Talk:Burali-Forti paradox

From Wikipedia, the free encyclopedia

Does this paradox rely on the fact that sets cannot contain themselves? If so, somebody should note that it does not apply to some of the post-ZFC theories which are consistent in the presence of self-containing sets; for example [Positive set theory] Megacz 03:21, 28 Sep 2004 (UTC)

I'm a little concerned about the definition of ordinal numbers as those which can be defined as "the set of all their predecessors". In formal set theory, the "predecessor" relation is often taken by definition to be just another name for the membership relation. So, "a is a predecessor of b" is just another way of saying "a is an element of b". In this case, to say that an ordinal number is one which is "the set of all its predecessors" would make every set an ordinal number, and this is clearly wrong.

One common definition for ordinal number is the following. A set A is element-transitive (or e-trans) if x in y in A (i.e. (x in y) and (y in A)) imply x in A. Then an ordinal number is an e-trans set, each of whose elements is also e-trans. From here, one can show that the empty set is an ordinal number, and that the successor of an ordinal number is an ordinal, and so on. This isn't the only possible definintion. You can also say an ordinal is a set that's totally ordered and every element is a subset.

Also, the paradox isn't quite the way it's stated, "if the ordinal numbers formed a set, that set would then be an ordinal number greater than any number in the set". The paradox is NOT so much that the set of all ordinals would be "greater" than any ordinal number it contains, but rather that the set of all ordinals would have to be a member of itself, and this ultimately violates the axiom of regularity, which implies that for all sets A, A is not in A.

I'm 90% certain that the paradox doesn't rely on the axiom of regularity, since it was formulated in the 19th century before this rather obscure axiom was an issue. Indeed, Frege's system patently denied the axiom of regularity, but the paradoxes that caused so much consternation (of which this was the first) were true contradictions only in Frege's system (the first formal system proposed in those days). -- Toby Bartels 04:15, 10 Mar 2004 (UTC)

The axiom of regularity is indeed a red herring here. Since we have an inherent antinomy, we can derive a contradiction to anything, be it true, false, axiom or not. Also as Toby says, the axiom of regularity was only introduced later. -- Mellum 12:03, 10 Mar 2004 (UTC)

The method of using classes, and then defining sets as those classes which are contained as an element of some other class is one way of skirting the paradox. I'm not an expert on mathematical logic, but I don't think the class-set way of doing things is the only acceptable alternative (although probably the most popular). If others know more about these alternatives, that would be good to know. Revolver

[edit] the basis of the paradox

The paradox does not depend on anything about membership per se. It depends on the supposed possibility of assigning an order type to every isomorphism type of well-orderings, combined with the observation that the order types are then naturally ordered, and each order type (ordinal number) is the order type of the segment it determines in the natural order on order types. See the generalization of the argument that I inserted.

Randall Holmes 20:46, 16 December 2005 (UTC)