This blog is about Simon, a young gifted mathematician and programmer, who had to move from Amsterdam to Antwerp to be able to study at the level that fits his talent, i.e. homeschool. Visit https://simontiger.com
This is the fourth video in Simon’s short series Infinities Driving You Mad. In this episode, Simon attempts to start to comprehend indescribable numbers. To Simon’s knowledge, no one has ever made a video about indescribable numbers on YouTube before. Simon is planning to record a follow-up to this video, something like part 4b.
This is the second part in a series of four videos that Simon is recording about Infinities Driving You Mad (on Set Theory) and is devoted to ordinal numbers. If you would like a little more explanation about what ω-one is, please see this short footnote video where Simon explains in more detail how he moves from the first infinite ordinal ω to ω-one:
This is the first part in a series of four videos that Simon wants to record about Infinities Driving You Mad. Don’t worry, you won’t go mad just yet! The first video is about cardinal numbers, enumerable infinite sets and Aleph Null. Simon also shows Georg Cantor’s proof of why real numbers are not enumerable and explains what Continuum Hypothesis is about.
Earlier Simon told me about the Continuum Hypothesis, that states that there’s no infinity between the size of the natural and the real numbers: “There is no proof for it. It’s what I like to call a superposition problem: the answer is both yes and no. We do know the answer but the answer is that we son’t know the answer. You can choose what you want the answer to be and the mathematics will still be consistent!”
Warning: The next part may make your mind overheat as Simon will hop over to ordinal numbers.
Simon showed me an interesting paradox that’s difficult to wrap my mind around. If you slice a cone (at a random height), the section is a circle. The chopped-off part (a small cone) also has a circle as its base. Are those circles equal? They are the same, because they result from the same cross section. Hence the difference between them is 0. Now imagine slicing the cone an infinite number of times. “The difference between the circles will come up an infinite number of times: zero times infinity”, – Simon explained. “But zero times infinity has no value (or has any value, it’s indeterminate). Zero times infinity is the same as infinity minus infinity, which means that it can be whatever you want. Riemann’s rearrangement theory makes this true.”
Today Simon learned that it was Euler who first came up with the idea that the infinite sum 1 + 2 + 3 + 4 +… converges to -1/12. Simon explained to me the other day that there are several ways of looking at an infinite sum. One way is looking at its partial sums and summing those up. Another way is averaging partial sums and see what their average converges to (or what the average of their averages converges to). “That’s called Cesaro Summation and it’s good for closely related sums like 1 – 1 + 1 – 1 + 1… but not for 1 + 2 + 3 + 4 +…”, Simon explained. “Then there is Ramanujan Summation – a Calculus way of looking at infinite sums using derivatives and gamma, etc. That is the only way 1 + 2 + 3 + 4 +… converges to -1/12. All possible infinite sums converge if you use Ramanujan Summation.”
“Simon, you don’t trust Ramanujan Summation, do you?” I asked.
“No. Only an infinitely small section of infinite sums converge using the standard method. Converging means it settles down. That’s what we call a fixed point. If an infinite sum doesn’t converge, it can either explode to infinity or it can have more than one fixed point or do something else weird like that. Sums that are not convergent are called divergent.”
“And Ramanujan had none of those, no divergent sums?”
“Yeah, he really made it to the extreme! It’s an infinitely large extreme. All infinite sums Ramanujan-converge.”
For Simon and me, this book (“Infinite Lives of Maisie Day” by Christopher Edge) has probably been one of our most profound experiences of the year. We read it together, sometimes, giggling with joy as we recognized Simon’s favorite topics interwoven in the plot (like that the main character also dreams of proving the Riemann hypothesis), and sometimes tears choking our throats as we went through the sad and scary bits of the story. And what a trip down the memory lane last night, at the Royal Institution in London, where we attended a lecture about the science behind “Infinite Lives of Maisie Day”! As Simon proudly told one of the lecturers (University College London’s cosmologist Dr Andrew Pontzen) after the show, he even predicted something important in the book. Simon recognized that Maisie turned into a mirror image of herself after she had traveled around the Mobius-shaped universe, just as depicted in Escher’s “the impossible staircase” painting . “But that’s only possible if you’re flat, a 2D object! So it’s not correct in the book, but they probably put that in to make it simpler,” Simon laughed. “You’re absolutely right! Keep doing science!” the cosmologist told him. @Ri_Science
Simon: Mom, there’s no number that’s close to infinity.
I: There isn’t, is there? But even when I realise it’s true, I have trouble imagining this.
Simon: This is the way I imagine it: if you collapse an infinite length into some non-zero finite number, then all the numbers should be infinitesimally close together. That is, infinitely close together but not the same.