
TREE vs. Graham’s Number [video] - eindiran
https://www.numberphile.com/videos/tree-v-grahams-number
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x3n0ph3n3
I really need a video on transfinite cardinals and ordinals beyond Aleph_0 and
Epsilon_0. The VSauce video [1] on the topic only made me more curious with an
elusive tree diagram displaying a few, and PBS Infinite Series [2] [3] just
barely touched it. The mention of the Veblen hierarchies really piqued my
interest, but much of what I've found on them has been a bit too dense for me.

1\.
[https://www.youtube.com/watch?v=SrU9YDoXE88](https://www.youtube.com/watch?v=SrU9YDoXE88)

2\.
[https://www.youtube.com/watch?v=uWwUpEY4c8o](https://www.youtube.com/watch?v=uWwUpEY4c8o)

3\.
[https://www.youtube.com/watch?v=oBOZ2WroiVY](https://www.youtube.com/watch?v=oBOZ2WroiVY)

~~~
gosub
[https://www.youtube.com/watch?v=QXliQvd1vW0&list=PL3A50BB9C3...](https://www.youtube.com/watch?v=QXliQvd1vW0&list=PL3A50BB9C34AB36B3)

enjoy!

~~~
speakeron
[https://www.youtube.com/watch?v=vq2BxAJZ4Tc](https://www.youtube.com/watch?v=vq2BxAJZ4Tc)
is also a very good series and gets into infinite collapsing ordinals towards
the end.

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IanCal
There's lots of things about Graham's number, but because I love the style I
thought I'd add in here Wait But Whys "from 1 to Graham's number":

[https://waitbutwhy.com/2014/11/from-1-to-1000000.html](https://waitbutwhy.com/2014/11/from-1-to-1000000.html)

[https://waitbutwhy.com/2014/11/1000000-grahams-
number.html](https://waitbutwhy.com/2014/11/1000000-grahams-number.html)

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nils-m-holm
I love numberphile, but I cannot watch a single episode, because all the
zooming and panning while someone writes on a sheet of paper makes me motion
sick in no time.

WHY? How does this improve the quality of the video?

~~~
baalimago
It's action-math. Not meant for teaching, but for showcasing math to the
general public. The general public wants action cams and cute animations.

~~~
spacehome
Numberphile isn't that great if you're already interested in math. They talk a
lot about math words, but they hardly ever do any math. 3Blue1Brown is where
the good math is.

~~~
robinhouston
It isn’t a great way to learn the details, but it’s a great way to learn about
topics you might not have come across.

This video is about the hierarchy of fast-growing functions indexed by
ordinals, and it includes enough details for a competent mathematician to be
able to work out the rigorous general definition. For someone unfamiliar with
the topic, it’s an enjoyable and low-effort way to learn enough about it to
recognise whether it’s interesting enough to learn about properly.

There are several examples of cutting-edge mathematical work having been
directly inspired by numberphile videos, most recently the work of Andrew
Booker and Drew Sutherland on expressing integers as sums of three cubes.

3blue1brown concentrates on giving clear, elegant, rigorous explanations of
established subjects. So perhaps ironically, despite being typically more
rigorous, I don’t know any examples of it inspiring new mathematical research.

~~~
333c
> There are several examples of cutting-edge mathematical work having been
> directly inspired by numberphile videos, most recently the work of Andrew
> Booker and Drew Sutherland on expressing integers as sums of three cubes.

I don't know if this is why you made this comment or not, but Brady Haran
(creator of Numberphile) gave a talk on this exact topic a week ago today at
the MSRI in Berkeley.

~~~
robinhouston
Oh, interesting! I knew Brady Haran had given a talk at MSRI, but I didn’t
know what the talk was about. Was it filmed?

~~~
333c
I don't know if it was filmed. I was there but I didn't record it.

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umvi
Let t(n, i) be a function where n is the number of recursive TREE calls and i
is an integer.

So t(1,3) = TREE(3)

t(2,3) = TREE(TREE(3))

etc.

Now you can do:

t(TREE(3), 3)

or

t(t(t(t(t(t(t(t(t(TREE(3),3),3),3),3...)

~~~
contravariant
The way TREE grows it's very much possible that all these are inferior to
simply f(n) = TREE(n+1).

~~~
x3n0ph3n3
Unlikely since we are piping the output of TREE back into TREE, which is much
larger than n+1.

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hidroto
[https://oeis.org/A028444](https://oeis.org/A028444) would grow faster right?

does it make sense to compare the growth rates of non-computable functions?

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Eliezer
TREE(Graham's Number) is just comically inefficient Googology. It makes me
wince just to look at. Graham's Number isn't even on the same plane of
existence as TREE. You'd just use TREE(TREE(3)) if you wanted to move up in
the world. Somebody needs to tell them about the fast-growing hierarchy and
how Graham's Number lives around \omega+1 while TREE hangs out with the small
Veblen ordinal.

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Ravengenocide
Maybe I'm missing something obvious here, but that's what the video is about?

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speakeron
The video is a basic introduction to the fast-growing hierarchy. They do state
where Graham's Number sits in the hierarchy, but the explanation doesn't go as
far as to say where TREE sits.

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empath75
It does actually. It says it grows faster than anything in the hierarchy.

~~~
SAI_Peregrinus
Faster than anything in the Wainer hierarchy (which ends at
$f_{\epsilon_{0}}(n)$. But it's reasonably easy to define a fast-growing
hierarchy that uses the Veblen hierarchy of ordinals for the subscripts. Then
TREE is $f_{svo}(n)$ where $svo$ is the limit of the finitary veblen sequence
as the number of elements goes to $\omega$ (aka the small veblen ordinal).

