2Well-orderings and ordinals
II Logic and Set Theory
2.4 Successors and limits
In general, we can divide ordinals into two categories. The criteria is as follows:
Given an ordinal
α
, is there a greatest element of
α
? i.e. does
I
α
=
{β
:
β < α}
have a greatest element?
If yes, say
β
is the greatest element. Then
γ ∈ I
α
⇔ γ ≤ β
. So
I
α
=
{β}∪I
β
.
In other words, α = β
+
.
Definition (Successor ordinal). An ordinal
α
is a successor ordinal if there is a
greatest element β below it. Then α = β
+
.
On the other hand, if no, then for any
γ < α
, there exists
β < α
such that
β > γ. So α = sup{β : β < α}.
Definition (Limit ordinal). An ordinal
α
is a limit if it has no greatest element
below it. We usually write λ for limit ordinals.
Example. 5 and
ω
+
are successors.
ω
and 0 are limits (0 is a limit because it
has no element below it, let alone a greatest one!).