Limit ordinal
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A limit ordinal is an ordinal number which is neither zero nor a successor ordinal.
Various equivalent ways to express this are:
- It cannot be reached via the ordinal successor operation S; in precise terms, we say λ is a limit ordinal if and only if λ > 0 and for any β < λ, there exists γ such that β < γ < λ.
- 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[1] while others exclude it[2].
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[edit] Examples
Because the class of ordinal numbers is well-ordered, there is a smallest infinite limit ordinal; denoted by ω. This ordinal ω is also the smallest infinite ordinal (disregarding limit), as it is the least upper bound of the natural numbers. Hence ω represents the order type of the natural numbers. The next limit ordinal above the first is ω + ω = ω·2, and then we have ω·n, for any natural number n. Taking the union (the supremum operation on any set of ordinals) of all the ω·n, we get ω·ω = ω2. This process can be iterated as follows to produce:
In general, all of these recursive definitions via multiplication, exponentiation, repeated exponentiation, etc. yield limit ordinals. All of the ordinals discussed so far are still countable ordinals; it can be proved that there exists no recursively enumerable scheme of naming just all the countable ordinals.
Beyond the countable, the first uncountable ordinal is usually denoted ω1. It is also a limit ordinal.
Continuing, one can obtain the following (all of which are now increasing in cardinality):
In general, we always get a limit ordinal when taking the union of a set of ordinals that has no maximum element.
The ordinals of the form ω²α, for α > 0, are limits of limits, etc.
[edit] Properties
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 or definitions by transfinite recursion. 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.
If we use the Von Neumann cardinal assignment, every infinite cardinal number is also a limit ordinal (and this is a fitting observation, as cardinal derives from the Latin cardo meaning hinge or turning point!): the proof of this fact is done by simply showing that every infinite successor ordinal is equinumerous to a limit ordinal via the Hotel Infinity argument.
Cardinal numbers have their own notion of successorship and limit (everything getting upgraded to a higher level!).
[edit] References
- ^ Thomas Jech, Set Theory. Third Millennium edition. Springer.
- ^ Kenneth Kunen, Set Theory. An introduction to independence proofs. North-Holland.
