Barber & Russell Paradoxes (History of Undecidability Part 2) - Computerphile

The Barber Paradox: Professor Brailsford continues the history of undecidability.

This video was filmed and edited by Sean Riley.

[without]

I could listen to this bloke all day, he explains so well, and so effortlessly.

[stolenmuppets]

Wikipedia has a better description of Russel's paradox:

According to naive set theory, any definable collection is a set. Let R be the set of all sets that are not members of themselves. If R is not a member of itself, then its definition dictates that it must contain itself, and if it contains itself, then it contradicts its own definition as the set of all sets that are not members of themselves. This contradiction is Russell's paradox.

[mustafaadam]

It's not easy being entertaining while talking about mathematical logic, but this video and the one before has managed to do that beautifully. More videos on these kinds of topics please :)

[sudevsen]

I can imagine Godel trolling everyone while saying :
'Problem?"

[JahMusicTube]

I'd have loved to have a professor like this guy teaching me this stuff! Great video!

[addictedtoflames]

I'd really love to hear more about Goedel's incompleteness theorem. I know this is only tangentially relevant to computer science but if you guys and Professor Brailsford have time I'd really love to see a full length video on that. I learned about it in the first year of my philosophy degree but I feel like I'd really love to hear professor Brailsford talk about it as he explains things so well (far better than any professor I've ever been taught by). Obviously if you don't have time that's fine but I was kinda sad that this video ended where it did and I'd love to hear a bit more about it

[Demokritos]

Great video. These result are known for me for a long time but it was really interesting to hear how different mathematician reacted to them. Thank you very much for keeping this channel going.

[KdOUR]

You know it ain't tobacco in the pipe when a few guys are discussing the twoness of 2.

[paulorugal]

I loved this series of videos <3 

Thank you so much guys!

[amerhijazi]

" To me and you we'd be like wut? " literally laughed out loud. I love how mathematicians never forget how humbling math can be and just in one second and relentlessly take the position of a completely clueless person, that is their power.

[anon]

The statement describing Godel's theorem is actually wrong. Godel showed that if a first-order logic together with a set of axioms can derive a kind of simple arithmetic, then  such an axiomatic system can express a statement which is not provable by the system. -- The video mistakenly claims that the system only has to be able to have "one thing derive from another" which is true of all axiomatic systems. But not all axiomatic systems are amenable to Godel's construction and in fact infinitely many such systems are decidable. "Decidable" means that every true statement expressible in that system has a proof in that system.

[davidjesus]

Correct me if I'm wrong, but the Barber's paradox and the parallel lines are two distinct problems. In the first, who shaves the barber is not undecidable. Both options lead to a paradox, hence both are wrong. This is because the set of axioms (or as you call them, propositions), is inconsistent.
Whereas, in the parallel lines problem, the axiomatic is too weak to decide whether the statement is true or false, and the need for another axioms emerges.
The main difference is that in the first case, addition of axioms with never solve the problem, and in the last case it does.
Talking about the other form of the barber's axiom: The set problems. This was solved by creating a completely new theory, called set theory, which solves this problem, very vaguely said, by increasing the "dimension" of what is understood by a set, whenever you talk about a set that contains all sets. Hence it does not need to contain itself, since it is not called a set, but something else.

[messyhair]

Hofstadter's G.E.B. is one of the best books I've ever read. It's got a wonderful proof of Godel's imconpleteness theorem

[peterroberts]

I love reading Russell. Maybe not profoiund, but highly provocative with his logical analysis in the vein of Lewis Carroll.

[dmitryalexandersamoilov]

This reminds me of dividing by zero, or taking the square root of a negative number...

If you divide not by zero, but by numbers approaching zero, you get closer and closer to infinity, whatever that is.

If you represent the sqrt of -1 as i .. you get a complex number which represents an extension of dimension in a coordinate system.

What if this paradox is not a problem at all, what if it's a solution?

What if.. the undecidability is actually what causes existence to exist in the first place?

[rotflmaopmpqxyz]

Consider a set of all the sets that have never been considered...
Wait now they're all gone.

[toxictype]

I could listen to that voice for hours

[chrisg]

My solution: We're speaking of transitive verbs, shave -  praise - follow - paint - play badminton with - include in a set (whatever you want it doesn't matter) that have a logical tense. This kind of tense isn't necessarily overtly marked by any grammatical suffix like "-ed" in English, or by a word like "then" or "now" or "tomorrow". But it is indicated by the relative position of the verb.  An artist paints all and only those who don't paint themselves. Each of the two occurrences of "paint" have a different logical tense since they occur at different times in this sequence of clauses, a difference that can be made explicit: An artist will paint tomorrow all and only those who didn't paint themselves yesterday. So if the artist didn't paint herself, then she will. And if she did, then she won't. Simple. Logical. The fallacy in the paradox consists in failure to differentiate with respect to logical tense.

[razan]

interesting but complicated
I really enjoyed listening to the story from you
thank you

[JalapenoStudios]

Or perhaps the barber lives outside of town.