Hurkyl said:Firstly, there are 'too many' ordinals to fit in a set, so you'd have to talk about the class of ordinals.
Now, the class of ordinals doesn't have any arithmetic operations on it -- which did you mean:
(1) You want to know if the multiplicative monoid of ordinals can be extended to a group.
(2) You want to see if there is any binary product on the class of ordinals (or an extension of them) that turns them into a group. (I assume you want associativity)
If you mean the former, then clearly no extension exists; the multiplicative monoid of ordinals is not right-cancellable.
Ordinals are mathematical objects that represent the order or sequence of a set. They are important in mathematics because they provide a way to compare the size or magnitude of different sets, and they can be used to construct larger and more complex mathematical structures.
To extend ordinals beyond the finite numbers, we use a mathematical concept called transfinite induction. This allows us to define and construct ordinals that are infinitely large, such as omega (ω) and the aleph numbers (ℵ).
The Cantor Normal Form is a way of representing ordinals as a sum of smaller ordinals. It is used in extending ordinals because it allows us to express large ordinals in a more compact and manageable form.
The inverse of an ordinal is constructed by taking the reciprocal of the ordinal in terms of ordinal addition. This means that for any ordinal α, the inverse of α is defined as 1/α, where 1 is the smallest ordinal.
Yes, ordinals can be used to compare the size of infinite sets. In fact, the cardinality (size) of a set is often defined in terms of the smallest ordinal that can be put into a one-to-one correspondence with the set. This allows us to compare the size of any two sets, including infinite ones.