Do we still run into Burali-Forti paradox even if we use Zermelo's definition of ordinals?

Question

Will we still have Burali-Forti paradox if we use Zermelo's definition of ordinals instead of the Von Neuman's definition? Because I think when we define the ordinals as {}, {{}}, {{{}}}, ... (using Zermelo's definition) then the set of ordinals which is { {}, {{}}, {{{}}}, ... } can not contain itself by definition hence prevents the paradox. If I am correct then what is it about Von Neuman's definition that raises Burali-Forti paradox?

Answer 1

The point of the Burali-Forti paradox is that there is no largest well-ordered set.

Simply because if there is such a set, then by collecting all the initial segments of such a well-ordering (including the set itself), we obtain a well-ordering which is strictly longer, by one point to be precise, and thus we get a contradiction to the maximality of the order.

The actual definition of an ordinal is irrelevant. We just need to be able to run the basic comparison argument on well-orders (i.e. given two well-ordered sets, one of them is isomorphic to an initial segment of the other), and that given a linearly ordered set, we can form the set of its initial segments.

Of course, using the full power of $\sf ZF$, we don't care about arbitrary well-orders, just about the ordinals, so the paradox is seemingly about the class of ordinals. But it really isn't, it's a paradox about having a largest well-ordered set. Just like Cantor's paradox is about having a largest cardinal.

Answer 2

Zermelo's definition only works for finite ordinals - what should $\omega$ be? "$\{\{\{\{...\}\}\}\}$" isn't a set (think about the axiom of foundation). Burali-Forti works because we can prove that the set of ordinals (assuming it exists) is an ordinal; this fails for Zermelo ordinals.