The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's Lemma?

Jerry Bona

I sometimes speak to people who reject the axiom of choice, or who say they would rather only accept weaker versions of the axiom of choice, like the axiom of dependent choice, or most commonly the axiom of countable choice. I think such people should stop being silly, and realize that obviously we need the axiom of choice for modern mathematics, and it's not that weird anyway! In fact, it's pretty natural.

So what is the axiom of choice? The axiom of choice simply says that given any indexed collection of (non-empty) sets, you can make one arbitrary choice from each set. It doesn’t matter how many sets you have — you could have an uncountable number of sets. You can still make a choice from all of the sets, saying “I’ll take this element from the first set, that element from the second set, this element from the third set…” and so on.

The axiom of choice is the only explicitly non-constructive axiom of set theory[1], and for that reason, in the 1920s and 1930s, it was [...]

---

Outline:

(01:32) Why do people reject the Axiom of Choice?

(01:56) Why paradoxes are not that bad

(02:35) Banach-Tarski Paradox

(05:21) Vitali sets

(12:02) Why we should keep the axiom of choice?

The original text contained 11 footnotes which were omitted from this narration.

---

First published:
December 13th, 2025

Source:
https://www.lesswrong.com/posts/t2wq3znQuPP66DpMT/the-axiom-of-choice-is-not-controversial

---

Narrated by TYPE III AUDIO.

---

Images from the article:

Two-panel comic: dog in burning room says

Apple Podcasts and Spotify do not show images in the episode description. Try Pocket Casts, or another podcast app.

“The Axiom of Choice is Not Controversial” by GenericModel

LessWrong (30+ Karma)

“The Axiom of Choice is Not Controversial” by GenericModel