Why do paradoxes like Russel's paradox and the barber paradox occur? Is something wrong with the definition or what?

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Such paradoxes exist because we are able to construct sentences to which there may or may not be any grounds in reality. We can ask all sorts of silly questions as well.
"What colour is Wednesday?" for example.

In the case of the barber paradox, though, the assumptions lead to a contradiction. This implies that one or more of the assumtions is false. Essentially, such a town does not exist and cannot exist.

Exactly ,even I think that the question itself is silly. But then why are people trying to modify the definition of set theory to avoid the paradox?

Exactly ,even I think that the question itself is silly. But then why are people trying to modify the definition of set theory to avoid the paradox?
Who is trying to modify set theory?

My understanding was that Cantor's set theory was proven to be inconsistent (by Russel and Zermelo and perhaps others) so a more axiomatic approach was necessary.

The issue is this: a set is defined by it's elements.
So I can come up with all sorts of inconsistent sets eg Russle's paradox.

So maybe a set is not defined by it elements? Or maybe we have to be careful about what elements define a set. I believe that is what other approaches to set theory are about (eg Quinian set theory).

Exactly ,even I think that the question itself is silly. But then why are people trying to modify the definition of set theory to avoid the paradox?
Regarding the set of all sets:

If you allow unrestricted set formation, you get a paradox. So the solution is to disallow unrestricted set formation.

That is: Let P be a predicate. A predicate is just a statement that is either true or false about some particular object. So if P(x) is the statement, "x is an even number" then P(2) is true and P(47) is false.

Now we make a rule: For any predicate P, we can form the set {x : P(x)}. In other words given any predicate, we can form a set made up of exactly those objects for which the predicate is true.

But that rule fails!! If we let P(x) mean, "x is not an element of x" and we form the set

R = {x : P(x)}

then R is an element of R if and only if R is NOT an element of R. We have a contradiction. So our "rule" was a disaster. It led to an inconsistent system of set theory. (I'm using R in honor of Russell, of course.)

The solution (or more accurately, ONE possible solution, and the solution in common use today) is to disallow arbitrary set formation. The new rule, let's call it Zermelo's rule, is that to form a set using a predicate, the objects over which we test the predicate must already be elements of some other set U.

So if P is a predicate, and U is a set, then

Z = {x $\in$ U : P(x)}

does not lead to a contradiction. Problem solved! And that's why we don't allow unrestricted set formation: because it immediately leads to a contradiction. From now on, when constructing new sets of objects that satisfy some predicate, we can only test objects that are already known to be elements of some other set.

That fixes the problem.

[Technical note: It's more accurate to say that Zermelo's idea does not lead to a contradiction as far as we know. We can't know for sure if set theory is free of contradictions. That's a technical issue that's not relevant here, but I didn't want to lie by claiming that set theory can prove itself consistent; because it can't.]

Now, as far as the barber who shaves all those who don't shave themselves: Clearly he is shaved by Occam's razor!!

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What he said.

Thank you guys!