Absolute Infinity - Numberphile

Asaf Karagila takes us deep into the world of Infinity - from lazy eights to aleph to omega to tav. More links & stuff in full description below ↓↓↓

NUMBERPHILE

Video by Brady Haran and Pete McPartlan

[PhilBagels]

"In mathematics, you don't understand things, you just get used to them." - John von Neuman
I never heard this quote before, but I love it!

[asymptoticspatula]

Brady is low-key one of the best interviewers and students ever. I always get the feeling that he is way more knowledgeable than he lets on just by the quality of his questions and the way he steers the conversations.

[patton]

"Just infinity. You say it like it's just a trivial thing"

"YES."

[CinemaRockPizza]

I love that Numberphile combines both modern quality of presentation and old school vibe of filming which is quite comforting in a way.

[unvergebeneid]

Missed opportunity to tell Brady that his improvised term "graspable" has a formal equivalent, which is "countable". The natural and rational numbers are countably infinite; the real numbers are not.

[andrewkepert]

On Brady’s “why 2” question. Yes it doesn’t matter numerically, but it is not arbitrary. It represents the cardinality of a power set - the set of all subsets of a set. To form a subset of a set X you need to make a binary choice (in/out) for each element of X. So 2^X is a common notation for the power set, and then |2^X|=2^|X|

[laju]

9:53
- It gets bigger and bigger until eventually you "run out of sets".
- How can you ran out?
- Exactly!

Hilarious!

[julian]

Just want to quickly mention that the "2 to the Aleph_0" IMO comes from taking power sets. Take a (finite) set X, and then consider the set of all subsets of X. This new set is called the power set and has precisely 2^|X| elements, with |X| denoting the number of elements of X. And this will always be strictly larger than the original set; even when considering infinite once. Hence the 2 to the power of...part :)

[f_f_f_]

The example I would give as a response to the question whether this is used: Turing showed that the size of the computable numbers is aleph 0. This immediately implies that non-computable real numbers exist for example the diagonal numbers of the computable numbers. And if you look for the reason you can not compute these you discover the Halting problem.

[quinn]

15:13 "Now I'm asking you, a set theorist, who deals with infinity every day, and throws around infinity like pieces of candy..." Legendary

[AlanKey]

3:56
Brady is always so good at asking the most interesting questions... I'd never think to question that but I'm so glad he did!

[drdca]

2:58 : “not to scale ... obviously” : haha

[jacksonstarky]

True comprehension of infinity is beyond us... but attempting to turn one's own brain into a black hole is always a worthy pursuit, and comprehending infinity is the most fun way to do that, in my opinion.

[TheUniversalist]

The quality of numberphile = absolute infinity

[oserodal]

The least controversial statement in the video at 4:27

> "There is nothing between aleph null and aleph one."

[Cashman]

17:30 Brady's so eloquent, but we all know he's known the answer for quite some time :D

[theepicosity]

asaf is such a wonderful presenter, i feel like he could answer any question brady throws at him!

[funktorial]

hope we get another session with Asaf about the axiom of choice!

[vonmatrices]

Yeah, well, whatever the thumbnail is, +1. I win

[djsmeguk]

17:30 a fun fact I'll never not keep repeating. The rationals have the same size as the natural numbers. Because of the way you measure sizes when you're playing with infinite sets and measures, this means that they have size ZERO in the set of reals. BUT, they are also dense in the reals, meaning you can find a rational number arbitrarily close to any real number. So they're nowhere but also everywhere at the same time in the set of real numbers. 😂🤯