For instance, consider the well-
order consisting of two copies of
N, one after the other:
(N+N,≤):01<11<21<⋯<02<12<22<⋯
Every element has a ‘next’ element; after
11 is
21 and then
31, so on; after
22 is
32 and then
42, so on. Generally, after
n1 is
(n+1)1 and after
n2 is
(n+1)2.
Generally, constructing such ‘sucessor’ elements can be done in any well-
order by defining a successor
function S as taking a value to the
least element greater than it. In notation, we have for some well-
order (A,≤)
S(a)=min({b:b>a})
Since
A is a well-
order, then every subset has a
least element; notably, the
set {b:b>a} has a
least element, so this is well-
defined. The only exception is if
a happens to be the
maximum element of
A, in which case it will have no successor.
The same cannot be said about constructing a ‘predecessor’
function which takes an element to its ‘previous’. The example
(N+N,≤) above is a good counter-example. Many elements of this well-
order do have a predecessor: before
51 is
41; before
122 is
112 and in
most cases before
nk is
(n−1)k. But what of
01 and
02? Well,
01 is the
least element of the whole
order, so it has no predecessor.
02 is more interesting. It is
not least—it has elements below it—but also has no predecessor. There are an infinite number of elements less than
02, and none is the
maximum, so there is no good notion of one single element ‘before’
02.