Solving a finite number problem using infinities

Here's a problem about sequences of finite (natural) numbers, subject to a simple finite rule, and all of these sequences have finite length... but we can't use the elementary language of finite numbers to prove this!

Recommended prerequisite:

Addenda:
(4:07) This argument might seem like a complete proof, but it's incomplete! Pure base expansions of a number can have incredibly recursive structure, which makes the intuition difficult to formalise without something with comparable expressive strength to ordinals. Consider: how do we measure the "complexity" of the shape of a base expansion, so that this "complexity" strictly decreases? Given the arbitrary nesting of these expansions in general, how do we know that a descending sequence of "complexity" must inevitably reach the shape for 0 in *finitely* many steps?

(11:24) That every descending sequence of ordinals is necessarily finite follows immediately from how ordinals are well-ordered. The demonstration in the video was an attempt to make it more believable (success in that matter is questionable).

(9:31) Please pay no mind to the number line. (Thanks @pudy2487.)

__________
Timestamps:
00:00 - Introduction
02:35 - Defining the sequence's rule
03:45 - Proof idea
04:43 - Overview on ordinals
05:51 - Using ordinals to describe "shape"
06:31 - The ordinal "shapes" decrease!
08:42 - Ordinals are well-founded
09:06 - Finite example
09:30 - First infinite example
10:06 - A bigger infinite example
11:22 - General case
12:06 - Putting it all together
12:32 - Loose ends
13:08 - Thx 4 watching

[klembokable]

you know an advanced math video is going to be good if we start by reminding ourselves how reading numbers works

[redsalmon]

A 727 reference is the last thing I expect to see in a video like this one

[dwalsh]

"There’s this emperor, and he asks the shepherd’s boy how many seconds in eternity. And the shepherd’s boy says, ‘There’s this mountain of pure diamond. It takes an hour to climb it and an hour to go around it, and every hundred years a little bird comes and sharpens its beak on the diamond mountain. And when the entire mountain is chiseled away, the first second of eternity will have passed.’ You may think that’s a hell of a long time. Personally, I think that’s a hell of a bird."

[legendgames]

As both a math enthusiast and an Undertale fan, I can say the -1 in the Goodstein sequence fills me with DETERMINATION.

[minecrafting_il]

My hype moments where when you revealed that the -1 makes the sequence always converge, and when I realized the very definition of ordinals force every decreasing sequence to converge

[MartinKleins]

727 WYSI IN MATH CONTENT NO WAYYYY LOL

[JorgexD]

As a PhD in modelling and prediccton I can assure this is going to become my favourite channel

[null_st]

always happy to see a video that starts with decimal expansions, thanks g!

[ModusTollendoTollens]

Finally, someone beats me to the convergence of all goodstein sequences theorem, super nais example. Really recommend the Kirby and paris paper where it is first proven indecidable in peano aritmethic, by generalizing with the hydra game.

[KylerAce]

Always nice to see a new sequence that seemingly against all common sense, converges

[HEHEHEIAMASUPAHSTARSAGA]

Using strong induction to prove that ordinals are well-ordered seems backwards, cause usually you define the ordinals to be well-ordered and then use that fact to prove strong induction works. But I know these videos are more of an intuitive introduction to ordinals than a rigorous treatment

[Testgeraeusch]

I guess the largest mental barrier for the proof is the idea that "descending" implies you substract -1 every step. But subtracting -1 from omega will not get you to any specific number, so whenever you "jump" one omega lower, you actually leap over inifitely many numbers in between. Otherwise you could not define the descending sequence in the first place. So yes, you could "avoid zero" for as long as you want, but not with a strictly descending sequence of ordinals, as the definition forces you to take ridiculous leaps every now and then (but only finitely many ^^ ).

[soqjqxobfw]

Only some teachers can generate inspiration, you are one of them. Awesome way you started this video.

[errorpixelyt]

"Solving a finite number problem using infinities" Isn't that just calculus

[TheArtOfBeingANerd]

THANK YOU I've been looking for a good series about infinite ordinals forever! Finally something to live for

(If anyone else knows some good infinite ordinal videos/series, please let me know)

[FullPwned]

I remember when I saw this problem in PBS Infinite Series. It is good to see a deeper treatment, while still being able to understand it

[SoggyCat-bn]

0:54 I FOUND THE WEBSITE AGAIN THANKS TO YOU

[lumi]

ever since i knew about goodstein sequences, their construction seemed so convoluted for me, compared to stuff like the primitive sequence system and kirby-paris hydras, systems that are much simpler, but equally strong (ε_0 level in the fast growing hierarchy). those deserve much more attention

[karig]

Cant believe he said wysi after saying 727….. based math youtuber

[VaraNiN]

This channel gonna be big! Great videos - currently binge watching all of them!