Difference Between Cantor's Ordinals and Von Neumann Ordinals?
"By the numbering [Anzahl] or the ordinal number of a well-ordered set MM 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.
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 ∈.
- Badges
- Science topic
- Similar topics
- Mathematics
- Applied Mathematics