It's Russell's Paradox:
http://plato.stanford.edu/entries/russell-paradox/It caused a revolution in the foundations of mathematics in the early 1900s.
I can also relate it to the topics in this thread in this sense: Russell's Paradox is actually quite easy to understand.
Topics like economics or climate are nowhere near this simple. If someone claims to have certainty on issues like fiscal/monetary policy or climate change, then a good test is to see if they can understand something simple like Russell's Paradox. If they can't, then it's very likely they are simply parroting the expected lines of their social group.
I understand the paradox although I can't understand what flaw in the system of sets gave birth to it, or how to prevent it from happening. Could you explain?
In the letter he was explaining how the paradox can be used to prove Frege's system inconsistent -- a system that's a bit removed from set theory. However, the paradox applies quite generally and is often presented in terms of set theory, so I'll focus on sets.
Suppose we are working in a naive set theory where whenever we have a property P(x) we can form a set {x|P(x)} -- the set of all x satisfying P(x). Russell's Paradox gives a particular instance that leads to a problematic set: R := {x| x is not a member of x}. R is in R if and only if R is not in R, a contradiction. In short: this kind of naive set theory with all sets like {x|P(x)} is inconsistent.
There were many proposals how to put set theory on a sound footing. The one that gained the most followers is ZF set theory. In ZF you can make sets using certain ZF primitives. Thes primitives allow sets like like {x in X | P(x)} -- separating out the elements from a preexisting set X -- and sets like {f(x)|x in X} -- replacing the elements from a preexisting set X. You can't form a set like {x|P(x)} unless you can express it in terms of the ZF primitives.
Thanks for your interest. I enjoy thinking about these things.
I don't think lack of understanding of some mathematical paradox is proof that therefore you don't understand politics and economics. Someone could study one thing and not the other.
Well, if something like Russell's Paradox is new to someone, then it's fair that they need time to understand it. I'll agree with you to that extent.
What I was trying to say is that if someone can't understand Russell's Paradox it betrays an inability to comprehend basic logic (like Modus Ponens). If someone cannot comprehend basic logic, when they study other subjects what they are probably doing is the kind of rote learning that allows them to repeat things. Basic logic is required to analyse a statement and start asking the right questions to justify or refute the statement.
