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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
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)
Article could be improved
I am a mathematician (but not a set theorist) who has just been reading up on cardinals, ordinals and in particular the Burali-Forti paradox, and I find the current article to be a bit obscure and also a bit anemic. On the one hand, I can't figure out why the paradox is initially stated in terms of von Neumann ordinals -- as mentioned later, it was not the way Burali-Forti expressed it, and it tempts one to think that issues of "foundation" are involved in the paradox. (In fact, I had thought about this material a little while ago and written up notes. When I looked back at my notes, I saw that I had derived the claim o \in o, noted its suspectness, and then constructed von Neumann ordinals to show that for them "o \in o" is not "paradoxical" but just false. But this is unnecessary and misses the spirit of it, it seems to me now.) I wonder why the section "Stated more generally" does not appear first, written up slightly more carefully. For instance, it says that the association of an order-type to a well-ordered set is done in an "unspecified" way, but we should at least specify the key point: we assign the same order-type to two well-ordered sets iff they are order-isomorphic!
Also the paradox is not even given a careful statement, which is strange. I think the statement should be something like: "The order-isomorphism types of well-ordered sets do not themselves form a set; equivalently, there is no single set S of well-ordered sets such that every well-ordered set is order-isomorphic to at least one element of S." (The equivalence should be explained.) Then the proof can be given, which basically consists of unpacking the statement and citing the known theorem that no well-ordered set is order-isomorphic to one of its initial segments. (This is of course referred to in the current draft, but it could be made even more clear.) Saying that the proof "can be carried out in naive set theory" seems to raise unnecessary questions (i.e., what is naive set theory, exactly).
I also think that it should be pointed out that just like Cantor's "paradox" is really an important theorem about cardinalities, Burali-Forti's paradox proves the following important fact: the collection of order-isomorphism types of well-ordered sets whose cardinalities are at most \kappa (for any cardinality \kappa) itself does not have cardinality at most \kappa. In particular, this shows the existence of uncountable well-ordered sets. The point here (as I understand it -- I hope I will be corrected if I am wrong!) is that this fact is true _without the axiom of choice_, whereas the fact that there is a well-ordered set of every cardinality is of course equivalent to AC. To those who would say that one should not worry about the use of AC in "naive set theory", I can only say that I myself am not especially interested in doing deductions from the ZF/C axioms, but I find it very interesting to know which of the theorems I know in set theory (and other areas of mathematics) are equivalent to AC. And anyway isn't this exactly the sort of thing you would want to know if you were trying to decide whether you wanted to work in ZFC? -- P.L. Clark
May not be due to Burali-Forti
In his talk 1999 talk at UMass-Lowell, Gregory Chaitin said that "if you look at the paper by Burali-Forti, you don't see the paradox". (http://www.umcs.maine.edu/~chaitin/lowell.html) Apparently the paradox occurred to Russell while reading the paper.Scorwin (talk) 01:18, 15 January 2011 (UTC)
- Perfectly true. My understanding has for many years been that Burali-Forti was unaware of any paradox, and I have just checked his paper, and confirmed that it makes no mention of any paradox. Burali-Forti merely proved a theorem which contradicted a pre-existing theorem unknown to him. I shall edit the article accordingly. The editor who uses the pseudonym "JamesBWatson" (talk) 14:29, 10 May 2014 (UTC)