In the context of ZFC, one normally uses von Neumann's definition of the ordinals. However, originially an ordinal was just the order-type of a well-ordered set (where "order-type of A" may for example be defined to be the equivalence class of all ordered sets that are order-isomorphic to A; this definition is of course no longer allowed in ZFC, but was common in pre-ZFC naive set theory).
I am now looking for a complete exposition of Burali-Forti paradox, with the original definition of ordinal. One can in a number of papers and books find expositions similar to the following one cited from Wikipedia:
"The "order types" (ordinal numbers) themselves are well-ordered in a natural way, and this well-ordering must have an order type Ω. It is easily shown in naïve set theory (and remains true in ZFC but not in New Foundations) that the order type of all ordinal numbers less than a fixed α is α itself. So the order type of all ordinal numbers less than Ω is Ω itself. But this means that Ω, being the order type of a proper initial segment of the ordinals, is strictly less than the order type of all the ordinals, but the latter is Ω itself by definition. This is a contradiction."
To complete this exposition, we need proofs for the following two facts:
- The ordinals are well-ordered under their natural ordering.
- The order type of all ordinal numbers less than a fixed α is α itself.
The book "Grundbegriffe der Mengenlehre" by Gerhard Hessenberg (1906) (which can be read online at http://www.archive.org/stream/grundbegriffede00hessgoog#page/n79/mode/1up) presents proofs for these facts, which to me however seem invalid (I do not understand why he may conclude "und umgekehrt ist jeder Zahl ν<μ ein Abschnitt in M eindeutig zugeordnet" on page 550).
I have found a complicated proof for the first fact, which however is based on induction (over the natural numbers) and three applications of the Axiom of Choice. It seems to me that the second fact may be proven using transfinite induction (transfinite induction, it seems to me, may only be used once one has established the first fact). So in principle, I think I can complete the Burali-Forti paradox as stated above, but this completed derivation would be very lengthy and involved.
So what I am actually looking for is a more concise or less involved complete derivation of the Burali-Forti paradox. Can anyone present such a derivation here, or point me to an existing one in the literature?
Best Answer
I have now found a textbook that provides a complete proof of the Burali-Forti paradox without making use of von Neumann's definition of ordinals: "Basic Set Theory" by Azriel Levy. Before providing von Neumann's definition, he works just on the assumption that some "order types" can be defined such that the order type of two well-ordered sets is identicall iff they are order-isomorphic. Based only on this assumption, and, importantly for my concern, without using the Axiom of Foundation, he shows that the class of ordinals cannot be a set.