Absolute Infinity - Numberphile
355,951
Published 2024-03-19
Asaf is a UKRI Future Leaders Fellow. Asaf's blog - karagila.org/
More videos and Numberphile podcast featuring Asaf - • Asaf Karagila on Numberphile
Infinity Videos: • Infinity on Numberphile
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All Comments (21)
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More videos and Numberphile podcast featuring Asaf - youtube.com/playlist?list=PLt5AfwLFPxWJyt0zdvzvDoe…
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3:00 "not to scale, obviously" I'm glad it was made clear
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"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!
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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.
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I love that Numberphile combines both modern quality of presentation and old school vibe of filming which is quite comforting in a way.
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"Just infinity. You say it like it's just a trivial thing" "YES."
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9:53 - It gets bigger and bigger until eventually you "run out of sets". - How can you ran out? - Exactly! Hilarious!
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Yeah, well, whatever the thumbnail is, +1. I win
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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|
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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
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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.
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2:58 : “not to scale ... obviously” : haha
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The quality of numberphile = absolute infinity
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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 :)
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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.
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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!
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17:30 Brady's so eloquent, but we all know he's known the answer for quite some time :D
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The least controversial statement in the video at 4:27 > "There is nothing between aleph null and aleph one."
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asaf is such a wonderful presenter, i feel like he could answer any question brady throws at him!
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hope we get another session with Asaf about the axiom of choice!
