Cantor’s diagonal argument, and other ways to unravel infinities

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Daily Campus Columnist, Sharon Spaulding, gives a math lesson as she explains "Cantor's Diagonal Argument", and how the lessons it teaches can be applicable to our own perceptions. Photo by Andrea Piacquadio on Pexels.com

There’s a neat trick in math called Cantor’s diagonal argument. It was one of the ways which proved, in a sense, that there are different “sizes” of infinity. Intuitively, it makes sense that if we only think about whole numbers like 1, 2, 3, 4 and onwards, there would be “less” of them than if we were to think about all the numbers, including fractions, negative numbers and other numbers like pi — but how would we actually prove this, since it’s impossible to actually write out two infinite lists of numbers?  

Cantor’s diagonal argument answers that question, loosely, like this: Line up an infinite number of infinite sequences of numbers. Label these sequences with whole numbers, 1, 2, 3, etc. Then, make a new sequence by going along the diagonal and choosing the numbers along the diagonal to be a part of this new sequence — which is also infinite. Then, swap out every single number in this new sequence for a different number. Now, you have a sequence that is different from every single one of the other sequences you had before — this final sequence has a different first number than the first sequence, a different second number than the second sequence, and so on. But wait! We started out with an infinite number of sequences, which we labeled with all the whole numbers, and now, we just created a new sequence that isn’t one of those other sequences — so how could this sequence even exist, if we already listed an infinite number of sequences? This brings us to the conclusion that “infinite” doesn’t necessarily mean “all”, in the sense that when we wrote down an infinite number of sequences in the beginning we didn’t necessarily write down all the possible sequences — and so, there must be a “larger” infinity than we had originally. 

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Cantor’s Diagonal argument brings us to the conclusion that “infinite” doesn’t necessarily mean “all”, in the sense that when we wrote down an infinite number of sequences in the beginning we didn’t necessarily write down all the possible sequences — and so, there must be a “larger” infinity than we had originally. Photo by Philippe Donn on Pexels.com

I encountered Cantor’s diagonal argument in another proof a few months ago. The goal of the proof was to show that if you have a bunch of sequences that are of a certain type, you can find a subsequence from them that is “nice” in certain mathematically relevant ways. In the proof, as we did above, the sequences were all lined up, and labeled with whole numbers. Then all of sudden, Cantor’s diagonal argument is applied, and you end up with the exact subsequence you wanted. It was aggravatingly magical.  

Cantor’s diagonal argument is neat because it provides us with a clever way to confront infinities which can’t be avoided. Infinities are present in other areas of life, too — in systems of innumerable levels of connection, and in circumstances with unfathomable magnitudes of consequence. And despite the challenge of encapsulating them with familiar tools of expression or understanding, as actors in these systems and circumstances, we must, in some way, confront these infinities, which most often present themselves as questions rather than answers: 

“Cantor’s diagonal argument is neat because it provides us with a clever way to confront infinities which can’t be avoided. Infinities are present in other areas of life, too — in systems of innumerable levels of connection, and in circumstances with unfathomable magnitudes of consequence.”

How do we make sense of pervasive injustice in societal systems — particularly when we are inside those systems? How do we even begin to properly see this contamination when it is obscured by cultures and histories which we are parts and extensions of?  

How do we resolve the logical inconsistencies of our own imaginations to truthfully encapsulate the vastness of humanity? 

How do we become consistently attentive and dutifully reactive to the limits and provisions of our geographies? 

How do we render our personal awareness of beauty and fallibility — despite limited understandings of depths and probabilities — into political opinions that are diligently responsive to life in worlds apart from our own?  

Cantor’s diagonal argument instructs us that although we might not be able to fully understand the scope of infinity, we can puzzle over its properties and consequences. The solution to dealing with infinities in Cantor’s diagonal argument is not to avoid infinities altogether — it’s to realize that what we’re trying to prove isn’t dependent on enumerating infinity, only on our own willingness to unravel it in the most carefully bold and truthful ways. And so we must. 

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