The Unknowability of the Number Line


My Favorite Theorem guest Skip Garibaldi. Credit: Skip Garibaldi

On this episode of our podcast My Favorite Theorem, my cohost Kevin Knudson and I had the pleasure of talking with Skip Garibaldi, a mathematician at the Center for Communications Research in La Jolla. You can listen to the episode here or at kpknudson.com, where there is also a transcript.

Dr. Garibaldi decided to talk about a theorem he calls the unknowability of irrational numbers. Many math enthusiasts are familiar with the idea of countable versus uncountable infinities. A collection of objects is countably infinite if you can list the objects so you know where each one is on the list and what comes next. The prototypical example is the set of counting numbers. Other sets of numbers, such as prime numbers and rational numbers, are also countable because with a little ingenuity they can be listed. 

The set of all real numbers—all points on the number line—is uncountable, as Georg Cantor proved using a beautiful argument called diagonalization. The basic idea is that any list of real numbers will be incomplete: if someone tells you they’ve listed the real numbers, you can cook up a number their list omits.

If the entire number line is uncountable, and the rational numbers are countable, then it stands to reason that the irrational numbers must be uncountable. (Otherwise, if both the rationals and irrationals were countable, you could count the real numbers by alternating between rationals and irrationals.) We can keep drilling down: the irrational numbers can be subdivided further. For example, an algebraic number is the solution to a polynomial equation. This opens up a lot of new numbers, like 2 and -1/31/12, but the set of algebraic numbers is once again countable. That means the rest of the irrationals (called transcendental numbers) must again be an uncountable collection of numbers. 

The end result is, in Dr. Garibaldi’s words, sort of hideous. Any classes of numbers you can describe explicitly end up being merely countably infinite. Even with heaping helpings of logarithms, trigonometry, and gumption, the number line is more unknown than known.

Dr. Garibaldi recommended a few resources to learn more about this idea, which is sometimes called something like describable or closed-form numbers. One is Timothy Chow’s article What Is a Closed-Form Number? and one is Closed Forms: What They Are and Why We Care (pdf) by Jonathan Borwein and Richard Crandall.

On each episode of the podcast, we ask our guest to pair their theorem with something: food, beverage, art, music, or other delight in life. Dr. Garibaldi chose the TV show Twin Peaks as his pairing. You’ll have to listen to the episode to learn why it’s the perfect accompaniment for our vast ignorance of the world of numbers.

You can find Dr. Garibaldi at his website. In addition to books and articles about his research, he has written about the mathematics of lotteriesSome people have all the luck and Finding Good Bets in the Lottery, and Why You Shouldn’t Take Them.

You can find more information about the mathematicians and theorems featured in this podcast, along with other delightful mathematical treats, at kpknudson.com and here at Roots of Unity. A transcript is available here. You can subscribe to and review the podcast on iTunes and other podcast delivery systems. We love to hear from our listeners, so please drop us a line at myfavoritetheorem@gmail.com. Kevin Knudson’s handle on Twitter is @niveknosdunk, and mine is @evelynjlamb. The show itself also has a Twitter feed: @myfavethm and a Facebook page. Join us next time to learn another fascinating piece of mathematics.

Previously on My Favorite Theorem:

Episode 0: Your hosts’ favorite theorems
Episode 1: Amie Wilkinson’s favorite theorem
Episode 2: Dave Richeson’s favorite theorem
Episode 3: Emille Davie Lawrence’s favorite theorem
Episode 4: Jordan Ellenberg’s favorite theorem
Episode 5: Dusa McDuff’s favorite theorem
Episode 6: Eriko Hironaka’s favorite theorem
Episode 7: Henry Fowler’s favorite theorem
Episode 8: Justin Curry’s favorite theorem
Episode 9: Ami Radunskaya’s favorite theorem
Episode 10: Mohamed Omar’s favorite theorem
Episode 11: Jeanne Clelland’s favorite theorem
Episode 12: Candice Price’s favorite theorem
Episode 13: Patrick Honner’s favorite theorem
Episode 14: Laura Taalman’s favorite theorem
Episode 15: Federico Ardila’s favorite theorem
Episode 16: Jayadev Athreya’s favorite theorem
Episode 17: Nalini Joshi’s favorite theorem
Episode 18: John Urschel’s favorite theorem
Episode 19: Emily Riehl’s favorite theorem
Episode 20: Francis Su’s favorite theorem
Episode 21: Jana Rordiguez Hertz’s favorite theorem
Episode 22: Ken Ribet’s favorite theorem
Episode 23: Ingrid Daubechies’s favorite theorem
Episode 24: Vidit Nanda’s favorite theorem
Episode 25: Holly Krieger’s favorite theorem
Episode 26: Erika Camacho’s favorite theorem
Episode 27: James Tanton’s favorite theorem
Episode 28: Chawne Kimber’s favorite theorem
Episode 29: Mike Lawler’s favorite theorem
Episode 30: Katie Steckles’ favorite theorem
Episode 31: Yen Duong’s favorite theorem
Episode 32: Anil Venkatesh’s favorite theorem
Episode 33: Michèle Audin’s favorite theorem



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