Difference Between Cantor's Ordinals and Von Neumann Ordinals?

Question

Cantor's Naive Set Theory allows the construction of the set of all ordinals, which contains itself, which triggers the Burali-Forti Paradox. ZFC both disallows a set of the size of all ordinals and typically uses Von Neumann's definition of ordinals. Under Von Neumann's definition, a set $\alpha$ is an ordinal number iff

1. If $\beta$ is a member of $\alpha$, then $\beta$ is a proper subset of $\alpha$;

2. If $\beta$ and $\gamma$ are members of $\alpha$ then one of the following is true: $\beta=\gamma$, $\beta$ is a member of $\gamma$, or $\gamma$ is a member of $\beta$;

and 3. If $\beta$ is a nonempty proper subset of $\alpha$, then there exists a $\gamma$ member of $\alpha$ such that the intersection $\gamma \cap \beta$ is empty. (Definition from Wolfram Mathworld).

The first rule implies that no ordinal is an element of itself, hence even if the axiom of foundation and axiom of replacement did not exist, the set of all ordinals could not be an ordinal, since it would violate the first rule, and the Burali-Forti Paradox would therefore still be resolved. So... there seems like their must be a significant difference between Cantor's definition of ordinals and Von Neumann's, since Cantor's Naive Set Theory still allows the set of all ordinals to be an ordinal. Cantor's ordinals can be elements of themselves. Why can Cantor's ordinals be elements of themselves? What is it about his definition of ordinals that allows them to be elements of themselves?

Answer 1

"By the numbering [Anzahl] or the ordinal number of a well-ordered set $\frak M$ I mean the general concept or universal [Allegemeinbegriff, Gattungsbegriff] which one obtains by abstracting the character of its elements and by reflecting upon nothing but the order in which they occur."

(J.W. Dauben, Georg Cantor: His Mathematics and Philosophy of the Infinite pp. 127–128; the quote is attributed to G. Cantor "Mitteilungen zur Lehre vom Transfiniten" on p.388 in the 1932 collection "Gesammelte Abhandulngen mathematischen und philosophischen Inhalts".)

This is not a rigorous definition, but to me, the only modern way of interpreting this is to consider ordinals either as an abstract category, or more likely at the time, as a collection of equivalence classes of well-ordered sets under the order-isomorphism equivalence relation.

The latter interpretation seems to be more in the spirit of the time, so I will stick with it.

This means that an ordinal is not a set, in modern perspective, since for any non-empty well-ordered set, there is a proper class of well-ordered sets which are isomorphic to this given order.

Therefore the main difference between the von Neumann ordinals and Cantor's ordinals, is that the former are sets, and the latter are not. Moreover, the collection of Cantorian ordinals is not even a proper class, in modern terms, since its "elements" are not sets.

Cantor's ordinals were not elements of themselves. Even if you take the naive set theoretic approach, just because a set can be a member of itself, doesn't mean that every set will be a member of itself.

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Let me also make a tangential remark, that a complete name of the von Neumann ordinals should be "the von Neumann ordinal assignment". Because we simply show that there is a canonical choice from every Cantorian ordinal: a transitive set well ordered by $\in$.

Answer 2

The definition from Wolfram Mathworld that you cite is actually incorrect. The correct definition says "subset" instead of "proper subset" everywhere. (In (3), it also says $\gamma\in\beta$ rather than $\gamma\in\alpha$). So, no ordinal being a member of itself is actually not explicitly built in to the definition of von Neumann ordinals. It turns out to be a consequence of the definition (specifically, it follows from (3), since if $\alpha\in\alpha$ then the subset $\beta=\{\alpha\}$ would violate (3)), but this is not an important part of the motivation for the definition or a difference between von Neumann ordinals and Cantor ordinals.

(More generally, in my experience Mathworld is a highly unreliable source and you should always be skeptical of the details of anything you read on it.)