Papers unpicked: Chaotic Orderings of the Rationals and Reals

This video gives an overview of the ideas and proofs found in "Chaotic orderings of the rationals and reals" (Hayri Ardal, Tom Brown, and Veselin Jungić). This proves that there is a total ordering of real numbers without monotonic 3-term arithmetic progressions, thereby answering a question by Paul Erdős and Ronald Graham.

Music by Arulo (Mixkit)

[arongil]

Hi, I'm so happy I stumbled on your "Papers Unpicked" series. They are of a superb quality! Your visuals complement a genuinely well conceived lesson about research-level mathematics. I hope the YouTube algorithm directs many more viewers here soon. Fantastic work, again. Thank you very much for making these for lucky viewers like me!

[moocowpong]

König’s Lemma is one of my favorite tools in mathematics. Every time I see it used, it makes me happy.

[FineDesignVideos]

This was really beautiful, thank you so much for making it!

[tonatiuhm.wiederhold]

This is really high quality, please continue making these!

[samueljosephs]

This was great, cannot wait to see your future videos!

[WindyNight]

Found this series after I watched your Angel and devil video. This is so well done! I’m looking forward to seeing more of them!

[Omeomeom]

THANK YOU, local meadows, FOR Papers unpicked:. This channel is a gem and a true feat of humanity.

[camiloariasabad]

This is great! Please keep making videos.

[omhd]

Really nice explanation, thank you for sharing!

[shadamethyst]

Excellent video!
I spotted a small typo at 6:35 but I'm impressed by the overall quality

[viliml]

Is it possible to calculate the chaotic ordering of the rationals? That is, is there a finite algorithm to calculate the ordering of the first N rationals that is certainly included in the infinite branch?
Also, the axiom of choice is stupid.

[hassanalihusseini]

Really high quality video, even I could not follow it totally.

[umnikos]

6:27 The standard ordering of the reals is not a well-ordering relation, therefore you cannot always pick the smallest h (aka. the leftmost as you phrased it) for which the coefficients of a and b differ. The reals are a well-ordered set tho, so there does exist an ordering for which the rest of the proof is valid

[xdman]

Thats some nice piece of maths. Thank you!

[bentupper]

At 4:00, how do you determine all possible chaotic orders of the first 3 rationals? It seems like there should be more than two possibilities.

[vladthemagnificent]

of course, it is based on the axiom of choice!

[debblez]

I’m pretty skeptical about that step where you assume every real number can be written as some rational combination of real numbers. It just seems kinda sketchy to me, idk.

[htomerif]

Its a good video, unfortunately for me, though, I've never had it explained how the reals can be constructed from the rationals so I just had to sort of believe that part for the rest of it to make sense.

[columbusmyhw]

I wonder if you really need the axiom of choice for the result?

[astropgn]

I cant understand what the symbol sideways-Y is doing. A < B means is less than B. But what is the sideways-Y doing? The video just says "connected transitive irreflexive binary relation on the reals", but the author was not successful to predict that this might mean nothing to someone. The "<" is also (by what he explained before) a "connected transitive irreflexive binary relation", but for the sideways-Y (annoyingly he didn't even name it) he says "perhaps in the new order 2 proceeds 1"... Perhaps? This looks so arbitrary