningfortheinfinite.net
29 окт 2019
