A limit ordinal is an ordinal number which is neither zero nor a successor ordinal.
Another way of defining a limit ordinal is to say that λ is a limit ordinal if and only if:
- There is an ordinal less than λ and whenever β is an ordinal less than λ, then there exists an ordinal γ such that β < γ < λ.
So in the following sequence:
- 0, 1, 2, ..., ω, ω+1
ω is a limit ordinal because for any smaller ordinal (in this example, a natural number) we can find another ordinal (natural number) larger than it, but still less than ω.
Various other ways to define limit ordinal are:
- It is equal to the supremum of all the ordinals below it, but is not zero. (Compare with a successor ordinal: the set of ordinals below it has a maximum, so the supremum is this maximum, the previous ordinal.)
- It is not zero and has no maximum element.
- It can be written in the form ωα for α > 0. That is, in the Cantor normal form there is no finite number as last term, and the ordinal is nonzero.
- It is a limit point of the class of ordinal numbers, with respect to the order topology. (The other ordinals are isolated points.)
Some contention exists on whether or not 0 should be classified as a limit ordinal, as it does not have an immediate predecessor; some textbooks include 0 in the class of limit ordinals while others exclude it.
... Any ordinal number can be made into a topological space by endowing it with the order topology (since, being well-ordered, an ordinal is in particular ... as a topological space, then the class of all ordinals is also a topological space for the order topology.) The set of limit points of an ordinal α is precisely the set of limit ordinals less than α ... Successor ordinals (and zero) less than α are isolated points in α ...
... The classes of successor ordinals and limit ordinals (of various cofinalities) as well as zero exhaust the entire class of ordinals, so these cases are often used in proofs by transfinite induction ... Limit ordinals represent a sort of "turning point" in such procedures, in which one must use limiting operations such as taking the union over all preceding ordinals ... In principle, one could do anything at limit ordinals, but taking the union is continuous in the order topology and this is usually desirable ...
... Formally, if is a limit ordinal, then a set is closed in if and only if for every, if, then ... Thus, if the limit of some sequence in is less than, then the limit is also in ... If is a limit ordinal and then is unbounded in if and only if for any, there is some such that ...
... In mathematics, even and odd ordinals extend the concept of parity from the natural numbers to the ordinal numbers ... The literature contains a few equivalent definitions of the parity of an ordinal α Every limit ordinal (including 0) is even ... The successor of an even ordinal is odd, and vice versa ...
... of cardinal numbers, we can define a successor operation similar to that in the ordinal numbers ... This coincides with the ordinal successor operation for finite cardinals, but in the infinite case they diverge because every infinite ordinal and its successor have the same cardinality (a bijection can be ... easy to define for a cardinal number κ we have , where ON is the class of ordinals ...
Famous quotes containing the word limit:
“We live in oppressive times. We have, as a nation, become our own thought police; but instead of calling the process by which we limit our expression of dissent and wonder censorship, we call it concern for commercial viability.”
—David Mamet (b. 1947)