Weird things you believed as a child
Mar. 14th, 2006 12:13 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
jackOK, there's millions of things in the "Weird things you believed as a child" catgeory[1], but this amused me.
I was reading about Bertrand Russel (on plover.com/blog, linked to by Tony), and the quote:
Obligatory exposition. At some point, people and societies come to distinguish between "Jack Vickeridge" and "all people called Jack Vickeirdge"[3]. In fact, I am the only one afaik, but they mean very different things if you speak of them having another man in. Here we have the notion of a thing, x, and set, {x}. "Aha!" goes the new set enthusiast. {x} is a thing! So we can have a set with {x} in {{x},other stuff}. And so on! And we could have people that *aren't* jack: {people who aren't x}. And things that aren't in {x}, {things not jack}...
Lots of this was going on in the 1900's. For instance, the aforementioned Frege. "But Hold on one cotton picking minute!" says Russel. "You can't have *all* of those." Else, (this is the paradox) consider the set of things which are sets which don't contain themselves. Is it a member of itself? Both answers are wrong, you made a mistake somewhere...
Well, what can we do?
(i) The easy answer is not to have sets of sets at all, or at least, have sets that contain real things, and supersets that contain sets, etc. But this is boring (TM), and confusing if you're not sure if something is a supersupersupersupersuperset or a supersupersupersupersupersuperset
(ii) Well, maybe the mistake was "All things that aren't sets that..."[4] There's a hidden axiom that you can take two sets and find all things in one not in the other. Maybe that isn't true. But it's obvious! It's obvious for little sets. For ginormousinfinite sets, maybe not. This seems really odd, but we did see it in Part III.
(iii) Well, maybe the problem is "all things". We implicitely assumed that was a set. Maybe it isn't[5]. And then we can keep the axiom in (ii), but it can't find *all* things that aren't x. This is what everyone does nowadays, but it was quite hard work to think it up, which is why Russel's paradox is important and not just an annoying coincidence.
Of course, Russel's and later works on defining the whole of mathematics from simple axioms hit other speed bumps and brick walls eventually, courtesy of Godel's theorems, but that's for another time.</Sheherazade>
[1] Pun not intended :)
[2] Contrary to rumour I don't normally want to hug teutonic mathematicians, but I did there.
[3] I remember trying to explain in a maths lesson at school, that no, the empty set wasn't {Φ}, it was {}, called just Φ.
[4] I know where this is going. No you don't.
[5] We need a name for things that aren't a set. IIRC "class" is generally used for "something definied similarly to a set, ie. some lot of things, but that doesn't necessarily obey the set axioms" and "collection" for "anything we want to make absolutely no implications about at all."
I was reading about Bertrand Russel (on plover.com/blog, linked to by Tony), and the quote:
Hardly anything more unwelcome can befall a scientific writer than that one of the foundations of his edifice be shaken after the work is finished. I have been placed in this position by a letter of Mr Bertrand Russell just as the printing of the second volume was nearing completion[2]...However, in my head, I had the impression that this had been Russel's response to Godel's paradox torpedoing *his* system of arithmetic, whereas in fact, it was Russel's paradox torpedoing Frege's.
Obligatory exposition. At some point, people and societies come to distinguish between "Jack Vickeridge" and "all people called Jack Vickeirdge"[3]. In fact, I am the only one afaik, but they mean very different things if you speak of them having another man in. Here we have the notion of a thing, x, and set, {x}. "Aha!" goes the new set enthusiast. {x} is a thing! So we can have a set with {x} in {{x},other stuff}. And so on! And we could have people that *aren't* jack: {people who aren't x}. And things that aren't in {x}, {things not jack}...
Lots of this was going on in the 1900's. For instance, the aforementioned Frege. "But Hold on one cotton picking minute!" says Russel. "You can't have *all* of those." Else, (this is the paradox) consider the set of things which are sets which don't contain themselves. Is it a member of itself? Both answers are wrong, you made a mistake somewhere...
Well, what can we do?
(i) The easy answer is not to have sets of sets at all, or at least, have sets that contain real things, and supersets that contain sets, etc. But this is boring (TM), and confusing if you're not sure if something is a supersupersupersupersuperset or a supersupersupersupersupersuperset
(ii) Well, maybe the mistake was "All things that aren't sets that..."[4] There's a hidden axiom that you can take two sets and find all things in one not in the other. Maybe that isn't true. But it's obvious! It's obvious for little sets. For ginormousinfinite sets, maybe not. This seems really odd, but we did see it in Part III.
(iii) Well, maybe the problem is "all things". We implicitely assumed that was a set. Maybe it isn't[5]. And then we can keep the axiom in (ii), but it can't find *all* things that aren't x. This is what everyone does nowadays, but it was quite hard work to think it up, which is why Russel's paradox is important and not just an annoying coincidence.
Of course, Russel's and later works on defining the whole of mathematics from simple axioms hit other speed bumps and brick walls eventually, courtesy of Godel's theorems, but that's for another time.</Sheherazade>
[1] Pun not intended :)
[2] Contrary to rumour I don't normally want to hug teutonic mathematicians, but I did there.
[3] I remember trying to explain in a maths lesson at school, that no, the empty set wasn't {Φ}, it was {}, called just Φ.
[4] I know where this is going. No you don't.
[5] We need a name for things that aren't a set. IIRC "class" is generally used for "something definied similarly to a set, ie. some lot of things, but that doesn't necessarily obey the set axioms" and "collection" for "anything we want to make absolutely no implications about at all."