Ordinals- just what exactly are they?!
If ordinals have not confused you, you haven’t really made a serious attempt to understand them.
Let me illustrate this. If I have 5 fruits (all different) and 5 plates (all different), then I can bijectively map the fruits to plates. However, I arrange the fruits or plates, I can still bijectively map them.
Let’s suppose I have a set finite set , and don’t know its cardinality. But I know hat it bijectively maps to . This directly implies that however I arrange or , they will still bijectivey map to each other.
This intuition fails for infinte sets. In fact weird things start happening for infinite sets. Natural numbers can be bijectively mapped to rational numbers. What?!! Isn’t the set of rational numbers a superset of natural numbers?! Yes. Then how can there be a bijection between them? Bijection implies both sets contain the same number of elements.
No. That the cardinality should be the same is not part of the definition of bijection. Bijection is defined as an injective and surjective mapping between two sets. It is just that same cardinality is implied through bijection in the case of finite sets. For rational numbers, by cantor’s diagonalization argument, for every number, we cam find a unique pre-image amongst the natural numbers. Hence, we have a bijection.
Coming back to ordinals, let denote the ordered set of natural numbers. Does bijectively map to ? Not if you use the mapping . However, if you map and , then you’re done. Note the fact that if two infinite sets are bijective, that does not imply that every one-to-one mapping will be surjective. It just means that there exists *one* such mapping. This is in direct contrast with the case of finite sets, in which every injective mapping between two bijective sets is surjective.
What if you map to ? Note that as is ordered, this implies there is only *one* mapping we’re allowed to have: . We can clearly see can’t be bijectively mapped to .
Where most mathematical texts fail is actually explaining these finer points to students. Most of them just regurgitate the material present in “classics” of that subject. The most important thing to note here is that if we have two infinite sets which are not ordered, then there *might* be some bijective mapping between them, and finding one can be tricky sometimes. For example, Cantor’s diagonal mapping is brilliant, and non-trivial. Hence for non-ordinal infinite sets, we can’t be sure if they’re bijective with . However, in the case of ordinals, as there is only one mapping, determining whether the set is bijective with respect to is trivial.